Download Time-asymptotic wave propagation in collisionless plasmas

PHYSICAL REVIEW E 68, 026406 共2003兲
Time-asymptotic wave propagation in collisionless plasmas
Carlo Lancellotti
Department of Mathematics, City University of New York–CSI, Staten Island, New York 10314, USA
J. J. Dorning
Engineering Physics Program, University of Virginia, Charlottesville, Virginia 22903-2442, USA
共Received 8 April 2003; published 18 August 2003兲
We report the results of a new, systematic study of nonlinear longitudinal wave propagation in a collisionless
plasma. Based on the decomposition of the electric field E into a transient part T and a time-asymptotic part A,
we show that A is given by a finite superposition of wave modes, whose frequencies obey a Vlasov dispersion
relation, and whose amplitudes satisfy a set of nonlinear algebraic equations. These time-asymptotic mode
amplitudes are calculated explicitly, based on approximate solutions for the particle distribution functions
obtained by linearizing only the term that contains T in the Vlasov equation for each particle species, and then
integrating the resulting equation along the nonlinear characteristics associated with A, which are obtained via
Hamiltonian perturbation theory. For ‘‘linearly stable’’ initial Vlasov equilibria, we obtain a critical initial
amplitude 共or threshold兲, separating the initial conditions that produce Landau damping to zero (A⬅0) from
those that lead to nonzero multiple-traveling-wave time-asymptotic states via nonlinear particle trapping
(A⬅0). These theoretical results have important implications about the stability of spatially uniform plasma
equilibria, and they also explain why large-scale numerical simulations in some cases lead to zero-field final
states whereas in others they yield nonzero multiple-traveling-wave final states.
DOI: 10.1103/PhysRevE.68.026406
PACS number共s兲: 52.35.Fp, 52.35.Mw
I. INTRODUCTION
One of the oldest and most fundamental problems in
plasma kinetic theory is that of small-amplitude longitudinal
wave propagation in a collisionless plasma described by the
Vlasov-Poisson 共VP兲 equations
⳵f␣
⳵ f ␣ q␣ ⳵ f ␣
E
⫽0,
⫹v
⫹
⳵t
⳵x m␣ ⳵v
⳵E
⫽4 ␲
⳵x
兺␣ q ␣ 冕 d v f ␣ ,
共1a兲
共1b兲
where f ␣ (x, v ,t) is the distribution function for particle species ␣, ␣ ⫽1,...,N S , and E is the self-consistent longitudinal
electric field. This well-known model has played a major role
in the analysis of plasma instabilities and wave propagation
in a wide variety of settings, ranging from astrophysical,
solar, and magnetospheric plasmas to laboratory and fusion
plasmas. However, due to the extreme analytical difficulties
associated with the nonlinear Vlasov equation, much of the
classic theory of plasma waves 关1– 4兴 has been based on the
analysis of the linearized VP system. The most famous result
of the linear theory is Landau damping: according to Landau’s solution of the initial value problem, every small perturbation of a Maxwellian electron plasma with a fixed ion
background decays exponentially because of collisionless
absorption of electric field energy by the resonant particles,
i.e., the particles traveling at velocities close to the phase
velocity of a wave mode. In the early 1960s, Landau’s result
was generalized by others 关5–7兴 who proved that the solutions to the linearized VP system exhibit Landau damping
whenever the equilibrium distribution is single humped, i.e.,
1063-651X/2003/68共2兲/026406共31兲/$20.00
when it has only one maximum, as indeed is the case for the
Maxwellian. Accordingly, the equilibria in this class are often called ‘‘linearly stable.’’
An unfortunate fact is that the linearization of the VP
system is, in general, not uniformly valid in time 关7兴 even for
small initial perturbations. Thus, the validity of most of the
linear results is limited to a relatively short time scale. This is
the case, in particular, for Landau damping; as early as 1965
O’Neil 关8兴 argued that nonlinear effects can prevent the complete damping of a sinusoidal wave and lead to a fixedamplitude time-asymptotic traveling-wave mode, sustained
by the oscillations of the particles trapped in the wave’s potential well very much like the well-known BernsteinGreene-Kruskal 共BGK兲 关9兴 nonlinear traveling-wave solutions. The time scale on which these nonlinear effects
become important can be estimated as the time scale on
which a particle crosses the potential well and reaches a turning point. Close to the bottom of the well, the field is well
approximated by the harmonic field E(x,t)⫽⫺(ek ⑀ /m)x
共for a plasma of electrons with charge e and mass m兲, so that
the ‘‘trapping’’ 共or ‘‘bounce’’兲 time is ␶ b ⫽ 冑m/ek ⑀ . On the
other hand, Landau damping takes place on a different time
scale, which can be estimated as ␶ L ⫽1/␥ L where ␥ L is a
typical Landau damping coefficient. O’Neil 关8兴 observed that
there are two limiting cases.
共1兲 If ␶ L Ⰶ ␶ b , the field is damped before nonlinear effects
become relevant. The wave dies away before it can significantly distort the single-particle trajectories 共and the
background equilibrium兲, and the linear theory is basically
accurate.
共2兲 If ␶ L Ⰷ ␶ b , the reverse is true: the wave is not damped
before nonlinear effects become important; rather, these effects appear quickly and modify the distribution function in
the resonant region, invalidating the linear theory. This sec-
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©2003 The American Physical Society
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
ond case was the object of O’Neil’s study; since Landau
damping has very little time to affect the wave amplitude, he
argued that, to first order in ␶ b / ␶ L , the constant-amplitude
field E(x,t)⫽sin(kx⫺␻t) can be used to compute the particle
trajectories. This can be done in terms of Jacobi elliptic functions and leads to a wave amplitude that is initially damped
according to Landau, but later grows back as the trapped
particles start transferring energy back to the wave. Then,
damping and growing alternate until the field settles to some
finite amplitude smaller 共in the case of initial damping兲 than
its initial value but different from zero. The decrease of the
amplitude oscillations occurs because, as the trapped particles oscillate on orbits of different frequencies, they lose
phase coherence until there is no net density flux in phase
space across the wave phase velocity.
The limitation of O’Neil’s analysis, of course, is that the
particle trajectories are calculated by assigning a priori the
electric field to be a single sinusoidal wave of fixed amplitude. However, in many practical cases amplitude variations
and the presence of multiple wave modes do affect the particle trajectories significantly. To account for changing amplitudes, others 关10–13兴 have carried out semi-self-consistent
computations, which apply O’Neil’s general method to various more sophisticated Ansätze for the field. This latter is
assumed, again, to be a monochromatic wave, but the amplitude is allowed a slow variation; the crucial point is then the
solution of the resulting Newton equations via various
asymptotic methods 共averaging, adiabatic invariants, etc.兲.
These studies essentially confirm O’Neil’s basic result; however, their value is also limited by the restrictive Ansatz on
the field, which prevents a fully self-consistent treatment of
the VP initial value problem. In fact, no progress at all has
been made toward a satisfactory self-consistent analysis of
the nonlinear VP initial value problem in the general case,
which includes ␶ b and ␶ L being of the same order. In particular, nothing is known about the transition between the
initial conditions that lead to a zero electric field and those
that lead to nonzero time-asymptotic fields via particle trapping. Results for this transition are included in this paper.
Recently, rigorous nonlinear analyses 关14 –16兴 based on
BGK representations 关9兴 have shown that collisionless plasmas can sustain small-amplitude waves near single-humped
equilibria, in spite of the predictions of the linear theory. The
question is whether, and how, periodic traveling-wave solutions of this kind 关14,15兴 can be generated from various initial conditions. Recently, this question has been the subject
of much work, both analytical 关17–21兴 and numerical 关22–
24兴, and also of some controversy 关25兴. In order to clarify
some of the outstanding issues, in this paper we report the
results of a detailed analysis, some of which were summarized earlier in a brief communication 关18兴.
Our analysis is based on four key steps. 共a兲 The decomposition of the electric field E into a transient part T and a
time-asymptotic part A such that E(x,t)⫽A(x,t)⫹T(x,t).
This representation makes it possible to decompose the full
nonlinear VP problem itself into a transient part and a timeasymptotic part 共Sec. II A兲. 共b兲 The linearization of the Vlasov equation with respect only to T(x,t) but not A(x,t), i.e.,
⳵f␣
⳵ f ␣ q␣ ⳵ f ␣
q ␣ ⳵ F␣
⫹v
⫹
A
⫽⫺
T
,
⳵t
⳵x m␣ ⳵v
m␣ ⳵v
共2兲
where F␣ (x, v ) is the initial distribution function 共Sec. II B兲.
共c兲 The formal solution of Eq. 共2兲 in terms of the nonlinear
characteristics corresponding to A 共Sec. II C兲. 共d兲 The utilization of the solution of Eq. 共2兲 to obtain algebraic equations
for the mode amplitudes of A(x,t) 共Secs. II D and II E兲. Step
共d兲 is somewhat laborious: it involves focusing on the timeasymptotic part of the problem and studying it as a bifurcation problem for small-amplitude multiple-wave solutions
near the basic solution A⬅0 which corresponds to complete
Landau damping. We find that, if any nonzero solution for A
‘‘bifurcates’’ from A⬅0 共for a ‘‘critical’’ initial condition兲,
then it is given, at leading order, by a finite superposition of
wave modes whose phase velocities satisfy a Vlasov dispersion relation and whose amplitudes can be obtained from a
finite dimensional system of nonlinear algebraic equations
that depend on T and F␣ . It is crucial that the nonlinear
characteristics for a field A(x,t) of this kind can be determined explicitly via Hamiltonian perturbation theory 关15兴.
The explicit calculation of these characteristics, and thus the
development of the equations for the mode amplitudes of
A(x,t) is carried out in Sec. III, in the case of a two-wave
final state. Because the coefficients in these equations still
depend on T and F␣ , in Sec. IV we have to complete the
analysis by turning to the transient part of the problem and
introducing a standard perturbation technique to determine T
near a bifurcation point of the time-asymptotic problem. This
approach to the determination of T exploits the fact that its
decay properties neutralize most of the secularities that in the
past have plagued attempts at perturbative solutions for the
complete field E.
In this way, we obtain two main results: 共a兲 the threshold
共or ‘‘critical initial amplitude’’兲 below which initial field amplitudes are damped to zero and above which they evolve to
nonzero time-asymptotic solutions; and 共b兲 the dependence
of the time-asymptotic field amplitude upon the initial field
amplitude when the latter is above, but close to, the threshold
value. In particular, we analyze a case of very general interest, a sinusoidal initial perturbation 关8,10,11,13,26,27兴 and
obtain a complete picture of the time-asymptotic evolution of
this type initial condition for various initial distribution functions. Interestingly, our analytical calculation of the critical
initial amplitude 共as briefly summarized in 关18兴兲 has been
confirmed by recent numerical simulations 关24兴.
II. TIME-ASYMPTOTIC ANALYSIS
In order to obtain the long-time solution to the nonlinear
VP equations, in this section we develop an approximation
scheme that yields the long-time electric field formally, in the
sense that the time-asymptotic solution depends on the details of the transient evolution of the field, which is determined later 共in Sec. IV兲.
A. A-T decomposition
As already discussed, a linearly stable spatially periodic
initial disturbance in a collisionless plasma will either un-
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TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
dergo Landau damping to zero, before the trapping effects
become important, or evolve to a nonzero time-asymptotic
state. In the second scenario, particle trapping will create a
flat spot on the distribution function at the phase velocity of
each wave mode that is not Landau damped. This implies
that, in the time-asymptotic limit, these modes will become
unable to exchange energy with the resonant particles, and
will essentially travel with constant amplitude; thus, the electric field will tend asymptotically to a nonlinear superposition of traveling waves very much like the multiple ‘‘BGKlike’’ modes recently described by Buchanan and Dorning
关15兴. This suggests that we look for solutions in which the
electric field E(x,t) is an asymptotically almost periodic
共AAP兲 function of t 关28兴. For such solutions, we can write
E(x,t) as the sum of a transient part and a time-asymptotic
part
E 共 x,t 兲 ⫽T 共 x,t 兲 ⫹A 共 x,t 兲 ,
共3兲
where both T(x,t) and A(x,t) are spatially periodic and
limt→⬁ T(x,t)⫽0; and A(x,t) is an almost periodic 关29兴
function of t, i.e., a general superposition of modes of the
form A(x,t)⫽ 兺 k, ␻ i a k, ␻ i e ikx⫺i ␻ i t , where the frequencies ␻ i
can take a countable set of real values, unlike the wave numbers k, which are restricted to integer values by the requirement of exact 共spatial兲 periodicity. The amplitudes a k, ␻ i are
the Fourier-Bohr coefficients of A(x,t), given by
a k, ␻ i ⫽ lim
␴ →⬁
1
␴
冕
␴
dt
0
1
2␲
冕
⫹␲
⫺␲
dx e ⫺ikx⫹i ␻ i t E 共 x,t 兲 ,
共4兲
which combines a standard Fourier transform in space with a
Bohr transform in time, in which the averaging operator
⫹⬁
dt.
lim␴ →⬁ (1/␴ ) 兰 ␴0 dt replaces the usual Fourier integral 兰 ⫺⬁
This integral transform filters out the transient phenomena
and retains only the time-asymptotic behavior; thus, the
Fourier-Bohr series of an AAP function is a ‘‘projector’’ P a
that separates its transient and time-asymptotic parts. When
E is of the form in Eq. 共3兲, clearly P a E⫽A and (I⫺ P a )E
⫽T. Thus, applying P a and (I⫺ P a ) to Eq. 共1b兲 yields a set
of coupled equations for A and T:
⳵A
⫽4 ␲ P a
⳵x
兺␣ q ␣ 冕 d v f ␣共 A⫹T 兲 ,
⳵T
⫽4 ␲ 共 I⫺ P a 兲
⳵x
兺␣ q ␣ 冕 d v f ␣共 A⫹T 兲 .
共5a兲
共5b兲
From the definition of T(x,t), a solution to Eqs. 共5兲 must
satisfy the additional condition limt→⬁ T(x,t)⫽0. Equation
5共a兲 will be called the time-asymptotic equation, while Eq.
5共b兲 will be the transient equation. The notation f ␣ (A⫹T)
here emphasizes that the f ␣ depend nonlinearly on A and T
via the Vlasov equation.
B. Transient linearization
The A-T decomposition makes it possible to obtain an
approximate solution for f ␣ (A⫹T) via transient linearization, i.e., the linearization of the Vlasov equation only with
respect to T, keeping the nonlinearity in the term that contains A. As long as T decays fast enough 共before the distribution function deviates substantially from the initial condition兲, we can expect the transiently linearized approximation
to the Vlasov equation to be uniformly valid in time, unlike
the standard linearized problem. Moreover, if the characteristics associated with A can be computed, the transiently linearized Vlasov equation can be solved analytically. Writing
Eq. 共1a兲 in terms of A⫹T and introducing the ‘‘linearization’’ of T ⳵ f ␣ / ⳵ v about the initial distribution yields
⳵f␣
⳵ f ␣ q␣ ⳵ f ␣
q ␣ ⳵ F␣
⫹v
⫹
A
⫽⫺
T
,
⳵t
⳵x m␣ ⳵v
m␣ ⳵v
共6兲
where F␣ (x, v )⬅ f ␣ (x, v ,0)⫽F ␣ ( v )⫹h ␣ (x, v ). Here, the
initial condition F␣ has been written as the sum of its spatially uniform part F ␣ 共which will be taken to be a Vlasov
equilibrium兲 and a perturbation h ␣ which will play the role
of a running parameter in our analysis.
In the standard linear Vlasov equation the complete field
E interacts at all times with the fixed background distribution
F ␣ . That approximation is not uniformly valid in the
asymptotic time limit, since the nonlinear distribution f ␣ becomes qualitatively different from F ␣ . Conversely, the linearization in Eq. 共6兲 involves only the transient; hence, it
does not require F␣ to be a uniformly good approximation to
f ␣ as t→⬁, as long as T→0 ‘‘fast enough’’ in that limit. In
fact, the full nonlinear interaction between the asymptotic
field A and the distribution function is maintained through
the term A ⳵ f ␣ / ⳵ v . Clearly, Eq. 共6兲 can be solved exactly
whenever the characteristics 关 x A␶ (x, v ,t), v ␶A (x, v ,t) 兴 can be
determined and leads to the number density in Eq. 共7兲. Interestingly, Eq. 共6兲 includes as special cases both O’Neil’s
strong-trapping scenario 关8兴 and linear Landau damping 关2兴.
In the first case, under O’Neil’s assumption that the transient
part of the field has negligible effects on particle trajectories,
T⫽0 and Eq. 共6兲 reduces to the nonlinear Vlasov equation
with E⫽A, which O’Neil solved analytically for a singlemode sinusoidal wave. This case also includes all the exact
BGK 关9,14,16兴 and BGK-like 关15兴 solutions. At the opposite
extreme, whenever the electric field is damped to zero before
nonlinear effects become relevant, A⫽0 and Eq. 共6兲 becomes a linearized Vlasov equation with E⫽T, leading to
Landau’s exponentially damped solution for the electric field
关which he obtained under the even stronger assumption that
f ␣ (x, v ,0) could be approximated by F ␣ ( v ) 关2兴兴.
Whereas the traditional linearization relies on the assumed small amplitude of the electric field, the transient linearization introduced here is based on an assumption about
the decay rate of T, not about its amplitude. Specifically, it
requires that ␶ transⰆ ␶ dyn , where ␶ trans is the time scale over
which T becomes negligibly small, and ␶ dyn is the time it
takes the nonlinear dynamics to make the distribution function f ␣ significantly different from the initial distribution
F␣ . These time scales are defined more precisely in Appendix A, where a detailed discussion of the error involved in
the transient linearization is presented. It is interesting to
compare the condition ␶ transⰆ ␶ dyn , which involves the decay
rate of T only, with O’Neil’s condition ␶ L Ⰶ ␶ b for the valid-
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C. LANCELLOTTI AND J. J. DORNING
ity of the standard linear theory. Whereas O’Neil considered
the damping rate of the whole field E, here the asymptotic
part A has been subtracted. Hence, this condition can be satisfied even when the complete field E does not damp at all,
e.g., for a BGK traveling mode of the Holloway-Dorning
type 关14兴, where T⫽0 and therefore ␶ trans⫽0. In this case
␶ transⰆ ␶ dyn is trivially satisfied, whereas ␶ L ⫽⬁Ⰷ ␶ b , i.e., it
is a trapping-dominated situation that is completely outside
the domain of validity of the linear theory.
C. Critical initial conditions
Equation 共6兲 can be solved formally in terms of the A
characteristics 关 x A␶ (x, v ,t), v ␶A (x, v ,t) 兴 , which associate with
every phase point (x, v ) at time t a ‘‘starting point’’ 关 x ␶A , v A␶ 兴
at ␶ ⭐t along the trajectory determined by A. Integrating in v
then gives
冕
R
d v f ␣ 共 x, v ,t 兲 ⫽
冕
d v F␣ 共 x A0 , v A0 兲
⫺
q␣
m␣
R
冕 冕 ␶再
t
R
dv
d
0
T
⳵ F␣
⳵v
冎
共7兲
关 x ␶A , v ␶A 兴
for the number densities, which will be the foundation for
our analysis of the nonlinear VP initial value problem. Substituting Eq. 共7兲 into Eq. 共5a兲 and Fourier transforming gives
共for k⫽0)
A k共 t 兲 ⫽
2
ik
兺␣ q ␣ P a 冕⫺ ␲ dx e ⫺ikx
⫻
冕
⫹␲
R
2
⫺
ik
D. Time-asymptotic linearization
In order to study solutions for A near A⬅0, T⬅T 0 , h ␣
⬅h ␣0 , we first consider the time-asymptotic equation linearized about this critical initial state. This linearization of Eq.
共8兲 requires some mathematical care 共see Appendix C兲 and
yields
d v F␣ „x A0 共 x, v ,t 兲 , v A0 共 x, v ,t 兲 …
兺␣ m ␣ P a 冕⫺ ␲ dx e ⫺ikx
q ␣2
冕 冕 再
⫹␲
⳵ F␣
⫻ dv d␶ T
⳵v
R
0
t
冎
the h ␣ -A plane showing the time-asymptotic field vs the initial disturbance, the solutions with A(x,t)⬅0 constitute a
trivial branch that coincides with the horizontal axis. These
solutions will be called vanishing asymptotic states and do
not necessarily correspond to solutions of the complete system of equations for A and T, Eqs. 共5a兲 and 共5b兲. In fact, it is
not necessarily true that when A⬅0 is substituted into Eq.
共5b兲 the resulting equation ⳵ T/ ⳵ x⫽4 ␲ 兺 ␣ q ␣ 兰 d v f ␣ (T) will
possess a solution T that tends to zero as t→⬁. If this does
happen, we will call the corresponding ‘‘point’’ on the fundamental branch an accessible vanishing asymptotic state
共AVAS兲. A trivial example is given by the origin A⬅0, h ␣
⬅0 共from which T⬅0 follows兲. These states correspond to
initial conditions such that the field is completely Landau
damped before trapping effects come into play. As discussed
in the Introduction, there are other initial conditions 共of
larger amplitude兲 for which we expect the trapping effects to
lead to A⫽0; in these cases, the solution A⫽0 to the timeasymptotic equation is not accessible and does not correspond to the solution to the complete VP initial value problem. The nonzero solutions can be imagined as a nontrivial
branch in the h ␣ -A plane, which bifurcates from the trivial
solutions A⬅0 at some critical initial condition h ␣ ⬅h ␣0 as
the initial condition h ␣ is changed. Physically, h ␣0 marks the
transition between the two classes of initial conditions. The
determination of h ␣0 will follow from the nonlinear analysis
below; we now proceed to show how the general form of the
solution for A can be determined for initial conditions near a
generic 共given兲 critical initial state A⬅0, T⬅T 0 , h ␣ ⬅h ␣0 .
共8兲
关 x ␶A 共 x, v ,t 兲 , v ␶A 共 x, v ,t 兲兴
共where here and below the ⫾␲ in the limits of integration
have units of length, i.e., the wavelength is taken as unity兲.
This equation contains the characteristics for the field A,
which, of course, is still unknown. Our strategy is to show
that, at the transition between the initial conditions that lead
to complete Landau damping (A⬅0) and those that lead to
nonzero small-amplitude solutions for A, the general solution
for A can be determined a priori. Such a general solution
will depend on a finite set of unknown amplitudes that will
satisfy certain nonlinear algebraic equations derived from
Eq. 共8兲.
A preliminary step toward determining the general form
of small-amplitude solutions for A is to observe that Eq. 共8兲
in isolation has the exact solution A(x,t)⬅0 independent of
the transient field T and of the initial distribution function
F␣ (x, v ). The proof of this result is somewhat tedious and is
given in Appendix B. Schematically, if we draw a graph in
a k, ␻ i D共 k, ␻ i 兲 ⫽0,
共9兲
where D(k, ␻ i ) is the Vlasov dielectric function
4␲
D共 k, ␻ i 兲 ⬅1⫺
k
兺␣ m ␣ P 冕Rd v ␻ i ⫺k v .
q ␣2
T
F ␣0 ⬘ 共 v 兲
共10兲
Here D(k, ␻ i ) is not determined by the Vlasov equilibrium
F ␣ ( v ) that appears in the initial distribution function; rather,
T
it contains a time-asymptotic equilibrium F ␣0 which includes
the effects of the transient field T 0 that evolves from the
critical initial state 关see Eq. 共B4兲 in Appendix B for a precise
T
definition of F ␣0 ]. Of course, T 0 has to be obtained from the
transient equation, Eq. 共5b兲, with A⬅0; however, there is an
important case in which D(k, ␻ i ) turns out not to depend on
T 0 , namely, when the problem is reflection symmetric, i.e.,
even in x and v . Reflection-symmetric initial conditions occur in many interesting problems 关15,27兴 and lead to
reflection-symmetric solutions at all times. In these cases, it
T
easy to show 关33兴 that F ␣0 ( v )⫽F ␣ ( v ). Here, we will only
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Eq. 共9兲 is not about the initial equilibrium F ␣ ( v ) but about
T
the ‘‘final’’ equilibrium F ␣0 ( v ), and it is important for the
study of the nonlinear problem. Indeed, many nonlinear studies of the VP system 关30–32兴 have been greatly hampered by
troublesome singularities that do not appear to be intrinsically related to the physical nature of the problem but only to
the nonuniformity in time of the standard linearization.
E. Dimensional reduction
FIG. 1. The Vlasov dispersion curve for a Maxwellian e-p
plasma with T e ⫽T p , and the roots ␻ (k) of the corresponding dispersion relation for a basic wave number k 0 and its harmonics; k is
in units of the inverse Debye length k D ⬅1/␭ D and ␻ is in units of
the plasma frequency ␻ p .
These previous results make it possible to reduce the nonlinear time-asymptotic equation, Eq. 共8兲, to a finitedimensional system of nonlinear algebraic equations for the
N amplitudes a k, ␻ i in Eq. 共11兲. This is done by simply substituting Eq. 共11兲 in the right side of Eq. 共8兲,1 and taking the
Bohr transform with respect to ␻ 1 (k), ␻ 2 (k),..., ␻ N k (k) for
k⫽1,...,%. This is equivalent, of course, to projecting the
nonlinear equation onto the null space of the linearized operator given by Eq. 共9兲, and yields N equations:
a k, ␻ i ⫽
consider initial conditions of this kind. This also has the
advantage that we need to consider only spatial Fourier
modes with k⫽0 since, if the spatially uniform (k⫽0) component in the electric field is initially zero, it will remain zero
for all times.
Equation 共9兲 can be solved immediately: it requires that
a k, ␻ i ⫽0 except for all k and ␻ i that satisfy D(k, ␻ i )⫽0
Nk
A 共 x,t 兲 ⫽
兺 兺
k⭐k d j⫽1
a k, ␻ j e i 关 kx⫺ ␻ j 共 k 兲 t 兴 .
共11兲
More precisely, since the basic wave number here is k⫽1,
there will be %⫽ 关 k d 兴 共the integer part of k d ) possible wave
%
N k posnumbers before cutoff, leading to a total of N⫽ 兺 k⫽1
sible modes.
A crucial fact about Eq. 共9兲 is that the Fourier-Bohr transformation has introduced the limit t→⬁ before the linearization was carried out, so that the resulting linear equation is
uniformly valid in time, unlike the standard linearized VP
system. This is apparent in the fact that the linearization in
冕 dt e i␻ t 冕⫺␲ dx e ⫺ikx
兺␣ q ␣ ␴lim
→⬁ ␴ 0
⫻
冕
1
R
2
⫺
ik
⫻
T
F ␣0 .
which is the Vlasov dispersion relation determined by
The properties of the Vlasov dispersion relation are well
known, especially for physically relevant Vlasov equilibria.
For instance, for a Maxwellian the Vlasov dispersion curve
can be plotted easily. It shows 共Fig. 1兲 some qualitative features that hold in great generality, even though for a modified
T
Vlasov equilibrium, like F ␣0 , it may be necessary to compute the exact roots of D numerically 关14兴. Specifically, the
dispersion curve shows a cutoff wave number k d such that for
k⬎k d Eq. 共10兲 has no solution. Moreover, given any wave
number k⭐k d , Eq. 共10兲 defines implicitly a finite number N k
of simple real roots ␻ 1 (k), ␻ 2 (k),..., ␻ N k (k), which means
that the general solution of the linearized time-asymptotic
problem will be given by a finite superposition of wave
modes of the form
2
ik
⫹␲
i
d v F␣ „x A0 共 x, v ,t 兲 , v A0 共 x, v ,t 兲 …
冕 dt e i␻ t 冕⫺␲ dx e ⫺ikx
兺␣ m ␣ ␴lim
→⬁ ␴ 0
q ␣2
冕 冕 ␶再
dv
d
0
␴
1
t
R
␴
T
⳵ F␣
⳵v
⫹␲
i
冎
,
共12兲
关 x ␶A 共 x, v ,t 兲 , v ␶A 共 x, v ,t 兲兴
where the right side is determined by the N amplitudes a k, ␻ i
through the characteristics associated with A. In the next section we show how, near a critical initial state 共where a k, ␻ i
⬅0 ᭙ k, ␻ i ), these characteristics can be determined via
Hamiltonian perturbation theory.
In summary, the time-asymptotic equation has been reduced to a finite-dimensional problem for the (small) amplitudes of a set of traveling-wave modes that satisfy a Vlasov
dispersion relation. Both the dispersion relation and the am1
By substituting the linear solution for A, Eq. 共11兲, into the nonlinear time-asymptotic equation, Eq. 共8兲, we neglect possible
higher-order terms in A corresponding to wave modes a k, ␻ i e ikx⫺ ␻ i t
such that D(k, ␻ i )⫽0. The validity 共at leading order兲 of this approximation will become apparent in the course of our nonlinear
analysis, in which we shall obtain the scalar equation for the amplitude a of a two-mode time-asymptotic field with equal mode
amplitudes, Eq. 共24兲. That calculation could be extended to include
higher-order wave modes such that D(k, ␻ i )⫽0. It is then easy to
see that these modes must be O(a 3/2) and will generate terms of the
same order in the charge density. In principle, these terms are comparable to other terms that we are going to keep in Eq. 共12兲; however, these latter terms are orthogonal to the wave modes corresponding to D(k, ␻ i )⫽0, and will disappear under the action of the
Fourier-Bohr transform on the right side of Eq. 共12兲. Thus, the
leading-order contribution to Eq. 共12兲 from the modes such that
D(k, ␻ i )⫽0 will turn out to be of order a 2 , and negligible. This
specific result has been proved previously in a more general context
关20兴.
026406-5
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
plitudes depend on the initial condition and on the transient
field. In particular, nonzero small-amplitude solutions for the
wave modes may appear when the amplitude of the initial
disturbance is increased through certain critical values. In
other words, these nonzero undamped solutions may bifurcate 共as the initial condition is changed兲 from the trivial solution branch A⬅0, which corresponds to a completely
Landau-damped electric field as t→⬁. In fact, analogous
results have been established for the original 共not transiently
linearized兲 VP system 关20兴. In that case, however, it is too
difficult to carry out an explicit calculation of the timeasymptotic wave amplitudes, because of the complicated interaction between the transient electric field and the distribution function. Conversely, for the transiently linearized
equations developed here the complete calculation can be
carried out explicitly, as is done below.
III. THE TWO-WAVE TIME-ASYMPTOTIC PROBLEM
Let us now consider a sequence of physical problems in
which a transition occurs from the strongly Landau damped
regime (A⬅0) to the O’Neil regime where the nonlinear
effects sustain small-amplitude wave propagation. We expect
that, as the amplitude of the initial disturbance is increased
共or dF ␣ /d v is decreased at the ‘‘right’’ phase velocity兲, at
some point a ‘‘first’’ undamped nonzero time-asymptotic
state will branch off the zero-field solution. It is logical to
assume that this phenomenon will not take place for all the
wave numbers and frequencies at the same transition point.
Rather, according to the basic insights from the standard linear theory, the modes with the lowest wave number and
highest phase velocity damp most slowly. Hence, for a
single-humped 共symmetric兲 equilibrium, the first nonzero
state should be a pair of Langmuir modes on the upper
branch of the Vlasov dispersion relation 共Fig. 1兲, with a
‘‘fundamental’’ wave number k⫽k 0 determined by the initial
condition; thus, A has the two-wave form
A 共 x,t 兲 ⫽2a sin k 0 x cos ␻ t
⫽a sin共 k 0 x⫺ ␻ t 兲 ⫹a sin共 k 0 x⫹ ␻ t 兲 ,
共13兲
where k 0 and ␻ satisfy D(k 0 , ␻ )⫽0.2 For this two-wave case
Eqs. 共12兲 reduce to one equation for a, and 关from Eq. 共13兲兴
the asymptotic field belongs to the one-dimensional space
spanned by sin k0x cos ␻t. Hence, the projection procedure
reduces to a cosine Fourier-Bohr transform 共by symmetry兲,
and Eq. 共12兲 becomes
2
Without loss of generality we take a⭓0, since its sign can be
changed arbitrarily by introducing a constant phase shift ␲ in Eq.
共13兲. This phase will be left undetermined until later, when the
analysis of the transient problem will determine the phase that
‘‘connects’’ A(x,t) to the initial condition.
2a⫽
8
k0
冕 dt cos ␻ t 冕⫺␲ dx cos k 0x
兺␣ q ␣ ␴lim
→⬁ ␴ 0
⫻
冕
d v F␣ „x A0 共 x, v ,t 兲 , v A0 共 x, v ,t 兲 …
⫺
8
k0
冕 dt cos ␻ t 冕⫺␲ dx cos k 0x
兺␣ m ␣ ␴lim
→⬁ ␴ 0
⫻
冕 冕 ␶冋
1
R
q ␣2
R
d
0
⫹␲
␴
1
t
dv
␴
T
⳵ F␣
⳵v
册
⫹␲
,
共14兲
关 x ␶A 共 x, v ,t 兲 , v ␶A 共 x, v ,t 兲兴
where A is given by Eq. 共13兲.
The characteristics 关 x ␶A , v A␶ 兴 are determined by the dynamics of particles moving in small-amplitude two-wave
fields, which have been studied by Buchanan and Dorning
关15兴. They constructed small-amplitude multiple-wave solutions to the VP system by extending the invariants originally
used to generate BGK solutions 关9兴. The latter are based on
an exact invariant of the motion for the single-wave potential, namely, the single-particle energy. Buchanan and Dorning found approximate invariants for multiple-wave systems
and used them to obtain generalized BGK solutions. The
simplest case is precisely the two-wave field 关Eq. 共13兲兴. In
the upper half phase plane, where one of the two waves is
located, the effect of the other wave 共in the lower half plane兲
can be viewed as a small perturbation of the unperturbed
particle motion driven by the first wave. Application of Lie
transforms shows that the energy invariant for mode ⫹ in
isolation 共‘‘⫹’’ and ‘‘⫺’’ correspond to the upper and lower
half phase planes兲, E␣(⫹) ⫽ 21 m ␣ ( v ⫺ v p ) 2 ⫹(q ␣ a/k 0 )cos(k0x
⫺␻t), is modified by the presence of mode ⫺ to become 关15兴
Ē共␣⫹ 兲 ⫽E共␣⫹ 兲 ⫹
冉
冊
a2
q ␣a v ⫺ v p
cos共 k 0 x⫹ ␻ t 兲 ⫹O
,
k0 v⫹vp
共 v⫹v p兲2
共15兲
where v p ⬅ ␻ /k 0 . The denominator v ⫹ v p makes this invariant invalid in the phase region of the second wave; in this
region, however, the same procedure yields the analogous
invariant Ē␣(⫺) , which is obtained by switching v p to ⫺ v p
and ␻ to ⫺␻ in Eq. 共15兲. By combining Ē␣(⫹) and Ē␣(⫺) , it is
possible to construct a global first order invariant, whose
level curves are shown in Fig. 2共a兲, which gives all the information we need about the particles’ motion. Of course,
this invariant is not exact, as should be expected since the
Hamiltonian system corresponding to Eq. 共13兲 is not integrable. In fact, there are small regions in the phase plane
where no invariant curves exist, because the nonlinear resonances between the particles and the waves generate chaotic
trajectories. These stochastic layers are thin regions that
separate bounded and unbounded trajectories 关see Fig. 2共b兲兴.
By invoking the classic Kolmogorov-Arnold-Moser 共KAM兲
theorem, though, Buchanan and Dorning 关15兴 showed that
these layers are exponentially small in a and can be neglected in a study of the self-consistent VP system.
Below we adapt Buchanan and Dorning’s technique and
explicitly calculate 关 x A␶ , v ␶A 兴 . The restriction to a two-wave
field is not essential, but the generalization to the N-wave
case, which is straightforward, becomes increasingly tedious
026406-6
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
FIG. 2. 共a兲 Buchanan and Dorning’s 关15兴 approximate invariant curves for a two-wave system. 共b兲 Level curves, stochastic layers, and
higher-order resonance islands for the same system.
as N grows. Hence, it will be omitted in order to make the
analysis as clear as possible. For the same purpose, we shall
focus our nonlinear analysis of Eq. 共14兲 on one of the simplest, yet most fundamental and widely studied
关8,10,11,13,26,27兴, plasma-wave problems, the evolution of
a sinusoidal initial perturbation E(x,0)⫽â 0 sin k0x.3 The corresponding initial distributions are F␣ (x, v )⫽F ␣ ( v )
⫹ ⑀ h ␣ ( v )cos k0x, where F ␣ are 共normalized兲 initial Vlasov
equilibria, h ␣ are given 共normalized兲 even functions of v
with strong-decay properties as 兩 v 兩 →⬁, and ⑀ is related to
â 0 via the Poisson equation. These initial distributions
clearly are reflection symmetric and the initial field has no
spatially uniform part. Thus, as anticipated above, the k⫽0
spatial Fourier component of E will be zero at all times, and
we can restrict our study to k⫽0. For fixed F ␣ and h ␣ , the
complete initial condition can be parametrized by ⑀. Hence,
Eq. 共14兲 becomes an algebraic equation for the scalar unknown a in terms of the scalar parameter ⑀ 关and the transient
field T, which will be determined later from Eq. 共5b兲 for a
given a兴. Notwithstanding these simplifications, this initial
value problem includes all the essential features of the more
general case 共i.e., a generic spatially periodic initial perturbation兲. The analysis can be extended to more general problems, but only at the price of a considerable increase in the
algebra.
3
This implies that we are taking the leading-order time-asymptotic
field 关Eq. 共13兲兴 to have the same wave number as the initial perturbation, which is fully justified based on the analysis of the transient
equation in the next section. All the higher harmonics will be indexed by k⫽lk 0 with l⫽1,2,... 共since k 0 is the fundamental wave
number兲.
A. O’Neil terms
Our analysis of Eq. 共14兲 begins with the first term on the
right side, which corresponds to the asymptotic evolution of
the initial distribution as though it were forced by a field of
fixed amplitude, with no transient part. This term and those
that follow from it will be called ‘‘O’Neil’’ terms because
their structure is loosely analogous to that of quantities that
arose in O’Neil’s calculation 关8兴, although that work was
restricted to a single sinusoidal wave of constant amplitude.
Conversely, the second term on the right side of Eq. 共14兲
corresponds to the 共linearized兲 effects of the transient field
T(x,t) on the distribution function; for a⫽0 this term is
essentially the quantity that arises in Landau’s solution 关2兴.
Thus, it and its descendants will be called ‘‘Landau’’ terms.
We evaluate the O’Neil terms in three steps. 共1兲 We calculate explicitly the characteristics 关 x ␶A , v A␶ 兴 in terms of Jacobi elliptic functions, via the Buchanan-Dorning technique
关15兴. 共2兲 Then, we take the limit as ␴ →⬁ by extending an
idea of O’Neil’s, who noted 关8兴 that the distribution function
共corresponding, in his case, to the evolution along the trajectories in a single sinusoidal wave兲 can be replaced in the
time-asymptotic limit by a coarse-grained version, which is
obtained by averaging on each energy level of the wave. In
our case, we show that the limit as ␴ →⬁ can be carried out
by averaging on each energy level of the time-asymptotic
field. In practice, we do this by transforming the integrations
to action-angle variables and replacing the time averages by
phase-space averages over the lines of constant action. 共3兲
Finally, we expand the resulting expression asymptotically in
terms of half-integer powers of the small time-asymptotic
amplitude a. The details of this fairly complicated calculation are given in Appendix D. The resulting expansion for
the first term on the right side of Eq. 共14兲 is
026406-7
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
2 关 ␹ 1 a 3/2⫹ ␹ 2 ⑀ a 1/2⫹ ␹ 3 ⑀ a 3/2⫹aK 0 共 k 0 , ␻ 兲兴 ⫹O 共 a 2 兲 ,
C. The amplitude bifurcation problem
共16兲
Combining Eqs. 共14兲, 共16兲, and 共21兲 gives
a⫺ ␹ 1 a 3/2⫺ ␹ 2 ⑀ a 1/2⫺ ␹ 3 ⑀ a 3/2⫺aK 0 共 k 0 , ␻ 兲
where
冋 册冋
兩 q ␣兩 5
兺␣
␹ 1 ⬅⫺ ␴ 1
k 50 m ␣3
␹ 2 ⬅⫺ ␴ 1
␹ 3 ⬅⫺ ␴ 2
⫻
再
1/2
兺␣
s␣
兺␣
册
1 dF ␣
d 2F ␣
共vp兲 ,
2 共 v p兲⫺
dv
2vp dv
共17兲
s␣
冋 册
兩 q ␣兩 5
冋 册
兩 q ␣兩 3
1/2
共18兲
h ␣共 v p 兲 ,
k 30 m ␣
1/2
k 50 m ␣3
冎
d 2h ␣
1 dh ␣
1
共 v p 兲 ⫹ 2 h ␣ 共 v p 兲 , 共19兲
2 共 v p兲⫹
dv
vp dv
2vp
K 0共 k 0 , ␻ 兲 ⫽
4␲
k 20
兺␣ m ␣ P 冕Rd v v p␣⫺ v ,
F⬘ 共v兲
q ␣2
共20兲
⫽⫺a 1/2⌫ 共 ⑀ ,T 兲 ⫺a 3/2⌺ 共 ⑀ ,T 兲 ⫹O 共 a 2 兲 ,
an explicit relationship between the ‘‘final’’ field amplitude a
and the initial amplitude ⑀. From it we easily determine the
nature of the transition between field solutions that are Landau damped to zero and those that approach two-wave timeasymptotic states with amplitude a. By construction, k 0 and
␻ satisfy the time-asymptotic Vlasov dispersion relation 1
⫺K 0 (k 0 , ␻ )⫽0. Thus, Eq. 共24兲 reduces to
a 1/2关 ␹ 2 ⑀ ⫺⌫ 共 ⑀ ,T 兲兴 ⫹a 3/2关 ␹ 1 ⫹ ␹ 3 ⑀ ⫺⌺ 共 ⑀ ,T 兲兴 ⫽O 共 a 2 兲 .
共25兲
Near any accessible vanishing asymptotic state 共critical or
not兲, we write T as T 0 ⫹ ␦ T where T 0 is the transient field
corresponding to the AVAS itself. Since both ⌫ and ⌺ are
linear in T, Eq. 共25兲 becomes
a 1/2关 ␹ 2 ⑀ ⫺⌫ 共 ⑀ ,T 0 兲兴 ⫺a 1/2⌫ 共 ⑀ , ␦ T 兲
and ␴ 1 ⫽20.67 and ␴ 2 ⫽0.53 are numerical constants.
⫹a 3/2关 ␹ 1 ⫹ ␹ 3 ⑀ ⫺⌺ 共 ⑀ ,T 0 兲兴 ⫽O 共 a 2 ,a 3/2兩 ␦ T 兩 兲 .
共26兲
B. Landau terms
Because of symmetry, we can write T in the last term in
Eq.
共14兲
as
a
Fourier
sine
series
T(x,t)
⬁
T nk 0 (t)sin nk0x. Then, a calculation very similar to
⫽ 兺 n⫽1
the one that led to Eq. 共16兲, also given in Appendix D, reduces the Landau terms to
a 1/22⌫ 共 ⑀ ,T 兲 ⫹a 3/22⌺ 共 ⑀ ,T 兲 ⫹O 共 a 2 兲 ,
共21兲
The possible bifurcation values, i.e., values of the parameter
⑀ where nonzero solutions cross 共‘‘branch off’’兲 the trivial
solution a 1/2⫽0, can be found by ‘‘lifting’’ the branch a 1/2
⫽0 共here simply dividing by a 1/2) and setting a⫽ ␦ T⫽0 and
⑀ ⫽ ⑀ 0 , where ⑀ 0 is the amplitude of the initial perturbation
corresponding to the AVAS; since ⌫( ⑀ 0 ,0)⫽0 this gives the
bifurcation condition
␹ 2 ⑀ 0 ⫽⌫ 共 ⑀ 0 ,T 0 兲 ,
where
⌫ 共 ⑀ ,T 兲 ⬅
兺␣
再
冋 册
兩 q ␣兩 5
1/2
k 30 m ␣3
⫻ ␳ 1共 T 兲
冎
dF ␣
⑀
dh ␣
共 v p 兲 ⫹ ␳ 2共 T 兲
共vp兲 ,
dv
2
dv
共22兲
⌺ 共 ⑀ ,T 兲 ⫽
兺␣
冋 册再
兩 q ␣兩 7
k 50 m ␣5
⫹ ⑀ ␳ 4共 T 兲
1/2
2 ␳ 3共 T 兲
d 3F ␣
共vp兲
dv3
␭ 2共 T 兲 d 2F ␣
d 3h ␣
共vp兲
3 共 v p兲⫺
dv
4vp dv2
⫺⑀
␭ 4共 T 兲 d 2h ␣
␭ 1 共 T 兲 dF ␣
共vp兲
2 共 v p兲⫺
8vp dv
8 v 2p d v
⫺⑀
␭ 3 共 T 兲 dh ␣
共vp兲 .
16v 2p d v
冎
共24兲
共23兲
The operators ␳ i (T) and ␭ i (T), i⫽1,...,4, are defined, respectively, in Eqs. 共D55兲 and 共D57兲 and are linear in T.
共27兲
which also will be called the threshold equation, since ⑀
⫽ ⑀ 0 represents a critical initial amplitude, i.e., a value of ⑀
where the time-asymptotic behavior of the electric field may
change from that of a vanishing state to that of a nonzero
state 共or vice versa兲. This equation has a clear physical
meaning: it expresses the balance between the effects of the
initial transient 关contained in ⌫, which depends on
(dF ␣ /d v )( v p ) and (dh ␣ /d v )( v p )], and the long-time
trapped-particle effects generated by the initial perturbation
关contained in ␹ 2 , which depends on h ␣ ( v p )]. In the case of
single-humped unperturbed equilibria, ⌫ measures the
strength of the Landau damping rate, while ␹ 2 expresses the
ability of the initial perturbation to generate a plateau at the
phase velocity via particle trapping.
Taking ⑀ 0 to be known from Eq. 共27兲, a local analysis can
be performed to determine the bifurcating solution branch.
Expansion of Eq. 共26兲 about ⑀ ⫽ ⑀ 0 yields
关 ␹ 2 ⫺⌫ ⑀ 共 ⑀ 0 ,T 0 兲兴 ⌬ ⑀ ⫺⌫ 共 ⑀ 0 , ␦ T 兲 ⫹a 关 ␹ 1 ⫹ ␹ 3 ⑀ 0 ⫺⌺ 共 ⑀ 0 ,T 0 兲兴
⫽O 共 a 3/2,a 兩 ␦ T 兩 ,a⌬ ⑀ , 兩 ␦ T 兩 ⌬ ⑀ 兲 ,
where ⌬ ⑀ ⬅ ⑀ ⫺ ⑀ 0 and
026406-8
共28兲
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
3兲. However, a itself ultimately depends on ⌬⑀, so that both
T and a must be expanded in ⌬⑀. Since the expansion of the
time-asymptotic equation generated half-integer powers of a,
half-integer powers of ⌬⑀ may be needed 共at higher order兲 to
have consistent expansions for a and T. Thus, our focus being on first order terms, we introduce the truncated expansion for T:
T 共 x,t 兲 ⫽T 共 0 兲 共 x,t 兲 ⫹⌬ ⑀ T 共 1 兲 共 x,t 兲 ⫹O 共 ⌬ ⑀ 3/2兲 .
共31兲
Substituting Eq. 共31兲 into Eq. 共28兲 关with T 0 ⬅T (0) and ␦ T
⬅⌬ ⑀ T (1) ⫹O(⌬ ⑀ 3/2)] gives4
a⫽⫺
FIG. 3. Local analysis at a critical initial amplitude ⑀ 0 in the ⑀ -a
plane.
⳵⌫
1
⌫ ⑀ 共 ⑀ 0 ,T 0 兲 ⬅
共 ⑀ ,T 兲 ⫽
⳵⑀ 0 0
2
兺␣
冋 册
兩 q ␣5 兩
k 30 m ␣3
1/2
dh ␣
␳ 2共 T 0 兲
共 v p兲.
dv
共29兲
Equation 共28兲 gives an explicit approximation to Eq. 共26兲;
thus it could be solved to provide an approximate solution to
Eq. 共14兲. In fact, it shows the qualitative dependence of a on
⑀: clearly, a undergoes a transcritical bifurcation at the critical initial amplitude ⑀ 0 , with the nonzero solutions for a
crossing the basic branch a⫽0 at a finite angle. In order to
have quantitative results, however, we have to calculate the
coefficients in Eqs. 共27兲 and 共28兲, which depend on T(x,t),
treated thus far as a parameter; indeed, Eq. 共14兲 must be
understood as part of a coupled system of equations for a and
T, Eqs. 共5兲. Hence, we now turn to the transient part of the
VP problem.
A. Small critical initial amplitudes
We first calculate the transient field along the basic branch
a⫽0, which provides T 0 ⬅T (0) and also is necessary to determine ⑀ 0 from Eq. 共27兲. Even for a⫽0, the leading-order
expansion of the transient equation is rather tedious; hence, it
is developed in Appendix E 1 which leads to the following
equation for the Fourier-Laplace-transformed zeroth-order
transient electric field T̃ (0)
k (p):
0兲
0兲
D k 共 p 兲 T̃ 共k0 兲 共 p 兲 ⫺ ⑀ 0 C k 共 p 兲关 T̃ 共k⫹k
共 p 兲 ⫹T̃ 共k⫺k
共 p 兲兴
0
0
⫽ ⑀ 0 ␦ k,k 0 N k 共 p 兲 ,
N k 共 p 兲 ⫽⫺
Substituting Eq. 共7兲 into Eq. 共5b兲 yields the transient
equation, which we solve via a perturbation expansion of
T(x,t) in powers of the deviation of the initial condition
from the AVAS under consideration. Because A(x,t) has
been subtracted from E(x,t) and its nonlinear effects on the
particle trajectories computed analytically, the perturbation
solution for T(x,t) will be spared the disruptive secularities
that arise in standard perturbation treatments of the VP system which use the complete field E(x,t).
Here we shall not assume a priori 关as in Eq. 共13兲, see
footnote 3兴 that the time-asymptotic field has the same wave
number k 0 as the initial condition; rather, we shall write
A 共 x,t 兲 ⫽2a sin k f x cos ␻ t
共30兲
and then prove that indeed k f ⫽k 0 . Below, the transient
equation will be expanded in the neighborhood of a given
critical amplitude ⑀ 0 in the small amplitudes a and ⌬⑀ 共Fig.
共33兲
where N k (p) and D k (p) are the same quantities that appear,
respectively, in the numerator and the denominator in Landau’s solution to the standard linearized initial value problem
关2兴 共for the initial condition F␣ ),
IV. TRANSIENT FIELD EXPANSION
⫽a sin共 k f x⫹ ␻ t 兲 ⫹a sin共 k f x⫺ ␻ t 兲
关 ␹ 2 ⫺⌫ ⑀ 共 ⑀ 0 ,T 共 0 兲 兲 ⫺⌫ 共 ⑀ 0 ,T 共 1 兲 兲兴
⌬ ⑀ ⫹O 共 ⌬ ⑀ 3/2兲 .
关 ␹ 1 ⫹ ␹ 3 ⑀ 0 ⫺⌺ 共 ⑀ 0 ,T 共 0 兲 兲兴
共32兲
4␲i
k2
4␲
D k 共 p 兲 ⫽1⫹ 2
k
␣
,
兺␣ q ␣ 冕Rd v v ⫺ip/k
共34兲
␣
,
兺␣ m ␣ 冕Rd v v ⫺ip/k
共35兲
h 共v兲
q ␣2
F⬘ 共v兲
and
2␲
C k 共 p 兲 ⬅⫺ 2
k
␣
.
兺␣ m ␣ 冕Rd v v ⫺ip/k
q ␣2
h⬘ 共v兲
共36兲
The sign of a 关taken to be positive after postulating an appropriate phase shift in Eq. 共13兲兴 depends here on the signs of ⌬⑀ and its
coefficient. In turn, ⌬⑀ must have the same sign as ⑀ 0 , when studying the transition from completely Landau damped solutions to
small-amplitude nonzero time-asymptotic solutions. Nevertheless,
if a in Eq. 共32兲 is negative, one can immediately find a solution with
a⬎0 simply by considering a perturbation ⫺⌬⑀ to a critical amplitude ⫺ ⑀ 0 . This is equivalent to applying the coordinate transformation x→⫺x, v →⫺ v , corresponding to the reflection symmetry.
4
026406-9
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
Equation 共33兲 is very similar to the equation that results from
Landau’s solution to the linear VP system 关2兴. The main
difference is that, because of the spatially nonuniform part of
the initial distribution F␣ , the equations for different values
of k are coupled. Hence, Eq. 共33兲 is an infinite system and its
solution will require an approximation. In general, numerical
approximation could be used, which then would lead to a
solution of Eq. 共27兲 for general 共not small兲 critical amplitudes. Here, however, we are interested in small initial perturbations, i.e., ⑀ Ⰶ1; hence ⑀ 0 Ⰶ1. Thus, further expanding
T̃ (0)
k in powers of ⑀ 0 ,
共 0,m 兲
T̃ 共k0 兲 ⫽ ⑀ 0 T̃ 共k0,1兲 ⫹ ⑀ 20 T̃ 共k0,2兲 ⫹¯⫹ ⑀ m
⫹¯ ,
0 T̃ k
共37兲
substituting into Eq. 共33兲, and solving at each order in ⑀ 0
straightforwardly yields
T̃ 共k0,1兲 共 p 兲 ⫽ ␦ k,k 0
N k0 共 p 兲
,
D k 0共 p 兲
T̃ 共k0,2兲 共 p 兲 ⫽ ␦ k⫹k 0 ,k 0
N k⫹k 0 共 p 兲 C k 共 p 兲
D k⫹k 0 共 p 兲 D k 共 p 兲
⫹ ␦ k⫺k 0 ,k 0
⫽ ␦ k,2k 0
T̃ 共k0,3兲 共 p 兲 ⫽ ␦ k,3k 0
0,1兲
␹ 2 ⑀ 0 ⫽ ⑀ 0 ␴ 1 S 共1,1
N k⫺k 0 共 p 兲 C k 共 p 兲
D k⫺k 0 共 p 兲 D k 共 p 兲
N k 0 共 p 兲 C 2k 0 共 p 兲
D k 0 共 p 兲 D 2k 0 共 p 兲
再
兺␣
冋 册
兩 q ␣兩 5
,
⫹
␴ 2 共 0,1兲
S
2 1,2
1/2
k 30 m ␣3
0,2兲
⫺4 ⑀ 20 ␴ 2 S 共2,2
N k 0 共 p 兲 C 2k 0 共 p 兲 C 3k 0 共 p 兲
dF ␣
共vp兲
dv
冋 册
冋 册
兺␣
s␣
兩 q ␣兩 5
k 30 m ␣3
兩q 兩5
兺␣ s ␣ k 30 m␣ ␣3
1/2
1/2
dF ␣
共vp兲
dv
冎
dh ␣
共 v p 兲 ⫹O 共 ⑀ 30 兲 .
dv
共39兲
D k 0 共 p 兲 D 2k 0 共 p 兲 D 3k 0 共 p 兲
⫹ ␦ k,k 0
N k 0 共 p 兲 C k 0 共 p 兲 C 2k 0 共 p 兲
D k 0 共 p 兲 2 D 2k 0 共 p 兲
.
共38兲
can be obtained by inverting the
The expressions for T (0,m)
k
Laplace transforms. 关In particular, T̃ (0,1)
(p) yields Landau’s
k
classic damped solution.兴
⑀ 共01 兲 ⫽
Clearly, there are initial conditions for which this solution
is not acceptable, because it does not vanish as t→⬁. For
linearly stable initial equilibria F ␣ , the standard results for
the zeros of D k (p) ensure that T̃ (0)
k (p) have no poles with
real parts ⭓0, so limt→⬁ T (0)
(t)⫽0.
But for initial conditions
k
such that the standard linear theory predicts traveling-wave
or growing-wave solutions, limt→⬁ T (0)
k (t)⫽0. This is perfectly reasonable, because in these cases we expect that the
asymptotic amplitude of the waves will be nonzero, no matter how small the initial amplitude is; thus, no solutions for T
can be found along the axis a⫽0 and the vanishing
asymptotic states are not accessible, which is consistent with
our earlier remark that the solution A⬅0 for Eq. 共5a兲 does
not necessarily correspond to a solution of the complete system Eqs. 共5兲 for both A and T.
The perturbation solution for T (0) , Eq. 共38兲, enables us to
compute explicitly the quantities that appear in the threshold
equation 共for k f ⫽k 0 ), Eq. 共27兲. The details, given in Appendix E 2, lead to the following equation for the 共small兲 critical
amplitude ⑀ 0 :
Clearly, this equation has the fundamental zero-field solution
⑀ (0)
0 ⫽0, corresponding to a zero initial amplitude. In this
case Eq. 共5b兲 has the trivial solution T 0 ⬅0, and the threshold
equation is obviously satisfied, since ⌫(0,0)⫽0. This just
corresponds to the trivial solution E⫽A⫽T⬅0, f ␣ ⬅F ␣ .
More interestingly, Eq. 共39兲 also yields
0,1兲
␴ 1 兺 ␣ s ␣ 关 兩 q ␣ 兩 3 /m ␣ 兴 1/2h ␣ 共 v p 兲 ⫹ ␴ 1 S 共1,1
兺 ␣ 关 兩 q ␣ 兩 5 /m ␣3 兴 1/2共 dF ␣ /d v 兲共 v p 兲
0,2兲
0,1兲
4 ␴ 2 S 共2,2
兺 ␣ s ␣ 关 兩 q ␣ 兩 5 /m ␣3 兴 1/2共 dF ␣ /d v 兲共 v p 兲 ⫹2 ␴ 2 S 共1,2
兺 ␣ s ␣ 关 兩 q ␣ 兩 5 /m ␣3 兴 1/2共 dh ␣ /d v 兲共 v p 兲
a nonzero critical amplitude, at which a nontrivial solution
branch for a crosses the axis a⫽0 共Fig. 3兲. Since Eq. 共40兲
was derived via an expansion for small ⑀ 0 , consistency re(0)
quires that ⑀ (1)
0 be close to ⑀ 0 ⫽0. Whenever the initial con(1)
dition is such that ⑀ 0 in Eq. 共40兲 does not satisfy 兩 ⑀ (1)
0 兩
Ⰶ1, there is no 共nonzero兲 small-amplitude critical point corresponding to k f ⫽k 0 , and all small sinusoidal perturbations
to a linearly stable Vlasov equilibrium undergo Landau
,
共40兲
damping to zero. Of course, this does not preclude the possibility of a nonsmall critical amplitude.
B. The critical amplitude ⑀ „00 … Ä0
We next determine the transient field for solutions with
a⫽0. We begin from the ‘‘trivial’’ critical amplitude ⑀ (0)
0
⫽0, and we seek small-amplitude time-asymptotic solutions
close to it, and look for a branch of time-asymptotic ampli-
026406-10
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
tudes that bifurcates from ⑀ 0 ⫽0, a⫽0 in the ⑀-a plane.
Whether such a branch exists is physically important because
it determines the time-asymptotic stability of the plasma. Indeed, such a branch would imply that there are arbitrarily
small initial conditions that do not damp and lead to timeasymptotic multiple-mode traveling-wave solutions of the
Buchanan-Dorning 关15兴 type 共in the coarse-grained sense described above兲. If no branch bifurcates from ⑀ (0)
0 , it will be
necessary to consider the nonzero critical amplitude ⑀ (1)
0 in
Eq. 共40兲 as another possible branching point for nonzero
time-asymptotic solutions, since in this case initial conditions with amplitudes ⑀ ⬍ ⑀ (1)
0 would have to be damped to
zero. Even then, though, the analysis at ⑀ 0 ⫽ ⑀ (0)
0 will be important, because it will provide the leading order term in the
expansion of the transient equation at ⑀ 0 ⫽ ⑀ (1)
0 with respect
(1)
to the small critical amplitude ⑀ 0 ⫽ ⑀ 0 Ⰶ1.
For ⑀ 0 ⫽0 and T 0 ⬅0 several terms in the transient equation are zero, and a tedious but straightforward extension
关33兴 of the calculation that led to Eq. 共33兲 yields 关for Re(p)
⭓0]
D k 共 p 兲 T̃ 共k1 兲 共 p 兲 ⫽ ␦ k,k 0 N k 共 p 兲 ⫺ ␦ k,k f D k 共 p 兲 Ã 共k0 兲 共 p 兲 , 共41兲
where A (0)
k (p) is the Fourier-Laplace transform of A in Eq.
共30兲 for ⑀ 0 ⫽0, i.e.,
à 共k0 兲 共 p 兲 ⬅
冕
⬁
0
dt e ⫺pt A k 共 t 兲 ⫽ ␦ k,k f
冋
a0
共 p⫹ik f v p 兲
⫹
a0
共 p⫺ik f v p 兲
册
共42兲
with
a 0 ⬅⫺
1
关 ␹ ⫺⌫ 共 0,T 共 1 兲 兲兴 ,
␹1 2
共43兲
which follows from evaluating Eq. 共32兲 at ⑀ 0 ⫽0 关where
T (0) ⬅0 and the ⌬⑀ has canceled with ⑀’s multiplying the first
two terms in Eq. 共41兲兴. Equation 共41兲 is similar to the equation that arises in Landau’s analysis 关2兴. In fact, it could be
obtained formally from Landau’s solution by simply decomposing the field into T⫹A, moving A to the right side, and
assuming it to be of the form in Eq. 共30兲. However, Eq. 共41兲
is an equation for T and a, not E and must be solved simultaneously with Eq. 共25兲 in such a way that limt→⬁ T(x,t)
⫽0.
(0)
If k f ⫽k 0 , for k⫽k f Eq. 共41兲 gives T̃ (1)
k f (p)⫽⫺Ã k f (p),
which is clearly unacceptable. Hence, the initial value problem cannot have a nonzero time-asymptotic solution of the
form in Eq. 共30兲 unless k f ⫽k 0 , as anticipated in the previous
section 共see footnote 3兲. Thus, let k f ⫽k 0 ; if k⫽k f ⫽k 0 , Eq.
共41兲 implies that T̃ (1)
k (p)⬅0, so that the transient has no
leading-order component with that wave number. We conclude that the only relevant spatial Fourier mode in T must
correspond to k⫽k f ⫽k 0 . Thus, the solution to Eq. 共41兲 is
T̃ 共k1 兲 共 p 兲 ⫽
0
N k 0共 p 兲
D k 0共 p 兲
⫺Ã 共k0 兲 共 p 兲 .
0
共44兲
Applying Landau’s procedure 关2兴 to this equation, the
transient field T is obtained as a sum of exponential terms,
each corresponding to a pole of 关 N k⫹ (p)/D k⫹ (p) 兴 ⫺Ã k(0) (p)
0
0
0
where N k⫹ and D k⫹ are the analytic continuations of N k 0 and
0
0
D k 0 关2兴. Under Landau’s assumptions 关2兴 on the analyticity
of F ␣ and h ␣ , this function has two kinds of pole: 共a兲 the
poles associated with the zeros of D k⫹ , and 共b兲 the poles of
0
à k(0) (p) at p⫽⫾ik 0 v p 关from Eq. 共42兲兴. The crucial point is
0
that the extra poles, corresponding to à k(0) (p), lie on the
0
imaginary axis. But since T is the transient field, it cannot
include undamped terms, and a necessary condition for the
existence of solutions to Eq. 共41兲 is that the residues of
(0)
关 N k⫹0 (p)/D ⫹
k 0 (p) 兴 ⫺Ã k 0 (p) at p⫽⫾ik 0 v p must be zero. We
classify the situations that lead to this condition according to
the nature of the roots of D k⫹ .
0
共I兲 First, all the roots of D k⫹ have Re(p)⬍0. Then, for the
0
residues at p⫽⫾ik 0 v p to be zero a 0 must equal zero. Hence,
all time-asymptotic solutions with initial amplitudes in the
neighborhood of ⑀ 0 ⫽0 coincide with the branch a⫽0 关with
errors of order O(e 3/2)] and correspond to complete Landau
damping, notwithstanding the trapping effects. Then nonzero
solutions for the time-asymptotic electric field will be possible only for initial amplitudes greater than some nonzero
in Eq. 共40兲. This situation will be
critical value, like ⑀ (1)
0
considered in the next subsection.
共II兲 Next, D k⫹ has roots with Re(p)⬎0. Then the solutions
0
for T contain growing modes, which is not acceptable. Thus,
for the choices of F ␣ that lead to this situation, there is no
small-amplitude time-asymptotic solution in the neighborhood of ⑀ 0 ⫽0, not even a⫽0. This leads us to the conjecture
that in these linearly unstable cases the time-asymptotic field
amplitude tends to a nonzero value as ⑀ →0. This problem,
although interesting, will not be further pursued here.
共III兲 Finally, D k⫹ has poles with Re(p)⭐0. This includes
0
poles on the imaginary axis, corresponding to a pair of eigenvalues embedded in the Van Kampen–Case 关3,4兴 continuous spectrum for the linearized VP system. These poles
produce undamped terms that must cancel with those coming
from à k(0) (p), in order to have a valid solution. Evaluating
0
the Landau dispersion relation on the imaginary axis and
separating the real and imaginary parts gives
1⫹
␣
␣
P冕 dv
⫽0,
2 兺
k 0 ␣ m ␣ R v ⫺␭/k 0
4␲
q2
F⬘ 共v兲
共45兲
q ␣2
兺␣ m ␣ F ␣⬘ 共 ␭/k 0 兲 ⫽0,
共46兲
where ␭⫽ip, ␭苸R. Equation 共45兲 is the time-asymptotic
Vlasov dispersion relation 共for symmetric initial conditions
and under the approximation of transient linearization兲. The
residues at p⫽⫾ik 0 v p can be zero only if 共i兲 ␭⫽⫾k 0 v p
⫹
satisfies Eqs. 共45兲 and 共46兲, and 共ii兲 the residues of N ⫹
k /D k
0
0
at p⫽⫾ik 0 v p are equal to those of à k(0) (p), i.e., equal to a 0 .
0
026406-11
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
In the context of the two-wave case, we consider the situation in which Eq. 共45兲 has just one pair5 of simple roots on
the imaginary axis p⫽⫿ik 0 v p or ␭⫽⫾k 0 v p , which satisfy
q ␣2
兺␣ m ␣ F ␣⬘ 共 ⫾ v p 兲 ⫽0.
a 0 and the imaginary part equal to zero yields the twofold
condition
⫾
共47兲
A Taylor expansion 关6兴 of D k⫹ about ⫾ik 0 v p yields the resi0
dues at these poles:
Res
冉
N⫹
k
0
,⫾ik 0 v p
D k⫹
0
⫽⫺
冊
兺 ␣ 共 q ␣2 /m ␣ 兲关 P兰 R d v F ␣⬙ 共 v 兲 / 共 v ⫾ v p 兲 ⫹i ␲ F ␣⬙ 共 v p 兲兴
.
共48兲
共Here and below the symmetries of F ␣ , h ␣ , F ␣⬘ , etc., allow
us to replace ⫾ v p by v p in their arguments.兲 This expression
must be set equal to a 0 , which is obtained from Eqs. 共43兲,
共17兲, 共18兲, and 共22兲. The expression for a 0 can be made
somewhat simpler if Eq. 共47兲 is replaced by the slightly
stronger condition (dF ␣ /d v )( v p )⫽0 ( ␣ ⫽1,2,...,N S ). In
many physical situations the two will be equivalent 共e.g., in
an ion-electron plasma, in which the charge-mass ratio for
the electrons is much larger than for all the other species
combined兲. Then, Eq. 共43兲 关with Eqs. 共17兲, 共18兲, and 共22兲兴
yields
k 0 兺 ␣ s ␣ 关 兩 q ␣ 兩 3 /m ␣ 兴 1/2h ␣ 共 v p 兲
兺 ␣ 关 兩 q ␣ 兩 5 /m ␣3 兴 1/2F ␣⬙ 共 v p 兲
.
共49兲
⫹
(0)
For 关 N ⫹
k 0 (p)/D k 0 (p) 兴 ⫺Ã k 0 (p) to have zero residues at
⫾ik 0 v p , Eqs. 共48兲 and 共49兲 must be equal, up to a possible
change in the sign of a 0 due to the choice of the phase of
A(x,t) in Eq. 共13兲.6 Setting the real part of Eq. 共48兲 equal to
In special cases Eq. 共45兲 could have N 共⬎1兲 pairs of roots on the
imaginary axis. Then, the Ansatz would be extended to include up
to 2N waves. If conditions 共i兲 and 共ii兲 were satisfied at all 2N poles,
we would associate a time-asymptotic wave mode with each root of
D⫹
k 0 on the imaginary axis. However, here we focus on the twowave case, both because cases with N⬎1 are exceptional and because the extension to those cases is straightforward, although very
tedious.
6
We remarked in footnotes 2 and 4 that a⫽a 0 ⌬ ⑀ can always be
made positive via an appropriate choice of the phase of A(x,t), and
that this might require us to replace ⌬⑀ by ⫺⌬⑀ if a 0 ⬍0. However,
until this point it was not clear why the sign of a should be chosen
positive or negative. Now, the transient analysis gives the natural
criterion to determine the phase of A(x,t) 共and the sign of a兲 for a
given initial condition: the phase of A(x,t) must be chosen so that
it is consistent with the phase of the corresponding linear modes
共with phase velocities ⫾ v p ), whose amplitude is given by the real
part of Eq. 共48兲. Indeed, we know that these linear modes correctly
describe the initial wave propagation, and that A(x,t) must ‘‘replace’’ them in the nonlinear regime. Hence, the sign of a 0 must be
the same as the sign of the real part of Eq. 共48兲.
5
兺 ␣ 关 兩 q ␣ 兩 5 /m ␣ 兴 1/2F ␣⬙ 共 v p 兲
⫽
⫽
k 0 兺 ␣ q ␣ 关 P兰 R d v h ␣ 共 v 兲 / 共 v ⫾ v p 兲 ⫹i ␲ h ␣ 共 v p 兲兴
a 0 ⫽⫺
兺 ␣ s ␣ 关 兩 q ␣ 兩 3 /m ␣ 兴 1/2h ␣ 共 v p 兲
兺 ␣q ␣h ␣共 v p 兲
兺 ␣ 共 q ␣2 /m ␣ 兲 F ␣⬙ 共 v p 兲
兺 ␣ q ␣ P兰 R d v h ␣ 共 v 兲 / 共 v ⫺ v p 兲
兺 ␣ 共 q ␣2 /m ␣ 兲 P兰 R d v F ␣⬙ 共 v 兲 / 共 v ⫺ v p 兲
,
共50兲
where the sign of the left-most term must be chosen to be
equal to the sign of the middle term to ensure that the nonlinear solution connects smoothly to the linearized initial solution, from which it evolves after particle trapping becomes
significant.
When both Eqs. 共47兲 and 共50兲 are satisfied, Eq. 共41兲 has
an acceptable solution for T (1)
k f , which goes to zero as t
→⬁. This solution is basically Landau’s solution for the
given initial condition, minus the ‘‘undamped’’ terms that
correspond to the poles on the imaginary axis. Indeed, the
inverse Laplace transform of Eq. 共44兲 yields a sum of exponentials corresponding to all the roots of Eq. 共50兲. The ‘‘linear’’ contributions due to these two roots are replaced exactly
by the time-asymptotic field A, which is given by Eq. 共30兲
with a⫽a 0 ⌬ ⑀ . Equation 共50兲 ensures that the timeasymptotic wave amplitude is equal to the amplitude of the
corresponding modes in the linear theory. Thus, as long as
Eq. 共50兲 is satisfied the electric field solution from the nonlinear theory is actually the same 共for case III兲 as that from
the linear analysis, not only initially but at all times. Clearly,
this happens only because we are considering the special
case of initial Vlasov equilibria with zero derivatives at the
phase velocities. Nevertheless, the solution for the distribution function, inside the v integrals in Eq. 共7兲, is completely
different from the solution to the linearized problem; in particular, it contains the trapping effects via the characteristics
for the time-asymptotic field A. Thus, Eq. 共50兲 gives a nonlinear criterion that determines which small-amplitude initial
conditions lead to time-asymptotic traveling-wave solutions.
According to the standard linear theory, this happens whenever Eqs. 共45兲 and 共46兲 hold. Unfortunately this is only an
initial time result, and it is not at all clear from the linear
theory whether the Landau modes corresponding to poles on
the imaginary axis can keep traveling unchanged in the nonlinear regime, i.e., when the trapping effects become relevant. Equation 共50兲, however, ensures precisely that the undamped modes generated by the poles p⫽⫾ik f v p are
consistent with the nonlinear dynamics, and will keep traveling at the same amplitude in the time-asymptotic limit.
An important example of an initial condition that satisfies
Eq. 共50兲 is provided by Buchanan and Dorning’s undamped
two-wave BGK-like solutions 关15兴 共see Appendix F here兲.
Thus, those solutions are a special case of the solutions developed here. What characterizes BGK and ‘‘BGK-like’’ solutions is that the distribution function is constant along the
level curves of suitable invariants for single-particle motion
共such as the energy in true BGK solutions or Buchanan and
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TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
Dorning’s two-wave approximate invariants in BGK-like solutions兲, so that there is no energy exchange between the
plasma and the field. Equation 共50兲 provides a generalized
BGK condition on the initial distribution in the sense that,
even when the resulting solution is not really BGK-like, the
resonant wave-particle interactions are sufficiently ‘‘gentle’’
that small-amplitude wave propagation still occurs. These
solutions provide the only nontrivial solution branch through
the origin in the ⑀ -a plane.
Interestingly, in the time-asymptotic limit the distribution
function becomes macroscopically equivalent to the coarsegrained BGK-like distribution obtained by averaging f ␣ over
the two-wave approximate invariant curves. By ‘‘macroscopically equivalent’’ we mean that, when integrated over
phase space, the two distribution functions produce the same
macroscopic quantities 共see Appendix D, Proposition 1兲. In
this coarse-grained sense, the solutions that we have obtained
evolve in time to reach BGK-like states as the outcome of
the non-linear dynamics, whereas the undamped solutions
obtained by Buchanan and Dorning 关15兴 have to be set up by
an initial distribution function that already has exactly the
structure of a BGK-like solution.
C. The nonzero critical amplitude ⑀ „01 …
In case I above we established that whenever the initial
Vlasov equilibrium is linearly stable the only solution branch
through ⑀ 0 ⫽ ⑀ (0)
0 ⫽0 is the basic branch a⫽0. Hence, we
must raise the question whether there are nonzero timeasymptotic solutions that branch from a⫽0 at a nonzero
critical initial amplitude. Equation 共40兲 gives the small nonzero critical initial amplitude ⑀ (1)
0 , which is a possible
threshold separating the initial conditions that result in Landau damping to zero from those that lead to traveling-wave
solutions with a⫽0. To find such nonzero time-asymptotic
solutions, we now study the transient equation in the neighborhood of ⑀ (1)
0 . This is also done via a perturbation expansion, first in powers of a and ⌬⑀, and then in powers ⑀ (1)
0 关to
invert the linear operator on the left side of Eq. 共33兲兴. Fortunately, all the leading-order information can be obtained
from a simple extension of the analysis carried out in Sec.
IV B for the ‘‘trivial’’ critical initial amplitude ⑀ (0)
0 ⫽0.
At zeroth order in ⌬⑀, of course, we have the purely transient field T 0 ⬅T (0) , which was already computed in Eq.
共38兲, but now with ⑀ 0 ⫽ ⑀ (1)
0 . At first order in ⌬⑀ we expand
(1)
the first order term T̃ k in powers of ⑀ (1)
0 , in the form
T̃ 共k1 兲 ⫽T̃ 共k1,0兲 ⫹ ⑀ 共01 兲 T̃ 共k1,1兲 ⫹¯ .
D k⫹ 共 ⫾ik 0 v p 兲
0
⫽1⫹
0
0
⫺Ã k(0) (p). Again, Ã k(0) (p) has poles ⫾ik 0 v p , whose resi0
dues must be zero to have a valid solution for T k(1,0) (t). Since
0
4␲
k 20
q ␣2
兺␣ m ␣
冋冕
P
R
dv
F ␣⬘ 共 v 兲
共v⫾vp兲
⫹i ␲ F ␣⬘ 共 ⫿ v p 兲
册
⫽O 共 ⑀ 共01 兲 兲 .
共52兲
Since ⫾ v p satisfy the Vlasov dispersion relation, this is
equivalent to
q ␣2
兺␣ m ␣ F ␣⬘ 共 ⫿ v p 兲 ⫽O 共 ⑀ 共01 兲 兲 .
共53兲
Obviously, this is a generalization of Eq. 共47兲: it indicates
that for there to be a small critical amplitude, the derivative
of the initial Vlasov equilibrium at the phase velocity must
be small, so that Landau damping is weak and even a small
initial disturbance has the possibility of trapping particles. In
what follows, Eq. 共53兲 will be taken to be a consequence of
the stronger condition
dF ␣
共 ⫾ v p 兲 ⫽O 共 ⑀ 共01 兲 兲
dv
␣ ⫽1,2,...,N s .
共54兲
Then, the residue calculation for N k⫹ /D k⫹ is the same as in
0
0
case III, since Eq. 共47兲 holds at zeroth order and leads to the
same expression as in Eq. 共48兲. Similarly, a 0 has the same
expression at zeroth order in Eq. 共49兲. Hence, the same argument as in the case ⑀ 0 ⫽0 shows that the initial distribution must satisfy 共now at leading order兲 the conditions in Eq.
共50兲, in order for small-amplitude nonlinear wave propagation to be possible in the neighborhood of the small critical
amplitude ⑀ (1)
0 . Then, the solution for T is
T 共 x,t 兲 ⫽ ⑀ 共01 兲 T 共 0,1兲 共 x,t 兲 ⫹⌬ ⑀ T 共 1,0兲 共 x,t 兲 ⫹O 共 ⑀ 共01 兲 ⌬ ⑀ 兲 ,
共51兲
is identical to Eq. 共41兲 共which correThe equation for T̃ (1,0)
k
sponds to the limit ⑀ (1)
0 →0). Hence, the same arguments
used before show that solutions with a 0 ⫽0 are possible only
if k⫽k 0 ⫽k f , and that T̃ k(1,0) (p) must be obtained from the
0
analysis of the poles of the function 关 N k⫹ (p)/D k⫹ (p) 兴
0
we are considering initial conditions near linearly stable
equilibria, the residue cannot be exactly zero if a 0 ⫽0, just as
in case I above. However, it can be approximately zero, with
(1,0)
itself. The necessary conthe same accuracy in ⑀ (1)
0 as T̃ k
dition for this is that ⫾ik 0 v p must be roots of the Landau
dispersion relation D k⫹ (p)⫽0 at zero order in ⑀ (1)
0 . 共They
0
cannot be exact roots because in the case being considered
all the roots have negative real parts.兲 Hence,
共55兲
where T (0,1) is Landau’s damping solution for the given initial condition and T (1,0) is the same Landau solution minus
the contributions from the poles that correspond 共at zeroth
order兲 to the time-asymptotic phase velocities ⫾ v p .
The field A is given, of course, by the two-wave solution
in Eq. 共30兲, with k f ⫽k 0 and amplitude a obtained from Eq.
共32兲 共see Appendix E 3 for the detailed calculation兲. To first
order in both ⌬⑀ and ⑀ 0
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C. LANCELLOTTI AND J. J. DORNING
再
a⫽ ⫺
⫹
k 0 兺 ␣ s ␣ 关 兩 q ␣ 兩 3 /m ␣ 兴 1/2h ␣ 共 v p 兲
兺 ␣ 关 兩 q ␣ 兩 5 /m ␣3 兴 1/2F ␣⬙ 共 v p 兲
⫹
0,1兲
1,0兲
兺 ␣ 关 兩 q ␣5 兩 /k 30 m ␣3 兴 1/2关 2s ␣ ⑀ 共01 兲 ␴ 2 S 共1,2
h ␣⬘ 共 v p 兲 ⫺ ␴ 1 S 共1,1
F ␣⬘ 共 v p 兲兴
␴ 1 兺 ␣ 关 兩 q ␣ 兩 5 /k 50 m ␣3 兴 1/2F ␣⬙ 共 v p 兲
0,1兲
0,1兲
␹ 2 ␹ 3 ⑀ 共01 兲 ⫹ ␹ 2 兺 ␣ 关 兩 q ␣ 兩 5 /k 50 m ␣3 兴 1/2兵 共 ␴ 1 /2v p 兲 F ␣⬘ 共 v p 兲 ⫺ ⑀ 共01 兲 共 兩 q ␣ 兩 /m ␣ 兲关 ␴ 2 S 共1,1
F ␣⵮ 共 v p 兲 ⫺ ␴ 1 S 共1,3
共 1/4v p 兲 F ␣⬙ 共 v p 兲兴 其
关 ␴ 1 兺 ␣ 关 兩 q ␣ 兩 5 /k 50 m ␣3 兴 1/2F ␣⬙ 共 v p 兲兴 2
冎
⌬⑀,
共56兲
where the S (m,n)
are given by Eqs. 共E9兲, 共E10兲, and 共E14兲.
i, j
Equation 共56兲 gives the nonzero final amplitude that E⫽A
⫹T reaches via particle trapping after slow initial Landau
damping. In the limit ⑀ (1)
0 →0 this branch reduces to Eq. 共49兲
(1)
关recall F ␣⬘ ( v p )⫽O( ⑀ 0 )], i.e., a solution branch coming
from the origin in the ⑀ -a plane and corresponding to a
共possibly infinitesimal兲 ‘‘flat spot’’ on the initial Vlasov equilibrium. The corresponding distribution function is again
given by the integrand in Eq. 共7兲 and is equivalent to a BGKlike distribution in the sense discussed in the previous section; namely, as t→⬁ f ␣ generates the same macroscopic
quantities as the coarse-grained function which is obtained
by averaging f ␣ in phase space along the ‘‘invariant curves’’
for the field A. In this sense, these time-asymptotic solutions
to the VP system can be viewed as a generalization of the
multiple-traveling-wave BGK-like solutions of Buchanan
and Dorning 关15兴. The solutions obtained here, of course,
exist only for initial field amplitudes above the threshold
value given by Eq. 共40兲 and have time-asymptotic amplitudes given by Eq. 共56兲.
V. CONCLUSION
Particle trapping effects are ubiquitous in plasma physics.
Even in simple situations, however, the theoretical understanding of these effects is severely limited by our inability
to analyze 共except possibly by large-scale numerical simulations兲 the nonlinear equations that model self-consistent
field-particle interactions. In this article we have studied
what is possibly the most fundamental example of nonlinear
wave-particle dynamics, the one associated with longitudinal
waves in a collisionless plasma. This classic problem, often
referred to as ‘‘nonlinear Landau damping,’’ is fairly well
understood in the two limiting cases of strong Landau damping 关2兴 and weakly damped trapping-dominated wave propagation 关8,9兴. Conversely, there is very little understanding of
the difficult ‘‘intermediate’’ regime in which the two time
scales associated with linear Landau damping and with particle trapping are of the same order of magnitude; hence, our
goal has been to study this intermediate regime. The fact that
the same type of initial disturbance 共e.g., a single-mode sine
wave兲 can, depending on its amplitude, be Landau damped to
zero or evolve to traveling-wave behavior 关8兴 suggests that
there must be a threshold 共a ‘‘critical initial amplitude’’兲
separating the initial conditions that lead to these two very
different time-asymptotic states. The existence of such a
threshold is strongly supported by numerical simulations
showing that small-amplitude electric fields are damped to
zero, whereas initial conditions of larger amplitude evolve to
nonzero multiple-wave final states 关26,27兴. In this article we
have developed analytical expressions both for the threshold
and for the amplitude of the time-asymptotic superimposed
traveling waves that evolve from initial conditions with amplitudes just above the threshold.
In the first part of the paper 共Sec. II兲 we developed a
general technique for studying self-consistent wave-particle
dynamics. This technique, which has wider applicability, is
based on two essential ideas. The first follows from the observation that, for initial conditions with amplitude just
above the threshold, the time-asymptotic electric field is
given by a superposition of small-amplitude BGK-like wave
modes, each of which satisfies a Vlasov dispersion relation.
Thus, the problem of solving the Vlasov-Poisson equation at
long times can be reduced to the determination of a finite
number of time-asymptotic amplitudes 共one for a symmetric
pair of waves兲. The second essential idea is that, since the
general form of the time-asymptotic field is a discrete superposition of small-amplitude waves, the long-time solutions to
the Vlasov-Poisson equations can be approximated via ‘‘transient linearization,’’ i.e., by linearizing only the interaction
between the distribution function and the transient part of the
electric field, while keeping the full nonlinear wave-particle
interaction in the limit t→⬁. Under this approximation, the
equations can be solved exactly via Hamiltonian perturbation
theory.
The detailed solution, given in Sec. III for the important
case of a sinusoidal initial disturbance 关8兴, shows that, as the
initial amplitude ⑀ of the perturbation is increased through a
certain threshold, the time-asymptotic wave amplitude a
changes transcritically 共i.e., at a finite angle兲 from zero
共complete Landau damping兲 to a nonzero value 共travelingwave propagation兲; the dependence of the final amplitude a
on the initial amplitude ⑀ near the threshold is given by Eq.
共32兲. The threshold itself satisfies a scalar equation, Eq. 共27兲,
which depends on the details of the initial distribution function. Naturally, the equations for the threshold and the timeasymptotic field amplitude have coefficients that depend on
the transient part of the electric field; hence, these general
equations must be combined with analysis of the transient
behavior. That analysis was presented in the last part of the
paper 共Sec. IV兲 for the more restricted but very important
case in which the critical initial amplitude 共or threshold兲 itself is small. Physically, this occurs when linear Landau
damping is weak, so that even a small initial perturbation
from zero is sufficient to cause particle trapping and evolution to self-sustained traveling-wave modes. In this case, the
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TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
transient behavior could be analyzed via straightforward perturbation expansions, which led to two types of long-time
solutions.
The solutions of the first type 共Sec. IV B兲 are those in
which the threshold in the initial amplitude is actually zero.
These solutions, for arbitrarily small perturbations of linearly
stable equilibria, are non-BGK-like solutions that are not
Landau damped to zero; rather, they evolve to undamped
multiple traveling waves. 共They include as a special case
undamped small-amplitude multiple-wave BGK-like solutions reported previously 关15兴.兲 The solutions of the second
type 共Sec. IV C兲 branch from the trivial zero-field timeasymptotic solution at a nonzero threshold and lead to nonzero final electric field states given by a nonlinear superposition of traveling-wave modes. The analysis yielded
completely explicit results for the threshold Eq. 共40兲 and for
the final amplitude of the time-asymptotic field generated by
initial perturbations just above the threshold Eq. 共56兲. Interestingly, recent large-scale numerical simulations 关24兴 have
already confirmed these results, which were summarized earlier in a brief communication 关18兴.
( ⳵ x A / ⳵ v̄ )(x̄, v̄ ,t)⫽d ␣A (x̄, v̄ ,t)t where d ␣A (x̄, v̄ ,t) is a uniformly bounded function whose detailed expression is not
important here.
To estimate the order of magnitude of R ␣ ,k , we restrict
the domain of the v̄ integration to a finite interval ⌰傺R that
represents the ‘‘width’’ of the distribution functions 共i.e., the
temperature of the plasma兲. Since F␣ and f ␣ are assumed to
have strong decay properties, ⌰ can be chosen so that the
error introduced by the restriction of the domain is of higher
order. Likewise, assuming tT(x,t) to be integrable in t, there
must be a positive ␦ such that ᭙ x
冕
0
ACKNOWLEDGMENT
R ␣ 共 x,t 兲 ⫽
q␣
m␣
冕 冕 ␶ 再 冋 ⳵⳵
t
R
dv
d
T
0
f ␣ ⳵ F␣
⫺
⳵v
v
册冎
q␣ k
m␣ i␲
冕
⫹⬁
␦
共A3兲
兩 tT 共 x,t 兲 兩 dt.
冕 冕 冕
⫻e ⫺ikx
During the final stages of this work C.L. was supported by
NSF Grant No. DMS-0207339.
To establish the range of validity of the transiently linearized Vlasov equation, Eq. 共6兲, we consider the error introduced in replacing f ␣ by F␣ in Eq. 共7兲:
兩 tT 共 x,t 兲 兩 dtⰇ
Clearly, ␦ is smaller if the rate of decay of T is greater. When
t⬎ ␦ , replacing R by ⌰, and using Eq. 共A3兲 and the uniform
boundedness of f ␣ ⫺F␣ and d Aa , Eq. 共A2兲 can be well approximated as
R ␣ ,k 共 t 兲 ⬃
APPENDIX A: RANGE OF VALIDITY
OF THE TRANSIENT LINEARIZATION
␦
␦
0
d␶
A x̄, , ␶ 兲
共 v̄
⫹␲
⫺␲
dx̄
⌰
d v̄
⳵xA
共 x̄, v̄ , ␶ 兲
⳵ v̄
T 共 x̄, ␶ 兲关 f ␣ 共 x̄, v̄ , ␶ 兲 ⫺F␣ 共 x̄, v̄ 兲兴 .
共A4兲
Here f ␣ (x̄, v̄ ,t)⫺F␣ (x̄, v̄ )⫽F␣ (x E0 , v E0 )⫺F␣ (x̄, v̄ ) where
关 x E0 (x̄, v̄ ), v E0 (x̄, v̄ ) 兴 are the inverse trajectories 共for the species ␣, with index omitted兲 associated with the total electric
field E. From the Newton equations it follows immediately
that, for ␶ 苸(0,␦ ),
兩 x E0 ⫺x̄ 兩 ⭐ ␦ 兩 v̄ 兩 ⫹
.
关 x ␶A 共 x, v ,t 兲 , v ␶A 共 x, v ,t 兲兴
q␣ 2
␦ 兩E兩,
m␣
兩 v E0 ⫺ v̄ 兩 ⭐
q␣
␦兩E兩,
m␣
共A1兲
The Fourier coefficients of the ‘‘residual’’ become, after exchanging the order of integration, performing the areapreserving transformation of the (x, v ) integration variables
to (x̄, v̄ )⫽ 关 x A␶ (x, v ,t), v ␶A (x, v ,t) 兴 , and integrating by parts,
R ␣ ,k 共 t 兲 ⫽
q␣ k
m␣ i␲
冕 ␶冕 冕
⫻e ⫺ikx
⫹␲
t
d
0
A x̄, , ␶ 兲
共 v̄
⫺␲
dx̄
R
d v̄
where 兩 E 兩 ⬅supx,t 兩 E(x,t) 兩 . Then, from the mean value theorem
冉
兩 f ␣ 共 x̄, v̄ , ␶ 兲 ⫺F␣ 共 x̄, v̄ 兲 兩 ⭐ ␤ x, ␣ ␦ 兩 v̄ 兩 ⫹
⳵xA
共 x̄, v̄ , ␶ 兲
⳵ v̄
⫹ ␤ v,␣
T 共 x̄, ␶ 兲关 f ␣ 共 x̄, v̄ , ␶ 兲 ⫺F␣ 共 x̄, v̄ 兲兴 .
共A2兲
Here x A (x, v , ␶ ) is the ‘‘direct’’ trajectory, i.e., the position at
time ␶ of a particle starting from the point x̄, v̄ at time zero.
For the multiple-wave time-asymptotic fields that we consider, the function x A (x̄, v̄ ,t) can be computed explicitly via
the Buchanan-Dorning perturbation method 关15兴, in which
the phase plane is divided into separate regions so that the
problem in each region can be reduced to motion in an autonomous, integrable system. Then the function x A can be
obtained explicitly in terms of elliptic functions and
共A5兲
q␣ 2
␦ 兩E兩
m␣
冊
q␣
␦兩E兩,
m␣
共A6兲
where ␤ ␩ , ␣ ⬅supx苸 关 0,2␲ 兴 supv 苸⌰ 兩 ( ⳵ F␣ / ⳵ ␩ )(x, v ) 兩 , ␩ ⫽x, v .
Hence, the order of magnitude of R ␣ ,k is
兩 R ␣ ,k 共 t 兲 兩 ⬃
冋 冉
kq ␣ 2
q␣ 2
␦ 兩 ⌰ 兩兩 T 兩兩 d ␣A 兩 ␤ x, ␣ ␦ 兩 ⌰ 兩 ⫹
␦ 兩E兩
m␣
m␣
⫹ ␤ v,␣
册
q␣
␦兩E兩 ,
m␣
冊
共A7兲
where 兩 T 兩 ⬅supx,t 兩 T(x,t) 兩 and 兩 d ␣A 兩 ⬅supx, v ,t 兩 d ␣A (x, v ,t) 兩 .
The same procedure applied to the approximate expression in Eq. 共7兲
026406-15
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C. LANCELLOTTI AND J. J. DORNING
Q ␣ 共 x,t 兲 ⬅
q␣
m␣
冕 冕 ␶再
t
R
dv
d
0
T
⳵ F␣
⳵v
冎
关 x ␶A 共 x, v ,t 兲 , v ␶A 共 x, v ,t 兲兴
f ␣T 共 x, v ,t 兲 ⫽F␣ 共 x⫺ v t, v 兲 ⫺
kq ␣ 2
␦ 兩 ⌰ 兩兩 T 兩兩 d ␣A 兩 .
m␣
␦ Ⰶ ␶ bx ,
␦ Ⰶ ␶ bv
d
T
0
冎
⳵ F␣
⳵v
.
关 x⫺ v共 t⫺ ␶ 兲 , v兴
共A9兲
f ␣T 共 x, v ,t 兲 ⫽F␣T 共 x⫺ v t, v 兲 ⫹g ␣T 共 x, v ,t 兲 ,
共A10兲
( ␣ ⫽1,...,N S ). Physically, ␦ is the time scale for the decay of
the transient T, whereas ␤ x, ␣ 兩 ⌰ 兩 , (q ␣ /m ␣ ) ␤ x, ␣ 兩 E 兩 , and
(q ␣ /m ␣ ) ␤ v , ␣ 兩 E 兩 measure the various effects that make f ␣
drift away from the initial condition F␣ . In particular,
␤ x, ␣ 兩 ⌰ 兩 corresponds to the zero-field advection 共in space兲 of
the spatially nonuniform part of the initial distribution, which
becomes relevant on the time scale ␶ ax, ␣ ⬇1/␤ x, ␣ 兩 ⌰ 兩 . The
term (q ␣ /m ␣ ) ␤ x, ␣ 兩 E 兩 measures the action of the field E on
the position of the particles and corresponds to the trapping
time scale ␶ bx, ␣ ⬇ 冑m ␣ /q ␣ ␤ x, ␣ 兩 E 兩 . Finally, (q ␣ /m ␣ ) ␤ v , ␣ 兩 E 兩
expresses the deviation imposed by E on the velocity of the
particles 共i.e., the advection in velocity兲 on the time scale
␶ b v , ␣ ⬇m ␣ /q ␣ ␤ v , ␣ 兩 E 兩 . Defining ␶ ax ⬅min␣␶ax,␣ , ␶ bx
⬅min␣␶bx,␣ , and ␶ b v ⬅min␣␶bv,␣ , Eq. 共A10兲 can be broken
into the three conditions
␦ Ⰶ ␶ ax ,
t
共B1兲
Under the assumption that tT(x,t) is integrable at infinity, by
writing the interval of integration as 关0,⫹⬁兲 minus 关 t,⫹⬁)
Eq. 共B1兲 can be written as
Clearly, for Eq. 共7兲 to be a good approximation to the full
Vlasov equation, 兩 R ␣ 兩 Ⰶ 兩 Q ␣ 兩 , or, from Eqs. 共A7兲 and 共A9兲,
q␣
q␣
␤ x, ␣ ␦ 兩 ⌰ 兩 ⫹ ␤ x, ␣ ␦ 2 兩 E 兩 ⫹ ␤ v , ␣ ␦
兩 E 兩 Ⰶ1
m␣
m␣
冕 ␶再
共A8兲
leads, for the Fourier coefficients, to
兩 Q ␣ ,k 共 t 兲 兩 ⬃
q␣
m␣
where
uniformly as t→⬁ and
F␣T 共 x, v 兲 ⬅F␣ 共 x, v 兲 ⫺
q␣
m␣
⳵ F␣
⳵v
⬁
d
T
0
冎
F ␣T 共 v 兲 ⫽F␣T ,0共 v 兲 ⫽F ␣ 共 v 兲
⫺
q␣ 1
m␣ 2␲
冕 冕 ␶再
⫹␲
⫺␲
⬁
dx
.
关 x⫹ v ␶ , v兴
d
T
0
⳵ F␣
⳵v
冎
.
关 x⫹ v ␶ , v兴
共B4兲
on the right side of
In particular, the charge density for
Eq. 共8兲 is now equal to the charge density for F ␣T . By definition, the initial Vlasov equilibrium F ␣ ( v ) has zero charge
density, as does the other term in Eq. 共B4兲 共this follows from
using spatial periodicity to eliminate the shift v t that arises
in the limits of integration in dx and an integration by parts兲.
Hence, F ␣T ( v ) is a charge-neutral equilibrium and Eq. 共8兲 is
identically satisfied by A⬅0.
f ␣T
APPENDIX C: LINEARIZATION OF THE
TIME-ASYMPTOTIC EQUATION
To develop the linearization of the time-asymptotic equation about A⬅0, we substitute into Eq. 共5a兲 for f ␣
f ␣ 共 x, v ,t 兲 ⫽F␣ 共 x⫺ v t, v 兲 ⫺
APPENDIX B: VANISHING TIME-ASYMPTOTIC
FIELD SOLUTIONS
To prove that A⬅0 is a solution to Eq. 共8兲, whether or not
it can be reached from a given initial condition, we note that
for A(x,t)⬅0 the 关 x A␶ , v ␶A 兴 are x 0␶ (x, v ,t)⫽x⫺ v (t⫺ ␶ ),
v ␶0 (x, v ,t)⫽ v , and the solution to Eq. 共6兲 is
冕 ␶再
共B2兲
共B3兲
From Eq. 共B2兲 it follows that, in the time-asymptotic
limit, f ␣T is macroscopically equivalent to a spatially uniform
Vlasov equilibrium, in the sense that there is an equilibrium
F ␣T ( v ) such that f ␣T and F ␣T generate the same macroscopic
quantities. To be precise, consider any integral of the form
兰 Rdu G( v ,u) f ␣T (x,u,t), which could be a charge or current
density 关 G( v ,u)⫽1,u 兴 , or any higher moment 关 G( v ,u)
⫽u n 兴 , or a filtered distribution function 关26兴. Substituting f ␣T
from Eq. 共B2兲 and taking the spatial Fourier transform, it is
easy to see that, for k⫽0, 兰 Rdu G( v ,u) f ␣T ,k (u,t)→0 as t
→⬁ by the Riemann-Lebesgue lemma 关because f ␣T ,k (u,t)
⫽F␣T ,k (u)e ikut plus terms derived from the transient functions g ␣ , which vanish in the time-asymptotic limit兴. Hence,
兰 Rdu G( v ,u) f ␣T (x,u,t)→ 兰 Rdu G( v ,u)F ␣T (u)
as
t→⬁,
where we have introduced the time-asymptotic equilibrium
共A11兲
for the validity of the transient linearization. They have been
derived for t⬎ ␦ , but the development can be adapted for t
⬍ ␦ . One obtains conditions just like those in Eqs. 共A11兲, but
with ␦ replaced by t; since t⬍ ␦ , it follows that Eqs. 共A11兲
provide a sufficient condition for the accuracy of the transient linearization at all times. Finally, ␶ transⰆ ␶ dyn follows by
defining ␶ trans⬅ ␦ and ␶ dyn⬅min(␶ax ,␶bx ,␶bv).
In practice, ␶ ax , ␶ bx , and ␶ b v 共and thus ␶ dyn) will usually
be determined by the electrons. Interestingly, the definitions
of the field-effect time scales ␶ bx and ␶ b v include the parameters ␤ x, ␣ and ␤ v , ␣ , which depend, respectively, on the spatial and velocity gradients of the initial distribution, whereas
O’Neil’s ␶ b contains the typical wave number k and therefore
depends only on the spatial gradient. In most physical situations, ␤ v , ␣ ⬃1 and ␤ x, ␣ ⬃k 兩 E 兩 ; thus, all three time scales on
the right sides of Eqs. 共A11兲 are proportional to 1/兩 E 兩 .
Hence, in the small-amplitude case ␶ dyn is larger than
O’Neil’s trapping time ␶ b , which is proportional to 1/兩 E 兩 1/2.
g ␣T (x, v ,t)→0
⫺
q␣
m␣
冕 ␶再
t
d
0
T
q␣
m␣
⳵ F␣
⳵v
冕 ␶再
t
d
0
冎
A
⳵f␣
⳵v
冎
,
关 x⫺ v共 t⫺ ␶ 兲 , v兴
共C1兲
关 x⫺ v共 t⫺ ␶ 兲 , v兴
which is obtained by integrating the transiently linearized
Vlasov equation along straight line trajectories (A⬅0). The
026406-16
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
first and third terms on the right side generate vanishing
quantities by the Riemann-Lebesgue lemma; hence, Eq. 共5a兲
becomes
2
ik
A k 共 t 兲 ⫽⫺
⫻
兺␣ m ␣ P a 冕⫺ ␲ dx e ⫺ikx 冕Rd v
q ␣2
冕 ␶再
t
d
A
0
⫹␲
⳵f␣
⳵v
冎
a k, ␻ i ⫽
共C2兲
.
a k, ␻ i ⫽⫺
2
ik
dt e i ␻ t 冕 dx 冕 d v
冕
兺␣ m ␣ ␴lim
␴
0
⫺␲
R
→⬁
⫻
冕
d ␶ e ⫺ikx⫺ik v共 t⫺ ␶ 兲 A 共 x, ␶ 兲
q ␣2
0
␴
1
⫹␲
i
⳵f␣
共 x, v , ␶ 兲 . 共C3兲
⳵v
Integrating by parts in t, introducing principal value integrals, and noting the cancellation of the associated half residues gives
q ␣2
a k, ␻ i ⫽⫺
2
k
⫻
冕
兺␣ m ␣ ␴ →⬁
⫹␲
⫺␲
2
⫹
k
⫻
1
P
␴
lim
冕
R
dv
e i␻i␴
␻ i ⫺k v
冕
␴
dt e ⫺ik v共 ␴ ⫺t 兲
0
⳵f␣
共 x, v ,t 兲
⳵v
dx e ⫺ikx A 共 x,t 兲
冕
⫹␲
⫺␲
e i␻it
␴
1
dx e ⫺ikx A 共 x,t 兲
⳵f␣
共 x, v ,t 兲 .
⳵v
2
k
兺␣
⫹
2
k
兺␣
⫻
冕
⫹␲
⫺␲
␴ →⬁
q ␣2
m␣
冕
冕 冕
␴
1
P
␴
q ␣ lim
1
lim
␴ →⬁
R
P
dv
共C4兲
R
e i␻i␴
f 共v,␴兲
␻ i ⫺k v ␣ ,k
␴
dv
0
dt
e i␻it
␻ i ⫺k v
⳵f␣
共 x, v ,t 兲 .
⳵v
dx e ⫺ikx A 共 x,t 兲
共C5兲
The first term on the right side is zero, since 1/␴ multiplies
bounded functions of ␴. Finally, another change in integration order yields
a k, ␻ i ⫽
2
k
q2
1
兺␣ m ␣␣ ␴lim
→⬁ ␴
⫻P
冕
R
dv
f ␣⬘ ,k
␻ i ⫺k v
冕 冕
␴
dt
0
,
⫹␲
⫺␲
冕
R
␴
1
⫹␲
i
T
dv
f ␣0,k⬘
␻ i ⫺k v
共C7兲
,
T
where f ␣0 is the distribution function at the critical state, Eq.
共B1兲 共with T⫽T 0 ).
T
Substituting f ␣0 from Eq. 共B2兲 into Eq. 共C7兲 yields Eq.
共9兲 as the linearized equation since the term containing the
T
transient g ␣0 gives no contribution to the time average and
T
all the spatial Fourier components of F␣0 with k⫽0 generate
oscillatory terms in the principal value integral, which go to
zero as t→⬁ by the Riemann-Lebesgue lemma 关20兴. Thus,
the only nonzero contribution comes from the spatially uniT
T
form part of f ␣0 ⬘ , i.e., the time-asymptotic equilibrium F ␣0
in Eq. 共B4兲.
APPENDIX D: EXPANSION OF THE TIME-ASYMPTOTIC
EQUATION
1. O’Neil terms
The first term on the right side can be simplified by noting
that it contains 共except for vanishing terms兲 the right side of
Eq. 共C1兲 evaluated at t⫽ ␴ . Hence,
a k, ␻ i ⫽⫺
dt 冕 dx e ⫺ikx⫹i ␻ t A 共 x,t 兲
冕
兺␣ m ␣ ␴lim
␴
0
⫺␲
→⬁
q ␣2
The details of the analysis that results in Eq. 共24兲, the
leading-order expansion of the two-wave time-asymptotic
equation, Eq. 共14兲, in terms of a, are given here.
P 冕 d v 冕 dt
兺␣ m ␣ ␴lim
␻ i ⫺k v
R
0
→⬁ ␴
q ␣2
2
k
⫻P
关 x⫺ v共 t⫺ ␶ 兲 , v兴
By periodicity, the integrand can be shifted by v (t⫺ ␶ ) in the
x variable; then, taking the Bohr transform of both sides
yields
t
which can be linearized about A⬅0 by simply replacing
f ␣⬘ (A⫹T) by its value at A⬅0. This gives the linearized
time-asymptotic equation
dx e ⫺ikx⫹i ␻ i t A 共 x,t 兲
We start from the first term on the right side in Eq. 共14兲
共O’Neil terms兲. To calculate the characteristics 关 x A0 , v A0 兴 , following Ref. 关15兴, we divide the phase plane into two halves,
one for each wave mode, and perform a sequence of canonical transformations that transform the dynamics in each half
plane into those for a single sinusoidal wave, which can be
calculated in terms of elliptic integrals. Instead of computing
关 x A0 , v A0 兴 directly, we shall use these canonical transformations to change the integration variables, working our way
backward. All such changes of variables will have Jacobian
determinants equal to 1 共at least to first order in a兲 because
the corresponding coordinate transformations are 共approximately兲 canonical, i.e., area preserving. We first consider the
half plane v ⭓0 that corresponds to the wave with v p ⫽
⫹ ␻ /k 0 . 关We shall not explicitly indicate all the infinitesimal
transformations in the lower limit of integration in v as the
integration variables are transformed, because in the end all
the contributions will combine into integrals on the whole v
axis. We shall simply write a plus sign on top of the integration symbol, to indicate a domain of integration that corresponds to a positive v semiaxis in the (x, v ) coordinates.兴
The first change of variables moves the problem to the
wave frame (p⫽m ␣ v , and the ␣ is suppressed兲
共C6兲
026406-17
␪ ⫽k 0 x⫺ ␻ t,
J⫽
1
共 p⫺m ␣ v p 兲 .
k0
共D1兲
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
Correspondingly, the ‘‘backward’’ values 关 x ␶A , v A␶ 兴 will be
transformed to 关 ␪ A␶ ,J A␶ 兴 , where
¯␪ ⫽ ␪ ⫺
1
q ␣k 0a
sin关 ␪ ⫹2 ␻ t 兴 ⫹O 共 a 2 兲 ,
2
m ␣ 关共 k 0 /m ␣ 兲 J⫹2 ␻ 兴 2
␪ A␶ 共 ␪ ,J,t 兲 ⫽k 0 x A␶ 共 x, v ,t 兲 ⫺ ␻ ␶ ,
J̄⫽J⫺
J A␶ 共 ␪ ,J,t 兲 ⫽
m␣ A
关v 共 x, v ,t 兲 ⫺ v p 兴
k0 ␶
共D2兲
For cos ␪␶A and J ␶A , the analog of Eq. 共D2兲 is found by evaluating the inverse equations at t⫽ ␶ ,
共␣ suppressed兲. Hence, the first term on the right side of Eq.
共14兲 becomes
8
k0
冕 dt cos ␻ t 冕⫺k ␲ d ␪
兺␣ m ␣␣ ␴lim
→⬁ ␴ 0
q
␴
1
⫻cos共 ␪ ⫹ ␻ t 兲
冉
⫹k 0 ␲
冕 再 冉
⫹
cos ␪ A␶ ⫽cos ¯␪ ␶A ⫺
0
dJ F ␣ v p ⫹
冊
k0 A
J 共 ␪ ,J,t 兲
m␣ 0
冎
冊
8
冕 dt cos ␻ t 冕⫺k ␲ d¯␪ 冕
兺␣ m ␣ ␴lim
→⬁ ␴ 0
q
k0
再
1
␣
␴
⫹k 0 ␲
⫹
J ␶A ⫽J̄ ␶A ⫹
冉
⫹
k 0m ␣ d v
k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A␶ 兴 2
q ␣a
1
k 20
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ ␶A 兴
cos2 ¯␪ A0
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兴
⫹
sin关 ¯␪ ⫹2 ␻ t 兴
q ␣a
k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
⫻ F ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲 ⫹ ⑀ h ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲 cos ¯␪ A0 ⫺
共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲
1
共D5兲
cos关 ¯␪ A␶ ⫹2 ␻ ␶ 兴 ⫹O 共 a 2 兲 .
共D6兲
Then Eq. 共D3兲 becomes 关with error O(a 2 )]
dJ̄ cos ¯␪ ⫹ ␻ t⫹
0
q ␣ ⑀ a dh ␣
q ␣a
⫻sin ¯␪ ␶A sin关 ¯␪ A␶ ⫹2 ␻ ␶ 兴 ⫹O 共 a 2 兲 ,
k0 A
J 共 ␪ ,J,t 兲 cos ␪ A0 共 ␪ ,J,t 兲 , 共D3兲
m␣ 0
where the shift ⫺ ␻ t in the ␪ integration limits has been
eliminated by periodicity.
The second coordinate change is 关15兴
⫹ ⑀ h ␣ v p⫹
q ␣a
1
cos关 ␪ ⫹2 ␻ t 兴 ⫹O 共 a 2 兲 .
k 0 关共 k 20 /m ␣ 兲 J⫹2 ␻ 兴
共D4兲
q ␣⑀ a
k 0m ␣
h ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲
q ␣ a dF ␣
k 0m ␣ d v
冊
共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲
sin2 ¯␪ A0
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兴 2
cos ¯␪ A0
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兴
冎
.
共D7兲
Here, (¯␪ A0 ,J̄ A0 ) 共with subscripts corresponding to ␶ ⫽0) have
arguments (¯␪ ,J̄,t); we have also ignored the effect of Eq.
共D4兲 on the limits of the integration in space 共since there also
is periodicity in ¯␪ ).
In (¯␪ ,J̄) the Hamiltonian is 关15兴
k 20
q a
¯J 2 ⫹ ␣ cos ¯␪ ⫽E,
H̄ 共 J̄,¯␪ 兲 ⬅
2m ␣
k0
only the case q ␣ ⬍0, and transform ¯␪ to ¯␪ ⫹ ␲ in the q ␣ ⬎0
terms in Eq. 共D7兲. For q ␣ ⬍0 the trajectories
关 ¯␪ A0 (¯␪ ,J̄,t),J̄ A0 (¯␪ ,J̄,t) 兴 that correspond to the Hamiltonian in
Eq. 共D8兲 can be obtained explicitly 关33兴 as
冋冉冏冊 册
冋 冉冏冊 册
¯␪ A0 共 ¯␪ ,J̄,t 兲 ⫽2 am F
共D8兲
which corresponds to particle motion in a single sinusoidal
wave with amplitude A⫽q ␣ a. When q ␣ ⬍0 this is the
Hamiltonian for a nonlinear pendulum. When q ␣ ⬎0 it is the
Hamiltonian for an upside-down pendulum, corresponding to
the fact that the positive particles oscillate in the downward
trough of the wave. Clearly, the Hamiltonian with q ␣ ⬎0 is
transformed into the Hamiltonian with q ␣ ⬍0 simply by
shifting the spatial variable ¯␪ by ␲. Hence, we shall study
J̄ A0 共 ¯␪ ,J̄,t 兲 ⫽
¯␪
t
m ⫺
,m ,
2
␬␶ b
共D9兲
¯␪
t
2m ␣ 1
dn
⫺F
m ,m , 共D10兲
␬␶ b
2
k 20 ␶ b ␬
where F(y 兩 m) is the incomplete elliptic integral of the first
kind and
␶ b ⫽ 冑m ␣ / 兩 q ␣ 兩 k 0 a,
026406-18
m⫽ ␬ 2 ⫽
a
共 k 30 /4兩 q ␣ 兩 m ␣ 兲 J̄ 2 ⫹a
.
sin2 ¯␪ /2
共D11兲
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
By definition, ␬ takes the same sign as J̄. The inverse trajectories depend on the particle species via ␶ b , ␬, and m; however,
we have suppressed the ␣ indices. Using ¯␪ →¯␪ ⫹ ␲ when q ␣ ⬎0 in Eq. 共D7兲 yields
8
k0
兺␣
s ␣q ␣
再
冕
␴
1
lim
m␣
␴ →⬁
␴
0
dt cos ␻ t
冕
⫹k 0 ␲
⫺k 0 ␲
d¯␪
冕
⫹
再
dJ̄ cos ¯␪ ⫹ ␻ t⫹
k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
⫻ F ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲 ⫹ ⑀ s ␣ h ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲 cos ¯␪ A0 ⫺
⫹
q ␣ ⑀ a dh ␣
k 0m ␣ d v
共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲
cos2 ¯␪ A0
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兴
⫹
sin关 ¯␪ ⫹2 ␻ t 兴
s ␣q ␣a
q ␣⑀ a
k 0m ␣
s ␣ q ␣ a dF ␣
h ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲
共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兲
k 0m ␣ d v
冎
sin2 ¯␪ A0
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兴 2
cos ¯␪ A0
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A0 兴
冎
,
共D12兲
where s ␣ ⬅⫺q ␣ / 兩 q ␣ 兩 mark the terms whose sign changes for
q ␣ ⬎0 and the shifted quantities are still denoted by ¯␪ ,¯␪ A0 .
When we substitute Eqs. 共D9兲 and 共D10兲 into Eq. 共D12兲 we
obtain a completely explicit expression.
Even though it appears quite complicated, Eq. 共D12兲 can
be simplified by exploiting the time-asymptotic properties of
the inverse phase flow 关 ¯␪ 0 (¯␪ ,J̄,t),J̄ 0 (¯␪ ,J̄,t) 兴 . Since the potential well in Eq. 共D8兲 is not harmonic, particles with different energies oscillate at different frequencies and ‘‘mix
up’’ the initial distribution. At long times, as filamentation
grows, we can expect the ¯␪ -J̄ integration in Eq. 共D12兲 to
average away all the high-frequency terms and leave only
some coarse-grained component. In practice, after perform-
ing a straightforward expansion of the trigonometric terms
appearing in the first line, Eq. 共D12兲 can be written as a
linear combination of terms of the form
lim
␴ →⬁
⫻
⫹
k 0m ␣ d v
共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兲
cos2 ¯␪
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
Then the following general result applies.
Proposition 1. Given an autonomous, periodic, and integrable one-degree-of-freedom system with ‘‘inverse characteristics’’ 关 ¯␪ ␶ (¯␪ ,J̄,t),J̄ ␶ (¯␪ ,J̄,t) 兴 , and two functions K and G0
as in Eq. 共D13兲,
lim
t→⬁
冕
⫹k 0 ␲
⫺k 0 ␲
⫽
冕
d¯␪
⫹k 0 ␲
⫺k 0 ␲
冕
⍀̄
d¯␪
⫹
q ␣⑀ a
k 0m ␣
⍀̄
dJ̄ K共 ¯␪ ,J̄ 兲 Ḡ0 共 ¯␪ ,J̄ 兲 ,
␴
0
⫹
dt w 共 t 兲
冕
⫹k 0 ␲
⫺k 0 ␲
d¯␪
dJ̄ K共 ¯␪ ,J̄ 兲 G0 关 ¯␪ 0 共 ¯␪ ,J̄,t 兲 ,J̄ 0 共 ¯␪ ,J̄,t 兲兴 ,
h ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兲
s ␣ q ␣ a dF ␣
k 0m ␣ d v
sin2 ¯␪
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兲
cos ¯␪
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
.
共D14兲
where ⍀̄ is an interval in R and Ḡ0 (¯␪ ,J̄) is the average in
phase space of the function G0 (¯␪ ,J̄) along the curves of constant energy of the system.
This is shown easily 关33兴 by transforming to action-angle
variables and invoking the Riemann-Lebesgue lemma. Equation 共D15兲 and Fréchet’s lemma 关28兴 imply that Eq. 共D13兲
can be rewritten as
dJ̄ K共 ¯␪ ,J̄ 兲 G0 关 ¯␪ 0 共 ¯␪ ,J̄,t 兲 ,J̄ 0 共 ¯␪ ,J̄,t 兲兴
冕
冕
冕
共D13兲
where w and K are continuous, bounded, and periodic in t
and ¯␪ , respectively, and
G0 共 ¯␪ ,J̄ 兲 ⬅F ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兲 ⫹ ⑀ s ␣ h ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兲 cos ¯␪ ⫺
q ␣ ⑀ a dh ␣
1
␴
共D15兲
026406-19
冕
⫹k 0 ␲
⫺k 0 ␲
d¯␪
冕
⫹
dJ̄ K共 ¯␪ ,J̄ 兲 Ḡ0 共 ¯␪ ,J̄ 兲 lim
␴ →⬁
1
␴
冕
␴
0
dt w 共 ␻ t 兲 .
共D16兲
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
Then, tedious manipulations reduce Eq. 共D12兲 to
4
k0
兺␣
s ␣q ␣
再
m␣
冕
⫹k 0 ␲
⫺k 0 ␲
⫻ cos ¯␪ ⫺
d¯␪
冕
For m⬎1 we define ␨ such that sin ¯␨⫽␬ sin ␰; then the
action-angle variables are 关33兴 ˜␪ ⫽ 关 ␲ /2K(1/m) 兴 F( ␨ 兩 1/m),
J̃⫽(8/␲ )(m ␣ / ␶ b k 2 ) 关 E(1/m)⫺(1⫺1/m)K(1/m) 兴 . Hence,
⫹
dJ̄
s ␣q ␣a
1
2k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
冎
冋冉冊 册
Ḡ0 共 ¯␪ ,J̄ 兲 .
共D17兲
The terms in Eq. 共D17兲 that depend on the asymptotic
field amplitude a only through the averaging process are
4
k0
兺␣
s ␣q ␣
m␣
冕
⫹k 0 ␲
⫺k 0 ␲
d¯␪
冕
⫹
dJ̄ cos ¯␪ H̄0 共 ¯␪ ,J̄ 兲 ,
where H̄0 (¯␪ ,J̄) is obtained by averaging
冉
H0 共 ¯␪ ,J̄ 兲 ⬅F ␣ v p ⫹
冊
冉
册
冋
and
冊
冋
册
˜␪ K 共 m 兲
,m
␲
共D20兲
Then
H̄0 共 ¯␪ ,J̄ 兲 ⫽
⫽
冋
4
k0
1
H̄0 共 ¯␪ ,J̄ 兲 ⫽
2␲
兺␣
冕 ␪再 冉
2␲
0
d˜ F ␣
⫹ ⑀ s ␣h ␣
冉
2 1
dn关 ˜␪ K 共 m 兲 / ␲ ,m 兴
v p⫹
k 0 ␶ b␬
2 1
dn关 ˜␪ K 共 m 兲 / ␲ ,m 兴
v p⫹
k 0 ␶ b␬
冎
⫻ 兵 1⫺2 sn 关 ˜␪ K 共 m 兲 / ␲ ,m 兴 其 .
2
冊
1 2˜␪ 1
2m ␣ 1
cn K
,
.
2
m ␲ m
k 0 ␶ b␬
冋冉冊 册
冕
2␲
0
再 冉
共D24兲
冋 冉 冊 册冊
冉
冋 冉 冊 册冊
冉 冋 冉 冊 册冊冎
1
2␲
d˜␪ F ␣ v p ⫹
1 2˜␪ 1
2 1
cn K
,
k 0 ␶ b␬
m ␲ m
2 1
1 2˜␪ 1
cn K
,
k 0 ␶ b␬
m ␲ m
1 2˜␪ 1
2 2
,
2 sn K
␬
m ␲ m
s ␣q ␣
m␣
冕
冕
⫹k 0 ␲
⫺k 0 ␲
1
2K 共 m 兲
冉
d¯␪
2K 共 m 兲
0
⫹ ⑀ s ␣h ␣ v p⫹
共D21兲
where m, ␬, and ␶ b were defined in Eq. 共D11兲 and ␰˙
⫽(1/␶ b ␬ ) 冑1⫺ ␬ 2 sin2 ␰ follows from the conservation of energy. Then,
⫽
共D25兲
.
Rescaling the integration variables in Eqs. 共D22兲 and 共D25兲
and substituting into Eq. 共D18兲 yields
⫻
册
2m ␣ 1
cos ␨
k 20 ␶ b ␬
⫻ 1⫺
2m ␣ ˙ 2m ␣ 1
冑1⫺ ␬ 2 sin2 ␰
␰⫽ 2
k 20
k 0 ␶ b␬
˜␪ K 共 m 兲
2m ␣ 1
dn
,m
2
␲
k 0 ␶ b␬
⫽
⫹ ⑀ s ␣h ␣ v p⫹
and
J̄⫽
2m ␣ 1
冑1⫺ ␬ 2 sin2 ␰
k 20 ␶ b ␬
J̄⫽
k0
k
¯J ⫹ ⑀ s ␣ h ␣ v p ⫹ 0 ¯J cos ¯␪
m␣
m␣
共D19兲
cos ¯␪ ⫽cos 2 ␰ ⫽1⫺2 sn2
冋冉冊 册
2
2
1 2˜␪ 1
2
2
,
2 sin ␨ ⫽1⫺ 2 sn K
␬
␬
m ␲ m
共D23兲
cos ¯␪ ⫽cos 2 ␰ ⫽1⫺
共D18兲
on the pendulum energy levels. These expressions correspond to particle dynamics in a single sinusoidal wave, while
the other terms in Eq. 共D17兲, temporarily set aside, represent
the effects of the second wave, with phase velocity ⫺ ␻ /k 0 .
To compute the phase-plane averages in Eq. 共D18兲, we transform from (¯␪ ,J̄) to the action-angle variables (˜␪ ,J̃) for the
nonlinear pendulum, where the averaging can be performed
more easily. For m⬍1 共untrapped particles兲, we define ␰
⬅¯␪ /2; then ˜␪ ⫽ 关 ␲ /K(m) 兴 F( ␰ 兩 m) and J̃⫽(4/␲ )(m ␣ /k 2 )
⫻(1/␶ b ␬ )E(m), where K and E are the complete elliptic
integrals of the first and second kinds, respectively. Hence,
˜␪ K 共 m 兲
␰ ⫽am
,m ,
␲
1 2˜␪ 1
,
,
m ␲ m
␨ ⫽am K
冕
⫹
m 共¯J ,¯␪ 兲 ⬍1
再 冉
dJ̄ cos ¯␪
dz F ␣ v p ⫹
2 1
dn关 z,m 兴
k 0 ␶ b␬
冊
冊
2 1
dn关 z,m 兴 共 1⫺2 sn2 关 z,m 兴 兲
k 0 ␶ b␬
冎
共D26兲
for m⬍1, and for m⬎1
冊
4
k0
兺␣
⫻
s ␣q ␣
m␣
冕
⫹k 0 ␲
⫺k 0 ␲
1
4K 共 1/m 兲
冉
冕
共D22兲
026406-20
冕
⫹
m 共¯J ,¯␪ 兲 ⬎1
4K 共 1/m 兲
0
⫹ ⑀ s ␣h ␣ v p⫹
d¯␪
dJ̄ cos ¯␪
再 冉
冋 册 冊冉
dz F ␣ v p ⫹
2 1
1
cm z,
k 0 ␶ b␬
m
冋 册冊
冋 册冊冎
2 1
1
cn z,
k 0 ␶ b␬
m
1⫺
2 2
1
2 sn z,
␬
m
共D27兲
.
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
To expand these quantities in powers of a, which appears
in ␶ b , ␬, and m 关Eq. 共D11兲兴, we shall break the velocity
integrals into regions that correspond to different types of
wave-particle interactions, distinguishing between resonant
and nonresonant particles based on the size of the coefficient
( ␶ b ␬ ) ⫺1 in the arguments of F ␣ and h ␣ . Here 1/␶ b is of
order 冑a, whereas 1/兩␬兩 can vary from 0 at the bottom of the
wave’s potential well to 1 on the separatrix to ⬁ far from the
wave in the phase plane. For 1/兩 ␬ 兩 ⱗ1, we perform ‘‘resonant’’ expansions of F ␣ and h ␣ around v p 共resonant particles兲. But when 1/兩␬兩 becomes larger, the coefficient
( ␶ b ␬ ) ⫺1 is no longer small and we must find a way to expand the functions around the free-streaming particle velocity instead of v p . This will be the case of nonresonant particles.
4
k0
s ␣q ␣
m␣
兺␣
⬁
⫻
兺
j⫽0
⫹⑀s␣
冕
⫺k 0 ␲
再
冋 冉 冊冉
⫹k 0 ␲
0
⫹⑀s␣
1
2
1
k 20
␬ 2 ␶ 3b
1
2
1⫹ ⫺1
3
m
2
1
k 20
␬ 2 ␶ 3b
I1
dJ̄ cos ¯␪
冕
2K 共 m 兲
dz
0
2 1 j 1 d jF␣
共 v p 兲 dn j 关 z,m 兴
k 0 ␶ b ␬ j! d v j
2 1 j 1 d jh␣
共 v 兲 dn j 关 z,m 兴
k 0 ␶ b ␬ j! d v j p
⫻ 共 1⫺2 sn2 关 z,m 兴 兲 ,
共D28兲
冉
冊
2m ␣
k 20 ␶ b
cos
冊冉
¯␪
¯␪
2m ␣
艛 ⫹ 2 cos ,⫹a 1/4 .
2
2
k 0␶ b
All terms with j odd are odd functions of J̄, since they are
odd in ␬, and ␬ has the same sign as J̄. Hence, since I 1 is
symmetric, these terms vanish. For n even both the A n (m)
and the B n (m) can be calculated via standard recursive formulas 关33兴. In Eq. 共D28兲 the small parameter a appears not
only in 1/␶ b but also in ␬ and m 共in the product ␶ 2b J̄ 2 ). To
deal with this we rescale J̄ by 2m ␣ /k 20 ␶ b 共and still use J̄, ␬,
and m for the transformed quantities兲 and arrive 关33兴 at the
leading-order terms in Eq. 共D28兲 as
冋 冉
E共 m 兲 d 2F ␣
⑀s␣
2 E共 m 兲
1⫹
⫺1
共 v p兲⫹
K共 m 兲 dv2
␶b
m K共 m 兲
E共 m 兲
⫺1
K共 m 兲
1
2K 共 m 兲
where
The resonant region will be further divided into two subregions, corresponding to the untrapped (m⬍1) and trapped
(m⬎1) resonant particles.
a1. Resonant untrapped particles (m⬍1). We shall consider as resonant the phase region such that 1⬎ 兩 ␬ 兩 ⬎a 1/4,
i.e., 1/兩 ␶ b ␬ 兩 ⬍a 1/4. Then Taylor expanding F ␣ and h ␣ about
v p in Eq. 共D26兲 gives
兺 s ␣ q ␣ 冕⫺k ␲ d¯␪ 冕˜I dJ̄ cos ¯␪
冕
冎
a. Resonant particles
k 30 ␣
d¯␪
再冋 册
冋 册
I 1 ⬅ ⫺a 1/4,⫺
8
⫹k 0 ␲
冊册
冊册
h ␣共 v p 兲
冎
d 2h ␣
共vp兲 ,
dv2
共D29兲
where E(m) is the complete elliptic integral of the second kind, now m⬅ ␬ 2 ⫽ 关 J̄ 2 ⫹sin2(¯␪/2) 兴 ⫺1 and Ĩ 1 ⬅„⫺a ⫺1/4,
⫺cos(¯␪/2)…艛„⫹cos(¯␪/2),⫹a ⫺1/4…. Now, ␬ and m do not depend on a, and the integrand in Eq. 共D26兲 has been written as a
1/2
series expansion in powers of ␶ ⫺1
b , i.e., powers of a .
The only inconvenience is the powers a ⫺1/4 that appear in Ĩ 1 . Let us add and subtract two terms and add a term that
integrates to zero in ¯␪ to rewrite Eq. 共D29兲 as
8
k 30
兺␣
⫹
s ␣q ␣
冕
⫹k 0 ␲
⫺k 0 ␲
d¯␪
冋 冉
冕
˜I
1
dJ̄ cos ¯␪
2 E共 m 兲
⑀s␣
1⫹
⫺1
␶b
m K共 m 兲
冊册
再 冋 冉
1 1
k 20 ␶ 3b
1⫹
h ␣共 v p 兲 ⫹ ⑀ s ␣
冊册
冋 冋 冉 冊冉
2 E共 m 兲
⫺1
m K共 m 兲
1 1
k 20 ␶ 3b
冉 冊
冊册 册
d 2F ␣
d 2F ␣
1 1 2
⫺1
⫹
兲
共
v
共vp兲
p
dv2
dv2
k 20 ␶ 3b m
2
2
1⫹ ⫺1
3m
m
E共 m 兲
⫺1
K共 m 兲
⫺
冎
1 d 2h ␣
共vp兲 .
4 dv2
共D30兲
All the functions in the square brackets are integrable as 兩 J̄ 兩 →⬁, i.e., 兩 m 兩 →0, since m⬃J̄ ⫺2 关see 关34兴, 17.3.11 and 17.3.12兴;
in the last term integrability was obtained by subtracting the spatially uniform term ⑀ 关 1/(4k 20 ␶ 3b ) 兴 (d 2 h ␣ /d v 2 )( v p ), which
vanishes under the ¯␪ integration兴. These terms will combine with corresponding terms from the nonresonant particle region,
allowing us to extend the domain of integration for these functions from Ĩ 1 to Ī 1 ⬅„⫺⬁,⫺cos(¯␪/2)…艛„⫹cos(¯␪/2),⫹⬁…. For
026406-21
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
the remaining term in Eq. 共D30兲, if we rescale J̄ back to its original definition and substitute the original expression for m from
Eq. 共D11兲, the part depending on J̄ cancels under the ¯␪ integration and 共noting that s ␣ q ␣ 兩 q ␣ 兩 ⫽⫺q ␣2 ) we have
⫺a
␣
冕 d¯␪ 冕I dJ̄ cos ¯␪
2兺
k ␣ m 2 ⫺k ␲
4
q2
0
␣
⫽a
⫽a
4␲
k0
0
1
兺␣ m 2 冕I dJ̄ d v 2
q ␣2
d 2F ␣
␣
4␲
k 20
⫹k 0 ␲
q ␣2
兺␣ m
冉
冊
¯␪
d 2F ␣
2 sin2 ⫺1
共vp兲
2
dv2
共 v p 兲 ⫽a
1
P
冕
1
dJ̄
I1
␣
J̄
冉
F ␣⬘ v p ⫹
4␲
k 20
k0
m␣
q ␣2
兺␣ m
␣
P
冕
兺␣
s ␣q ␣
m␣
⫹k 0 ␲
⫺k 0 ␲
d˜␪
冕
dJ̄
I2
再冋 册
冋 册
冉
冋 册冊冎
⬁
⫻
冕
cos ¯␪
4K 共 1/m 兲
冕
兺
⫻ 1⫺
2
1
2
2 sn z,
␬
m
8
兺 s ␣ q ␣ 冕⫺k ␲
k 30 ␣
⫻
冋
d¯␪
0
冕
˜I
2
册
dJ̄ cos ¯␪
再
1 1
k 20
␶ 3b
m␣
2 dv3
共vp兲
冋 册冎
k0
m␣
2
¯J
冎
冎
1 d 2h ␣
共vp兲 ,
4 dv2
共D33兲
where we have introduced the same constant factor in the last
term that was added to the corresponding term in Eq. 共D30兲
to make it integrable in J̄. This spatially uniform correction,
of course, vanishes under the ¯␪ integration. In Eq. 共D34兲 we
can write
冋
2 1 E 共 1/m 兲 m⫺1
⫺
m
k 20 ␶ 3b K 共 1/m 兲
1 1
k 20
␶ 3b
冋
2
册
冉 冊
册
E 共 1/m 兲
1 1 2
⫺1 ,
⫺1 ⫹ 2 3
K 共 1/m 兲
k0 ␶b m
where the first term connects continuously with the corresponding quantity in Eq. 共D30兲, and the second generates the
trapped particle contribution to the Vlasov dispersion integral 关as in Eq. 共D31兲兴.
b. Nonresonant particles
We also must consider the case 兩 ␬ 兩 ⬍a 1/4, i.e., 1/兩 ␶ b ␬ 兩
⬎a 1/4. Since 1/兩 ␶ b ␬ 兩 can become arbitrarily large as the distance from the wave in phase space increases, Taylor expansions around v p are not appropriate. Instead, since m⫽ ␬ 2
⬍ 冑a, we expand the elliptic function dn, which enables us
to expand F ␣ and h ␣ around the free-streaming velocity v p
⫹(k 0 /m ␣ )J̄ 关33兴 to obtain
a
␶ 3b
2 1
2
⫹ ⫺1
3 m
m
共vp兲
1 d 3F ␣
共D34兲
册
冊册
K 共 1/m 兲 m⫺1 d 2 F ␣
⑀ s ␣ E 共 1/m 兲
⫺
2
⫺1
2 共 v p兲⫹
K 共 1/m 兲
m
dv
␶b
K 共 1/m 兲
⫻h ␣ 共 v p 兲 ⫹ ⑀ s ␣
⫺
2 1
冋
再 冋 冉 冊冉
dv2
¯J ⫹
共D31兲
0
共D32兲
k 20
共 v p兲⫹
⫽
where I 2 ⬅„⫺(2m ␣ /k 20 ␶ b )cos(¯␪/2),⫹(2m ␣ /k 20 ␶ b )cos(¯␪/2)….
As in case a1, all the terms with odd j vanish, and the others
can be calculated following the same steps. Rescaling J̄ by
2m ␣ /k 20 ␶ b , redefining m as in case a1, and introducing the
modified domain Ĩ 2 ⬅„⫺cos(¯␪/2),⫹cos(¯␪/2)… yields at leading order
⫹k 0 ␲
dv
dz
冋 册
冋 册
,
再
4K 共 1/m 兲
2 1 j 1 d jh␣
1
共 v 兲 cn j z,
k 0 ␶ b ␬ j! d v j p
m
⫹⑀s␣
J̄
k0
冊
1
2 1 j 1 d jF␣
j
j 共 v p 兲 cn z,
k 0 ␶ b ␬ j! d v
m
j⫽0
dJ̄
d 2F ␣
¯J ⫹O 共 a 7/4兲 ,
where the first and third terms in the braces 共each of which
integrates to zero兲 have been added to form the truncated
Taylor series of F ␣⬘ ( v p ⫹(k 0 /m ␣ )J̄). The last line yields, at
leading order, the Vlasov dispersion integral 共in the wave
frame兲 restricted to the resonant region. As such, it will combine naturally with corresponding integrals from the other
regions in the phase plane.
a2. Trapped particles (m⬎1). All the trapped particles
are resonant and for them 兩 1/␶ b ␬ 兩 Ⰶ1 always. Thus, we can
expand F ␣ and h ␣ in Eq. 共D27兲 about v p , just as in case a1.
The analog of Eq. 共D28兲 then is
4
k0
I1
1 dF ␣
E 共 1/m 兲
⫺1
K 共 1/m 兲
4␲
q ␣2
0
␣
兺
k2 ␣ m
冕
⫹
I3
dJ̄
1 dF ␣
J̄ d v
冉
v p⫹
k0
m␣
¯J
冊
共D35兲
as the leading-order approximation to Eq. 共D26兲. Equation
共D35兲 is the contribution from the nonresonant particles to
the Vlasov dispersion integral.
Combining Eqs. 共D30兲, 共D34兲, and 共D35兲 yields the
leading-order terms in the expansion of the single-mode
O’Neil terms:
026406-22
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
⫺a ␴ 1
3/2
兺␣
冋 册
兩 q ␣兩 5
k 50 m ␣3
⫹ ⑀ a 1/2␴ 1
⫹a
4␲
k 20
1/2
兺␣ q ␣
q ␣2
兺␣ m
⫹ ⑀ a 3/2␴ 2
␣
dv2
冋 册
兩 q ␣兩
共vp兲
⫹
dJ̄
h ␣共 v p 兲
1 dF ␣
J̄ d v
冋 册
兩 q ␣兩 3
g 1 共 ¯␪ ,J̄ 兲 ⬅
1/2
k 30 m ␣
冕
兺␣ q ␣
d 2F ␣
1/2
k 50 m ␣3
冉
v p⫹
d 2h ␣
dv2
k0
m␣
¯J
冊
g 2 共 ¯␪ ,J̄ 兲 ⬅
共D36兲
共 v p兲,
⫹␲ ¯
¯ ¯
¯
where ␴ n ⬅8 兰 ⫺
␲ d ␪ 兰 RdJ̄ cos ␪gn(␪,J̄) and the integrals in ␪
over 关 ⫺k 0 ␲ ,⫹k 0 ␲ 兴 have been written as k 0 times integrals
over 关⫺␲,⫹␲兴. The functions g 1 and g 2 are
4
k0
s ␣q ␣
兺␣
m␣
⫹
⫺
冕
⫹k 0 ␲
⫺k 0 ␲
q ␣ ⑀ a dh ␣
k 0m ␣ d v
s ␣q ␣a
冉
d¯␪
冕
v p⫹
⫹
再 冋
dJ̄ cos ¯␪
k0
m␣
¯J
冊
k 0m ␣ d v
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
2k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
F ␣ v p⫹
where the terms with overbars must be averaged on the pendulum energy levels. This averaging can be carried out, and
the resulting quantities expanded in powers of a, using the
above techniques, which entails some tedious algebra but no
new ideas. Actually, we need to compute only terms through
order a 3/2, to be consistent with Eq. 共D18兲; we obtain
␴1
a 3/2
2vp
冋 册
兩 q ␣兩 5
兺␣
⫹⑀a
1/2
k 50 m ␣3
␴2
p
⫺a
4␲
k0
dv
兺 q␣
2v2 ␣
3/2
兺␣ m 2 冕
q ␣2
␣
dF ␣
冋 册冋
兩 q ␣兩 3
k 50 m ␣3
⫹
dJ̄
1/2
h ␣ 共 v p 兲 ⫹2 v p
F ␣ 共 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兲
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
2
.
dh ␣
dv
共vp兲
册
k0
共D37兲
E共 m 兲
⫺1
K共 m 兲
m⬎1,
冊册
冊册
2
E 共 1/m 兲
1⫹ 共 2⫺m 兲
⫺1
3m
K 共 1/m 兲
m␣
冉
v p⫹
⫺
冊
k0
m␣
q ␣⑀ a
k 0m ␣
2
1
⫺ ,
4
m⬍1,
1
⫺ ,
4
m⬎1,
冊
冉
冉
¯J
2
cos ¯␪
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
h ␣ v p⫹
¯J ⫹ ⑀ s ␣ h ␣ v p ⫹
k0
m␣
k0
m␣
¯J
冊
冊 册
sin2 ¯␪
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
¯J cos ¯␪ ⫹O 共 a 2 兲 ,
册
共D39兲
variable J̄ back to v , integrating by parts, and using the
symmetry of F ␣ ( v ).
2. Landau terms
⬁
Substituting T(x,t)⫽ 兺 n⫽1
T nk 0 (t)sin nk0x into the second
term on the right side in Eq. 共14兲 共Landau terms兲 and transforming to the wave frame via Eqs. 共D1兲 and 共D2兲 gives
⫹k 0 ␲
q ␣2
8
1 ␴
dt cos ␻ t
d ␪ cos共 ␪ ⫹ ␻ t 兲
2 lim
k 0 ␣ m ␣ ␴ →⬁ ␴ 0
⫺k 0 ␲
⫻
⫻
共D40兲
冕
冕 冕
⫹
t
dJ
再 冉
冉
⫹⑀
Equations 共D36兲 and 共D40兲 provide the leading-order terms
in the expansion of Eq. 共D17兲 in the upper half plane. Since
the initial condition is reflection symmetric, the lower halfplane contribution will be identical. Thus, the O’Neil term
becomes Eq. 共16兲, where the Vlasov dispersion term
K 0 (k 0 , ␻ ) is obtained by transforming the integration
m⬍1,
E 共 1/m 兲
2
⫺1,
K 共 1/m 兲
冋 冉 冊冉
冉
冋
兺
共vp兲
冊
共D38兲
⫺1
¯
and m⬅ ␬ ⫽ 关 J̄ ⫹sin (␪/2) 兴 . Numerical integration
yields ␴ 1 ⫽8⫻2.58⫽20.67 and ␴ 2 ⫽8⫻0.066⫽0.53.
We still must consider the terms in Eq. 共D17兲 that correspond to the interaction between the wave under consideration and the ‘‘other’’ wave:
s ␣ q ␣ a dF ␣
冋冉
冉
2 E共 m 兲
⫺1 ,
m K共 m 兲
2
2
1⫹ ⫺1
3m
m
2
cos2 ¯␪
1
再
再
1⫹
0
d␶
冕
兺n T nk 共 ␶ 兲 sin n 关 ␪ ␶A共 ␪ ,J,t 兲 ⫹ ␻ ␶ 兴
0
dF ␣
k0
v p ⫹ J A␶ 共 ␪ ,J,t 兲
dv
m
冊
冊
冎
dh ␣
k0 A
J 共 ␪ ,J,t 兲 cos关 ␪ A␶ 共 ␪ ,J,t 兲 ⫹ ␻ ␶ 兴 ,
v p⫹
dv
m␣ ␶
共D41兲
where ⫺ ␻ t in the ␪ integration limits has been eliminated by
periodicity. Then, using Eq. 共D4兲, the inverse relations Eqs.
共D5兲 and 共D6兲, and ¯␪ →¯␪ ⫹ ␲ when q ␣ ⬎0, Eq. 共D41兲 becomes, with order O(a 2 ) error,
026406-23
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
8
k0
冕 dt cos ␻ t 冕⫺k ␲ d¯␪ 冕
兺␣ m 2 ␴lim
→⬁ ␴ 0
q ␣2
冋
␴
1
⫹k 0 ␲
␣
冉
k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
s ␣ q ␣ a sin共 ¯␪ A␶ ⫹2 ␻ ␶ 兲 cos n 共 ¯␪ A␶ ⫹ ␻ ␶ 兲
k 0m ␣
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A␶ 兴 2
冉
冊
冉
sin关 ¯␪ ⫹2 ␻ t 兴
s ␣q ␣a
dJ̄ cos ¯␪ ⫹ ␻ t⫹
0
⫻ sin n 共 ¯␪ ␶A ⫹ ␻ ␶ 兲 ⫹n
册再 冉
冊冕
t
0
d␶
冊
冉
兺n T nk 共 ␶ 兲 s ␣ ,n
0
冉
冊
dF ␣
dh
k0 A
k
¯J ␶ ⫹ ⑀ s ␣ ␣ v p ⫹ 0 ¯J A␶
v p⫹
dv
m␣
dv
m␣
冊
cos共 ¯␪ ␶A ⫹2 ␻ ␶ 兲
⑀ q ␣a d 2h ␣
s ␣q ␣a d 2F ␣
k0 A
k0 A
¯
¯J
⫹
J
⫹
v
p
2
2 v p⫹
␶
A
k 0m ␣ d v
m␣
m␣ ␶
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ ␶ 兴 k 0 m ␣ d v
⫻cos共 ¯␪ A␶ ⫹ ␻ ␶ 兲 ⫹
⫻
⫹
cos共 ¯␪ A␶ ⫹2 ␻ ␶ 兲 cos共 ¯␪ ␶A ⫹ ␻ ␶ 兲
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A␶ 兴
冎
冊
⑀ q ␣ a dh ␣
k 0 A sin共 ¯␪ ␶A ⫹2 ␻ ␶ 兲 sin共 ¯␪ ␶A ⫹ ␻ ␶ 兲
¯J
⫺
,
v p⫹
k 0m ␣ d v
m␣ ␶
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ A␶ 兴 2
共D42兲
where s ␣ ,n is s ␣ for n even and unity for n odd. Since the variables with overbars satisfy Eq. 共D8兲, the 关 ¯␪ A␶ (¯␪ ,J̄,t),J̄ ␶A (¯␪ ,J̄,t) 兴
in Eq. 共D42兲 are easily obtained in terms of elliptic functions.
A straightforward extension 关33兴 of the arguments that led from Eq. 共D12兲 to Eq. 共D17兲 shows that Eq. 共D42兲 can be
rewritten as
4
k0
兺␣ m 2 冕⫺k ␲ d¯␪ 冕
q ␣2
⫹k 0 ␲
␣
⫹
再
dJ̄ cos ¯␪ ⫺
0
s ␣q ␣a
1
2k 0 m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
冎兺
n
Ḡ0,n 共 ¯␪ ,J̄ 兲 ,
共D43兲
where the Ḡ0,n (¯␪ ,J̄) are obtained trivially 共but tediously兲 by applying standard sum formulas to the trigonometric functions in
冋
T nk 0 共 ␶ 兲 s ␣ ,n sin n 共 ¯␪ ⫹ ␻ ␶ 兲 ⫹n
⫹
⫺
s ␣q ␣a d 2F ␣
k 0m ␣ d v
⑀ q ␣ a dh ␣
k 0m ␣ d v
2
冉
冉
v p⫹
v p⫹
k0
m␣
k0
m␣
¯J
s ␣ q ␣ a sin共 ¯␪ ⫹2 ␻ ␶ 兲 cos n 共 ¯␪ ⫹ ␻ ␶ 兲
¯J
冊
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
k 0m ␣
冊
cos共 ¯␪ ⫹2 ␻ ␶ 兲
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
⫹
册再 冉
dF ␣
dv
2
冉
⑀ q ␣a d 2h ␣
k 0m ␣ d v
sin共 ¯␪ ⫹2 ␻ ␶ 兲 sin共 ¯␪ ⫹ ␻ ␶ 兲
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
2
冎
v p⫹
v p⫹
k0
m␣
¯J
冊
k0
m␣
冊
¯J ⫹ ⑀ s ␣
dh ␣
dv
冉
v p⫹
k0
m␣
冊
¯J cos共 ¯␪ ⫹ ␻ ␶ 兲
cos共 ¯␪ ⫹2 ␻ ␶ 兲 cos共 ¯␪ ⫹ ␻ ␶ 兲
关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴
共D44兲
and then 共a兲 integrating the factors depending on ␶ from zero to infinity, and 共b兲 averaging the terms depending on (¯␪ ,J̄) on
the energy levels of the pendulum.
In Eq. 共D43兲 the single-mode terms, i.e., the terms that do not have an explicit dependence on a coming from multiple-mode
effects, are given by
4
k0
兺␣ m ␣2 冕⫺k ␲ d¯␪ 冕
q ␣2
⫹k 0 ␲
⫹
where the H̄0,n (¯␪ ,J̄) are obtained from the terms
T nk 0 共 ␶ 兲 s ␣ ,n sin n 共 ¯␪ ⫹ ␻ ␶ 兲
dJ̄ cos ¯␪
0
再 冉
冊
兺n H̄0,n共¯␪ ,J̄ 兲 ,
冉
共D45兲
冊
k0
k
dF ␣
dh
¯J ⫹ ⑀ s ␣ ␣ v p ⫹ 0 ¯J cos共 ¯␪ ⫹ ␻ ␶ 兲
v p⫹
dv
m␣
dv
m␣
冎
共D46兲
via the procedure described above. Standard trigonometric formulas 共and s ␣ ,n⫹1 ⫽s ␣ s ␣ ,n ) lead to
H0,n 共 ¯␪ ,J̄ 兲 ⬅s ␣ ,n 关 C n,n sin n¯␪ ⫹S n,n cos n ␪ 兴
冉
冊
dF ␣
⑀
k0
¯J ⫹s ␣ ,n⫹1 关 C n,n⫺1 sin共 n⫺1 兲¯␪ ⫹S n,n⫺1 cos共 n⫺1 兲¯␪
v p⫹
dv
m␣
2
⫹C n,n⫹1 sin共 n⫹1 兲¯␪ ⫹S n,n⫹1 cos共 n⫹1 兲¯␪ 兴
冉
冊
dh ␣
k0
¯J ,
v p⫹
dv
m␣
026406-24
共D47兲
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
where C n, j ⬅ 兰 ⬁0 d ␶ T nk 0 ( ␶ )cos j␻␶ and S n, j ⬅ 兰 ⬁0 d ␶ T nk 0 ( ␶ )sin j␻␶. For reference in Appendix E, these coefficients can be
expressed in terms of the Laplace transform T̃ n (p) of T n (t) as
1
C n, j ⫽ 关 T̃ nk 0 共 ⫺i j ␻ 兲 ⫹T̃ nk 0 共 i j ␻ 兲兴 ,
2
S n, j ⫽
1
关 T̃ 共 ⫺i j ␻ 兲 ⫺T̃ nk 0 共 i j ␻ 兲兴 .
2i nk 0
共D48兲
Now, each H0,n must be averaged on the energy levels of the nonlinear pendulum, via the techniques above. First, the
trigonometric functions of ¯␪ are expressed in terms of sin ␰ and cos ␰, where ␰ ⬅¯␪ /2: sin n¯␪⫽PSn(sin ␰,cos ␰) and cos n¯␪
⫽PCn (sin ␰,cos ␰), where P Sn and P Cn are (n⫹1)th degree polynomials. Then Eq. 共D45兲 becomes
4
k0
兺␣ m ␣2 冕⫺k ␲ d¯␪ 冕m共¯J ,¯␪ 兲⬍1 dJ̄ cos ¯␪ 2K 共 m 兲 冕0
q ␣2
⫻
⫹k 0 ␲
⫹
0
冉
2K 共 m 兲
1
冊
dz
兺n s ␣ ,n 关 C n,n P Sn共 sn,cn兲 ⫹S n,n P Cn 共 sn,cn兲兴
⑀
dF ␣
2 1
S
C
S
dn ⫹s ␣ ,n⫹1 兵 C n,n⫺1 P n⫺1
v p⫹
共 sn,cn兲 ⫹S n,n⫺1 P n⫺1
共 sn,cn兲 ⫹C n,n⫹1 P n⫹1
共 sn,cn兲
dv
k 0 ␶ b␬
2
C
⫹S n,n⫹1 P n⫹1
共 sn,cn兲 其
冉
2 1
dh ␣
dn
v p⫹
dv
k 0 ␶ b␬
冊
共D49兲
for m⬍1, where the elliptic functions sn, cn, and dn are understood to take the arguments 关 z,m 兴 . Similarly, we find
4
k0
兺␣ m ␣2 冕⫺k ␲ d¯␪ 冕m共¯J ,¯␪ 兲⬎1 dJ̄ cos ¯␪ 4K 共 1/m 兲 冕0
q ␣2
⫻
⫹k 0 ␲
⫹
0
冉
4K 共 1/m 兲
1
冊
dz
兺n s ␣ ,n 关 C n,n P Sn共 ␬ ⫺1 sn,dn兲 ⫹S n,n P Cn 共 ␬ ⫺1 sn,dn兲兴
⑀
dF ␣
2 1
S
C
S
cn ⫹s ␣ ⫹1,n 兵 C n,n⫺1 P n⫺1
v p⫹
共 ␬ ⫺1 sn,dn兲 ⫹S n,n⫺1 P n⫺1
共 ␬ ⫺1 sn,dn兲 ⫹C n,n⫹1 P n⫹1
共 ␬ ⫺1 sn,dn兲
dv
k 0 ␶ b␬
2
C
⫹S n,n⫹1 P n⫹1
共 ␬ ⫺1 sn,dn兲 其
冉
dh ␣
2 1
cn
v p⫹
dv
k 0 ␶ b␬
冊
共D50兲
for m⬎1, where now the elliptic functions take the arguments 关 z,1/m 兴 .
These expressions have the same general structure as Eqs. 共D26兲 and 共D27兲 and can be similarly expanded. In the resonant
region we simply expand dF ␣ /d v and dh ␣ /d v in Taylor series about v p , eliminate the odd terms in J̄, and rescale J̄ to obtain
8
k 30
q ␣2
兺␣ m ␣
冕
⫹k 0 ␲
⫺k 0 ␲
d¯␪
冕
册
兺 再冋
⬁
˜I 艛˜I
1
2
¯
dJ̄ ␶ ⫺1
b cos ␪
2 1
k 0 ␶ b␬
j⫽0
2j
冋
à 2 j 共 m 兲 d 2 j⫹1 F ␣
⑀ 2 1
2 j⫹1 共 v p 兲 ⫹
2 k 0 ␶ b␬
共 2 j 兲! dv
册
2j
冎
B̃ 2 j 共 m 兲 d 2 j⫹1 h ␣
共vp兲 ,
共 2 j 兲 ! d v 2 j⫹1
共D51兲
where Ĩ 1 , Ĩ 2 , and m are defined above, and
à l 共 m 兲 ⫽
B̃ l 共 m 兲 ⫽
冦
1
2K 共 m 兲
冕
2K 共 m 兲
0
冦
1
2K 共 m 兲
1
4K 共 1/m 兲
dz dnl 关 z,m 兴
冕
冕
2K 共 m 兲
0
4K 共 1/m 兲
0
dz dnl 关 z,m 兴
兺n s ␣ ,n S n,n P Cn 共 sn关 z,m 兴 ,cn关 z,m 兴 兲 ,
冋 册兺
1
dz cn z,
m
l
冕
冋 册兺 再
冉 冋 册 冋 册冊冎
4K 共 1/m 兲
0
dz cnl z,
C
⫹S n,n⫹1 P n⫹1
␬ ⫺1 sn z,
冉 冋 册 冋 册冊
1
1
1
sn z, ,dn z,
␬
m
m
共D52兲
,
m⬎1,
C
共 sn关 z,m 兴 ,cn关 z,m 兴 兲
兺n s ␣ ,n⫹1 兵 S n,n⫺1 P n⫺1
C
⫹S n,n⫹1 P n⫹1
共 sn关 z,m 兴 ,cn关 z,m 兴 兲 其 , m⬍1,
1
4K 共 1/m 兲
n
s ␣ ,n S n,n P Cn
m⬍1,
1
m
n
冉 冋 册 冋 册冊
C
s ␣ ,n⫹1 S n,n⫺1 P n⫺1
␬ ⫺1 sn z,
1
1
,dn z,
m
m
, m⬎1.
026406-25
1
1
,dn z,
m
m
共D53兲
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
Here we have used the fact that the z integrals containing
P Sn are zero, as follows from the symmetries of the elliptic
functions 共see 关34兴, Fig. 16.1兲. Indeed, P Sn and P Cn are, respectively, odd and even functions of each of their arguments, as can be seen by using the standard trigonometric
multiple-angle formulas and mathematical induction. In principle, the z integrals can be computed analytically by the
methods introduced above; in practice, as n grows this becomes very burdensome. Fortunately, in many concrete
cases, e.g., those discussed in Sec. IV, only very few spatial
Fourier modes T n are non-negligible. As above 关see the comments that follow Eq. 共D30兲兴, the J̄ integration in Eq. 共D51兲
is extended to R by combining the appropriate resonant and
nonresonant quantities, after eliminating from the resonant
terms certain spatially uniform quantities that are not integrable at infinity in J̄ but vanish under the ¯␪ integration.
In the nonresonant region Eq. 共D49兲 is expanded about
the free-streaming particle trajectories, exactly as for the corresponding O’Neil terms. However, here the Landau terms
give no contribution at leading order 关33兴. Hence, the
leading-order single-mode Landau terms from Eq. 共D51兲 are
a 1/2
兺␣
冋 册冋
兺冋 册 冋
册
兩 q ␣兩 5
k 30 m ␣3
⫹a 3/2
1/2
␳ 1共 T 兲
兩 q ␣兩 7
␣
1/2
k 50 m ␣5
dF ␣
⑀
dh ␣
共 v p 兲 ⫹ ␳ 2共 T 兲
共vp兲
dv
2
dv
册
M̃ i 共 m 兲 ⬅
␳ i 共 T 兲 ⬅8
冦
冕
1
4K 共 1/m 兲
2K 共 m 兲
4K 共 1/m 兲
dz
0
⫺␲
d¯␪
冕
dJ̄ cos ¯␪ R i 共 m 兲
⫺a 3/2
兺␣
⫹
冋 册冋
兩 q ␣兩 7
1/2
k 50 m ␣5
共D55兲
␭ 1 共 T 兲 dF ␣
␭ 2共 T 兲 d 2F ␣
共 v p兲⫹
共vp兲
2
4vp dv2
8vp dv
册
共D56兲
dJ̄ cos ¯␪ M̃ i 共 m 兲 ,
共D57兲
⑀ ␭ 3 共 T 兲 dh ␣
⑀ ␭ 4共 T 兲 d 2h ␣
⫹
共v 兲 ,
共vp兲
8vp dv2 p
16v 2p d v
␭ i 共 T 兲 ⬅8
冕
⫹␲
⫺␲
d¯␪
冕
R
兺n M i,n共 sn关 z,m 兴 ,cn关 z,m 兴 兲 ,
m⬍1,
兺n M i,n共 ␬
m⬎1,
dz
0
冕
⫹␲
and the R i (m), i⫽1,...,4, are, respectively, Ã 0 (m), B̃ 0 (m),
à 2 (m)/m, and B̃ 2 (m)/m. The functional dependence of ␳ i
on the field T has been indicated explicitly, and T⬅0 implies
C n, j ⫽S n, j ⫽Ã l ⫽B̃ l ⫽0, so that ␳ i (0)⫽0, i⫽1,...,4.
The same methods can be used to compute the multiplemode effects in Eq. 共D43兲. The result to leading order is 关33兴
共D54兲
1
2K 共 m 兲
冕
where
d 3F ␣
2 ␳ 3共 T 兲
共vp兲
dv3
d 3h ␣
⫹ ⑀ ␳ 4共 T 兲
共v 兲 ,
dv3 p
where
⫺1
sn关 z,m 兴 ,dn关 z,m 兴 兲 ,
共D58兲
⫺ 共 n⫹1 兲共 C n,n⫺1 P Sn ⫹S n,n⫺1 P Cn 兲兴 ,
and
共D59c兲
S
C
S
⫹S n,n⫹2 P n⫹1
⫺C n,n⫺2 P n⫺1
M 1,n ⬅s ␣ ,n 关 C n,n⫹2 P n⫹1
C
⫺S n,n⫺2 P n⫺1
兴,
共D59a兲
S
C
S
⫹S n,n⫺3 P n⫺2
⫹C n,n⫹3 P n⫹2
M 4,n ⬅s ␣ ,n⫹1 关 C n,n⫺3 P n⫺2
C
⫹S n,n⫹3 P n⫹2
⫹ 共 C n,n⫹1 ⫹C n,n⫺1 兲 P Sn
S
C
S
⫹S n,n⫹2 P n⫹1
⫹C n,n⫺2 P n⫺1
M 2,n ⬅s ␣ ,n 关 C n,n⫹2 P n⫹1
C
⫹S n,n⫺2 P n⫺1
兴,
S
C
⫹ 共 n⫹1 兲 S n,n⫹3 P n⫹2
M 3,n ⬅s ␣ ,n⫹1 关共 n⫹1 兲 C n,n⫹3 P n⫹2
S
C
⫺ 共 n⫺1 兲 C n,n⫺3 P n⫺2
⫺ 共 n⫺1 兲 S n,n⫺3 P n⫺2
⫹ 共 n⫺1 兲共 C n,n⫹1 P Sn ⫹S n,n⫹1 P Cn 兲
⫹ 共 S n,n⫹1 ⫹S n,n⫺1 兲 P Cn 兴 .
共D59b兲
共D59d兲
Equations 共D54兲 and 共D56兲 can be summed and extended 共by
symmetry兲 to the other half phase plane, yielding the totalLandau contribution to the nonlinear Poisson equation given
in Eq. 共21兲.
026406-26
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
APPENDIX E: TRANSIENT FIELD EXPANSIONS
⫻
Here we present the more tedious calculations needed for
the transient analysis in Sec. IV. Substituting Eq. 共7兲 into Eq.
共5b兲 and Fourier and Laplace transforming gives
T̃ k 共 p 兲 ⫽
4
k
兺␣ q ␣ 冕0 dt e ⫺pt共 I⫺ P a 兲 冕⫺ ␲ dx cos kx
⫻
R
4
⫺
k
R
d
0
T
⳵ F␣
⳵v
冎
,
兺␣ m ␣ 冕0 dt e ⫺pt共 I⫺ P a 兲 冕⫺ ␲ dx cos kx
q ␣2
⫹␲
⬁
4
k
兺␣ m ␣␣ 冕0 dt e ⫺pt共 I⫺ P a 兲 冕⫺k ␲ d¯␪ 冕
⫻
⫹
dJ̄cos
f
再冋
F ␣ 共 w A0 兲 ⫹ ⑀ h ␣ 共 w A0 兲 cos
冉
sin关 ¯␪ ⫹2 ␻ t 兴
k
q a
¯␪ ⫹ ␻ t⫹ ␣
kf
k f m ␣ 关 2 v p ⫹ 共 k 0 /m ␣ 兲 J̄ 兴 2
关v p ⫹w A0 兴
冉
冊册
⫺
冕
t
0
d␶
兺n T nk 共 ␶ 兲
0
q ␣ a sin共 ¯␪ A␶ ⫹2 ␻ ␶ 兲 cos关共 nk 0 /k f 兲共 ¯␪ ␶A ⫹ ␻ ␶ 兲兴
关v p ⫹w A␶ 兴 2
k 2f m ␣
冊
h ␣ 共 w A0 兲 sin ¯␪ A0 sin关共 k 0 /k f 兲¯␪ A0 兴
k 0 A q ␣ a F ␣⬘ 共 w A0 兲 cos ¯␪ A0
¯␪ 0 ⫹
⫺⑀
A
kf
k fm␣
关v p ⫹w 0 兴
关v p ⫹w A0 兴 2
h ␣⬘ 共 w A0 兲 cos ¯␪ A0 cos关共 k 0 /k f 兲¯␪ A0 兴
⫹nk 0
⫹
⫹k f ␲
⬁
q
⫹⑀
冊冋
冉
sin
nk 0 A
共 ¯␪ ␶ ⫹ ␻ ␶ 兲
kf
dF ␣ A
dh ␣ A
k0
共 w␶ 兲⫹⑀
共 w ␶ 兲 cos 共 ¯␪ ␶A ⫹ ␻ ␶ 兲
dv
dv
kf
冉
h ␣⬘ 共 w ␶A 兲 sin共 ¯␪ ␶A ⫹2 ␻ ␶ 兲 sin关共 k 0 /k f 兲共 ¯␪ ␶A ⫹ ␻ ␶ 兲兴
q ␣ a F ␣⬙ 共 ␻ A␶ 兲 cos共 ¯␪ A␶ ⫹2 ␻ ␶ 兲
⫺⑀
A
k fm␣
关v p ⫹w ␶ 兴
关v p ⫹w ␶A 兴 2
⫹⑀
h ␣⬙ 共 w A␶ 兲 cos共 ¯␪ A␶ ⫹2 ␻ ␶ 兲 cos关共 k 0 /k f 兲共 ¯␪ ␶A ⫹ ␻ ␶ 兲兴
关v p ⫹w ␶A 兴
冊 册冎
⫹ 共 corresponding terms from the other half phase plane兲 ,
where w A␶ (¯␪ ,J̄,t)⬅ v p ⫹(k f /m ␣ )J̄ ␶A (¯␪ ,J̄,t) and
冋
¯␪ A␶ 共 ¯␪ ,J̄,t 兲 ⫽⫺2 am
J̄ ␶A 共 ¯␪ ,J̄,t 兲 ⫽
共E1兲
关 x ␶A 共 x, v ,t 兲 , v ␶A 共 x, v ,t 兲兴
where k⫽lk 0 , l⫽1,2,..., F␣ (x, v )⫽F ␣ ( v )⫹ ⑀ h ␣ ( v )cos k0x,
and A is given by Eq. 共30兲. It is assumed that T k and the right
side of Eq. 共5b兲 are integrable. To obtain a more explicit
expression, we apply the procedure developed for the timeasymptotic equation to the two integral terms on the right
side. We carry out the same sequence of transformations in
the integration variables that led to Eqs. 共D17兲 and 共D43兲,
with k 0 in Eq. 共D1兲 replaced by k f . The result is
d v F␣ „x A0 共 x, v ,t 兲 , v A0 共 x, v ,t 兲 …
T̃ k 共 p 兲 ⫽
t
dv
⫹␲
⬁
冕
冕 冕 ␶再
冉冏冊 册
冋 冉冏冊 册
¯␪
t⫺ ␶
⫺F
m ,m ,
␬␶ b
2
¯␪
2m ␣ 1
t⫺ ␶
dn
⫺F
m ,m .
2
␬␶ b
2
k f ␶ b␬
共E3兲
asymptotic expansions of the various terms in Eq. 共E2兲 in
powers of ⌬⑀. We now substitute the perturbative expansion
for T, Eq. 共31兲, into Eq. 共E2兲 and solve the resulting hierarchy equations for the T (k) (x,t). We illustrate the simplest
situation, when the time-asymptotic field is zero.
共E4兲
The arguments of the trigonometric functions in Eq. 共E2兲
must be corrected to include a shift of ␲ in ¯␪ and ¯␪ A␶ when
q ␣ ⬎0. These corrections will be introduced in the
共E2兲
1. Transient expansion along aÄ0
Along the basic branch a⫽0, the equation for T̃ (0) (p) can
be obtained by setting a⫽0 in Eq. 共E2兲 and noting from Eq.
共E3兲 that ¯␪ 0␶ (¯␪ ,J̄,t)⫽¯␪ ⫺(k 2f /m ␣ )J̄(t⫺ ␶ ) and J̄ ␶0 (¯␪ ,J̄,t)⫽J̄,
where the superscripts correspond to A⬅0. We find
026406-27
PHYSICAL REVIEW E 68, 026406 共2003兲
C. LANCELLOTTI AND J. J. DORNING
T̃ 共k0 兲 共 p 兲 ⫽
4
k
兺␣
4
⫺
k
冕
q␣
m␣
⬁
dt e
⫺pt
⫺k f ␲
0
兺␣ m ␣2 冕0 dt e
q ␣2
冕
⫹k f ␲
⬁
再 冉
⫺pt
冕
d¯␪
⫹k f ␲
⫺k f ␲
冊
冕
⫹
d¯␪
冋 冉
冊
冉
冊 冉
k 2f
k
kf
kf
k0
¯
¯
¯
¯
dJ̃ cos 共 ␪ ⫹ ␻ t 兲 F ␣ v p ⫹
J ⫹ ⑀ 0h ␣ v p⫹
J cos
␪⫺
Jt
kf
m␣
m␣
kf
m␣
冕
⫹
k
dJ̄ cos 共 ¯␪ ⫹ ␻ t 兲
kf
冊 冋
冉
冕
t
0
d␶
兺n
T 共nk0 兲 共 ␶ 兲 sin
0
k 2f
dF ␣
dh ␣
kf
kf
k0
¯
¯
¯
⫻
J ⫹⑀0
J cos
␪⫺
J 共 t⫺ ␶ 兲 ⫹ ␻ ␶
v p⫹
v p⫹
dv
m␣
dv
m␣
kf
m␣
册冎
冋
k2
nk 0
¯␪ ⫺ f J 共 t⫺ ␶ 兲 ⫹ ␻ ␶
kf
m␣
⫹ 共 corresponding terms from the other half phase plane兲 .
册
冊册
共E5兲
The projector I⫺ P a in Eq. 共E2兲 becomes unnecessary in Eq. 共E5兲 because all the time-asymptotic parts vanish for a⫽0. Also,
no modifications are necessary in Eq. 共E5兲 for q ␣ ⬎0, since the straight line trajectories ¯␪ 0␶ and J̄ 0␶ do not depend on the sign
of q ␣ . Carrying out the ¯␪ integrations yields
T̃ 共k0 兲 共 p 兲 ⫽ ␦ k,k 0 ⑀ 0
⫻
4␲k f
k
兺␣
冕 冕 ␶再
⫹
t
dJ̄
d
0
q␣
m␣
冕
⬁
dt e ⫺pt
0
T 共k0 兲 共 ␶ 兲
冕
冉
⫹
dJ̄ cos k v p t⫹
冉
冊 冉
冊
冉
kf
k
4␲k f
¯J t h ␣ v p ⫹ f ¯J ⫺
m␣
m␣
k
冊
兺␣ m ␣2 冕0 dt e ⫺pt
q ␣2
dF ␣
⑀
kf
kf
0兲
0兲
¯J ⫹ 0 关 T 共k⫹k
¯J
v p⫹
共 ␶ 兲 ⫹T 共k⫺k
共 ␶ 兲兴 h ␣ v p ⫹
0
0
dv
m␣
2
m␣
冊冎 冋
sin k
⬁
kf
J 共 t⫺ ␶ 兲 ⫹ v p 共 t⫺ ␶ 兲
m␣
⫹ 共 corresponding terms from the other half phase plane兲 .
册
共E6兲
Transforming back to v ⫽ v p ⫹(k f /m ␣ )J̄, calculating the Laplace transform, and carrying out an identical calculation in the
other half phase plane gives
T̃ 共k0 兲 共 p 兲 ⫽ ␦ k,k 0 ⑀ 0
⫺
4␲
k
兺␣
q␣
冕
R
dv
ph ␣ 共 v 兲
4␲ 共0兲
T̃ 共 p 兲
2
2 2⫺
p ⫹k v
k k
q ␣2
⑀0 4␲ 共0兲
0兲
␶
兲兴 兺
关 T k⫹k 0 共 ␶ 兲 ⫹T 共k⫺k
共
0
2 k
␣ m␣
Since F ␣ and h ␣ are even, this becomes
冋
T̃ 共k0 兲 共 p 兲 1⫹
兺␣ m ␣␣ 冕Rd v k v␣⫺ip
q2
4␲
k
⫽⫺ ␦ k,k 0 ⑀ 0
4␲i
k
F⬘ 共v兲
册
⫹
冕
R
dv
兺␣ m ␣ 冕Rd v p 2 ⫹k␣ 2 v 2
kvF⬘ 共v兲
q ␣2
k v h ␣⬘ 共 v 兲
p 2 ⫹k 2 v 2
共E7兲
.
q ␣2 共 0 兲
⑀0 4␲
0兲
⫻兺
关 T̃ k⫹k 0 共 p 兲 ⫹T̃ 共k⫺k
共 p 兲兴
0
2 k
␣ m␣
冕
R
dv
h ␣⬘ 共 v 兲
k v ⫺ip
兺␣ q ␣ 冕Rd v k v␣⫺ip
h 共v兲
共E8兲
and Eq. 共33兲 follows.
2. The threshold equation
Next we derive the threshold equation, Eq. 共39兲, for a
small critical initial amplitude. If T (0) , Eq. 共38兲, is substituted into Eq. 共27兲, this threshold equation becomes explicit.
(0)
⫽O( ⑀ n0 ), and we shall need to keep only
From Eq. 共38兲, T nk
S 共2,j0,2兲 ⫽
⫽Im
0
(0,2)
T k(0,1) and T 2k
, which generate, respectively, the coeffi0
0
(0,2)
cients S (0,1)
1,j and S 2,j according to Eq. 共D48兲 共with errors of
order ⑀ 30 ). By symmetry,
S 共1,j0,1兲 ⫽
冋
册 冋
册
⫹
⫹
N k⫹ 共 ⫺i j ␻ 兲
1 N k 0 共 ⫺i j ␻ 兲 N k 0 共 i j ␻ 兲
0
⫺
⫽Im ⫹
,
2i D k⫹ 共 ⫺i j ␻ 兲 D k⫹ 共 i j ␻ 兲
D k 共 ⫺i j ␻ 兲
0
0
0
共E9兲
冋
冋
⫹
⫹
⫹
⫹
1 N k 0 共 ⫺i j ␻ 兲 C 2k 0 共 ⫺i j ␻ 兲 N k 0 共 i j ␻ 兲 C 2k 0 共 i j ␻ 兲
⫺
⫹
⫹
2i D k⫹ 共 ⫺i j ␻ 兲 D 2k
共 ⫺i j ␻ 兲 D ⫹
k 0 共 i j ␻ 兲 D 2k 0 共 i j ␻ 兲
0
0
⫹
N k⫹ 共 ⫺i j ␻ 兲 C 2k
共 ⫺i j ␻ 兲
0
0
⫹
D k⫹ 共 ⫺i j ␻ 兲 D 2k
共 ⫺i j ␻ 兲
0
0
册
,
册
共E10兲
where N k (p), D k (p), and C k (p) are defined in Eqs. 共34兲,
共35兲, and 共36兲, and the superscript ‘‘⫹’’ indicates analytic
continuation. These expressions, in conjunction with
(0,1)
(0,2)
⫺4 ⑀ 20 s ␣ ␴ 2 S 2,2
Eqs. 共D55兲, yield ␳ 1 (T 0 )⫽ ⑀ 0 ␴ 1 S 1,1
3
(0,1)
2
⫹O( ⑀ 0 ) and ␳ 2 (T 0 )⫽⫺4 ⑀ 0 s ␣ ␴ 2 S 1,2 ⫹O( ⑀ 0 ). Thus, to
leading order,
026406-28
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
0,1兲
⌫ 共 ⑀ 0 ,T 0 兲 ⫽ ⑀ 0 ␴ 1 S 共1,1
⫹
冋 册
兺 冋 册
兺␣
␴ 2 共 0,1兲
S
2 1,2
␣
兩 q ␣兩 5
1/2
k 30 m ␣3
s␣
再
dF ␣
0,2兲
共 v p 兲 ⫺4 ⑀ 20 ␴ 2 S 共2,2
dv
兩 q ␣兩 5
1/2
k 30 m ␣3
冎
兺␣
s␣
冋 册
兩 q ␣兩 5
1/2
k 30 m ␣3
dF ␣
共vp兲
dv
dh ␣
共 v p 兲 ⫹O 共 ⑀ 30 兲
dv
共E11兲
and Eq. 共27兲 becomes Eq. 共39兲.
3. Time-asymptotic field amplitude near the threshold
Finally, we present the details that lead from the general formula for the time-asymptotic amplitude, Eq. 共32兲, to the value
of a near the threshold, Eq. 共56兲. From Eq. 共32兲, to first order in both a and ⌬⑀, a⫽ ␮ ⌬ ⑀ where
␮ ⫽⫺
⫹
k 0 兺 ␣ s ␣ 关 兩 q ␣ 兩 3 /m ␣ 兴 1/2h ␣ 共 v p 兲
兺 ␣ 关 兩 q ␣ 兩 5 /m ␣3 兴 1/2F ␣⬙ 共 v p 兲
⫺
兺 ␣ 关 兩 q ␣5 兩 /k 30 m ␣3 兴 1/2关 21 ⑀ 共01 兲 ␳ 2 共 T 共 0,1兲 ⫹T 共 1,0兲 兲 h ␣⬘ 共 v p 兲 ⫹ ␳ 1 共 T 共 1,0兲 兲 F ␣⬘ 共 v p 兲兴
␴ 1 兺 ␣ 关 兩 q ␣ 兩 5 /k 50 m ␣3 兴 1/2F ␣⬙ 共 v p 兲
␹ 2 ␹ 3 ⑀ 共01 兲 ⫹ ␹ 2 兺 ␣ 关 兩 q ␣ 兩 5 /k 50 m ␣3 兴 1/2兵 共 ␴ 1 /2v p 兲 F ␣⬘ 共 v p 兲 ⫺ ⑀ 共01 兲 共 兩 q ␣ 兩 /m ␣ 兲关 2 ␳ 3 共 T 共 0,1兲 兲 F ␣⵮ 共 v p 兲 ⫺ 共 1/4v p 兲 ␭ 2 共 T 共 0,1兲 兲 F ␣⬙ 共 v p 兲兴 其
关 ␴ 1 兺 ␣ 关 兩 q ␣5 兩 /k 50 m ␣3 兴 1/2F ␣⬙ 共 v p 兲兴 2
.
共E12兲
From the definitions of ␳ i and ␭ i , Eqs. 共D55兲 and 共D57兲,
simple integrations lead to
1,0兲
␳ 1 共 T 共 1,0兲 兲 ⫽ ␴ 1 S 共1,1
,
1,0兲
0,1兲
␳ 2 共 T 共 1,0兲 ⫹T 共 0,1兲 兲 ⫽⫺4 ␴ 2 s ␣ 共 S 共1,2
⫹S 共1,2
兲,
1
0,1兲
␳ 3 共 T 共 0,1兲 兲 ⫽ ␴ 2 S 共1,1
,
2
0,1兲
␭ 2 共 T 共 0,1兲 兲 ⫽ ␴ 1 S 共1,3
, 共E13兲
where ␴ i , i⫽1,2, are given below Eq. 共D38兲, S (1,0)
1,j are zero
for j⫽1 since the only nonzero Fourier component in T (1,0)
(1,0)
corresponds to k⫽k 0 , S 1,1
is obtained by inserting T k(1,0)
0
into Eq. 共D48兲, and the S (0,1)
1,j , j⫽1,2,..., were already obtained in Eq. 共E9兲. Here, however, N k⫹ /D k⫹ in Eq. 共E9兲 is
0
0
⫹
replaced by the modified function 关 N ⫹
k (p)/D k (p) 兴
0
0
⫺Ã k(0) (p). This has a significant effect, because N k⫹ /D k⫹ has
0
0
1,0兲
S 共1,1
⫽
(0,1)
a singularity at p⫽⫾ik 0 v p , as ⑀ (1)
0 →0, which causes S 1,1
to be of order 1/⑀ (1)
0 . This can be seen from Eq. 共E9兲 for j
⫽1, since at ␻ ⫽⫾k 0 v p the denominator reduces to a term
proportional to 兺 ␣ (q ␣2 /m ␣ )F ␣⬘ ( v p )⫽O( ⑀ (1)
0 ). Thus, the second term in the numerator of Eq. 共40兲 is non-negligible at
leading order, whereas the first term in the denominator is
negligible at leading order due to Eq. 共54兲 关because there is
(0,2)
no singularity at p⫽⫾2ik 0 v p and S 2,2
⫽O(1)]. However,
(1,0)
the coefficient S 1,1 in Eqs. 共E13兲 and therefore 共E12兲 is
O(1) 关and the term that contains it is O( ⑀ (1)
0 ) due to Eq.
共54兲兴, because the singularities in 关 N k⫹ ( p)/D k⫹ (p) 兴
0
0
⫺Ã k(0) (p) at p⫽⫾ik 0 v p are removable, due to Eq. 共50兲.
0
⫹
Indeed, Taylor expanding N ⫹
k and D k in this modified func0
0
tion about ⫾ik 0 v p shows that the singular terms cancel, at
leading order in ⑀ (1)
0 . Taking the limit ␦ z→0 in that expansion and exploiting the symmetries yields
0
冋
册
i 兺 ␣ q ␣ 关 P兰 Rd v h ␣⬘ 共 v 兲 / 共 v ⫺ v p 兲 ⫹i ␲ h ␣⬘ 共 v p 兲兴
a0
⫺Im
,
2k 0 v p
兺 ␣ 共 q ␣2 /m ␣ 兲关 P兰 Rd v F ␣⬙ 共 v 兲 / 共 v ⫺ v p 兲 ⫹i ␲ F ␣⬙ 共 v p 兲兴
where a 0 was given in Eq. 共49兲. According to Eq. 共E13兲, Eq.
共E12兲 then becomes Eq. 共56兲.
Equation 共56兲 is accurate to order ⑀ (1)
0 ⌬ ⑀ , even though we
did not calculate the terms of order O( ⑀ (1)
0 ⌬ ⑀ ) in the solution for the transient field, Eq. 共55兲. This follows from Eq.
共54兲; in principle, the solution for a, Eq. 共32兲, contains a term
3 3 1/2
5
␳ 1 (T (1,1) )F ␣⬘ ( v p ), but
of the form ⑀ (1)
0 ⌬ ⑀ 兺 ␣ 关 兩 q ␣ 兩 /k 0 m ␣ 兴
共E14兲
2
this term is ‘‘pushed’’ to order O( ⑀ (1)
0 ⌬ ⑀ ) by Eq. 共54兲. In
cases in which only Eq. 共53兲 is satisfied 共and
兺 ␣ 关 兩 q ␣5 兩 /k 30 m ␣3 兴 1/2 F ␣⬘ ( v p )⫽0), one must add to Eq. 共56兲 the
contribution due to T (1,1) , by carrying out the perturbation
analysis of the transient equation through first order in both
⌬⑀ and ⑀ (1)
0 . This calculation, which is quite tedious, will be
omitted here.
026406-29
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C. LANCELLOTTI AND J. J. DORNING
F ␣⬘ 共 v 兲 F ␣⬘ 共 v 兲
k 0m ␣
2v
⬘ 共 v 兲⫽
h ␣共 v 兲 ⫽ 2
⫹
. 共F3兲
2 F␣
q␣
v⫹vp v⫺vp
v ⫺vp
APPENDIX F: BUCHANAN-DORNING SOLUTIONS
In this appendix we verify that the two-wave BGK-like
solutions discovered by Buchanan and Dorning 关15兴 satisfy
the nonlinear condition Eq. 共50兲, and thus are a special case
of the solutions developed in this paper. At t⫽0 the distribution function corresponding to the approximate invariants
E␣(⫾) 关see Eq. 共15兲兴 is
G␣
冉
m␣
q␣
q␣ v⫿vp
cos k 0 x⫹ ⑀
cos k 0 x
共 v ⫿ v p 兲2⫹ ⑀
2
k0
k0 v⫾vp
⫽G␣
⫽G␣
冉
冉
m␣
q␣ 2v
cos k 0 x
共 v ⫿ v p 兲2⫹ ⑀
2
k0 v⫾vp
冊
冊
冉
冊
⫽P
⫽P
共F1兲
where G␣ must satisfy certain criteria 关15兴 which in fact ensure that Eqs. 共45兲 and 共46兲 are satisfied. Clearly, the initial
condition in Eq. 共F1兲 is of the form F ␣ ( v )⫹ ⑀ h ␣ ( v )cos k0x
共at leading order in ⑀兲 with F ␣ ( v )⫽G␣ 关 (m ␣ /2)( v ⫿ v p ) 2 兴
and
h ␣ ( v )⫽(q ␣ /k 0 ) 关 2 v /( v ⫾ v p ) 兴 G␣⬘ 关 (m ␣ /2)( v ⫿ v p ) 2 兴 .
Differentiating F ␣ yields
冉
冊
关15兴
关16兴
F ␣⬘ 共 v 兲
共v⫿vp兲
F ␣⬙ 共 v 兲
v⫿vp
2
d v ⫹P
冕
F ␣⬘ 共 v 兲
共 v ⫹ v p 兲共 v ⫺ v p 兲
dv
共F4兲
dv,
F ␣⬙ 共 v 兲 ⫽m ␣2 共 v ⫿ v p 兲 2 G␣⬙
⫹m ␣ G␣⬘
冉
冉
m␣
共 v⫿v p兲2
2
冊
冊
m␣
共 v⫿v p兲2 ,
2
共F5兲
which at v ⫽⫾ v p gives
F ␣⬙ 共 ⫾ v p 兲 ⫽m ␣ G␣⬘ 共 0 兲 ⫽
k 0m ␣
h 共 ⫾v p兲.
q␣ ␣
共F6兲
共F2兲
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关10兴
关11兴
关12兴
关13兴
关14兴
h ␣共 v 兲
dv
v⫿vp
From Eqs. 共F4兲 and 共F6兲 it follows that Eq. 共50兲 is satisfied.
Then, substituting Eq. 共F6兲 into Eq. 共49兲 gives a 0 ⫽1, i.e.,
a⫽ ⑀ , which is what we should expect since we know that
these undamped ‘‘BGK-like’’ waves travel without changing
amplitude.
k 0 m ␣ 共 v ⫾ v p 兲共 v ⫿ v p 兲
⫽
h ␣共 v 兲 ,
q␣
2v
关1兴
关2兴
关3兴
关4兴
关5兴
关6兴
关7兴
关8兴
关9兴
冕
冕
冕
where the first term was integrated by parts, the second is
zero because the integrand is odd, and
m␣
共 v ⫿ v p 兲 2 cos k 0 x⫹O 共 ⑀ 2 兲 ,
2
m␣
F ␣⬘ 共 v 兲 ⫽m ␣ 共 v ⫿ v p 兲 G␣⬘
共 v⫿v p兲2
2
k 0m ␣
P
q␣
冊
m␣
q␣ 2v
共 v⫿v p兲2 ⫹⑀
2
k0 v⫾vp
⫻G␣⬘
Dividing by v ⫿ v p and taking the principal value integral
gives
026406-30
PHYSICAL REVIEW E 68, 026406 共2003兲
TIME-ASYMPTOTIC WAVE PROPAGATION IN . . .
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