Download Nonlinear wave mechanics and particulate self-focusing

Foundations of Physics, Vol. 10, Nos. 7/8, 1980
Nonlinear Wave Mechanics and Particulate Self-Focusing
Dan Censor 1,2
Received April 2, 1979
A previous model for treating electromagnetic nonlinear wave systems is
examined in the context of wave mechanics. It is shown that nonlinear wave
mechanics implies harmonic generation of new quasipartiele wave functions,
which are absent in linear systems. The phenomenon is interpreted in terms of
pail" (and higher order ensembles) coherence of the interacting particles. The
implications are jar-reaching, and the present approach might contribute
toward a common basis for diverse physical phenomena involving nonlinearity.
An intimate relationship connecting coherence, nonlocal interaction, and
nonlinearity has been previously noticed in the physics of superconductivity. It
is shown here that all these ingredients are consistently contained in the present
formalism. The present theory may contribute to elucidate a controversial
theory proposed by Panarella, who claims to have measured high-energy
photons due to high-intensity laser radiation, which cannot be predicted on the
basis of linear quantum theory. Panarella explains the new phenomena by
stipulating a nonlinear intensity-dependent photon energy. It is argued here
that nonlinearity, manifested in the presence of high intensity, may give rise to
high- and low-energy photons, the so-called "effective" and "'t&ed" photons,
respectively. However, the present explanation does not involve ad hoc assumptions regarding the foundations of quantum theory. In analogy with the electrodynamic model, the present theory leads to particulate self-focusing in highdensity streams of particles. Since such particulate beams are currently under
consideration in connection with fusion reactions, this might be of future
interest.
1. I N T R O D U C T O R Y REMARKS
M u t u a l influence between field theories in various branches of physics is
very c o m m o n . This is due to the fact that investigators are using similar
1 Department of Electrical Engineering and Computer Science, University of California-San Diego, La Jolla, California.
On leave of absence from the Department of Electrical Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel.
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0015-9018180/0800-0555503.0010 © 1980PlenumPublishingCorporation
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mathematical tools. Often this leads to new insight into the physics of the
pertinent fields of interest. Thus, presently a model for nonlinear electrodynamic wave systems is being investigated in connection with quantum
mechanical systems. Recently a ray formalism was proposed for analyzing
self-focusing of electromagnetic waves in lossless (1,2) and absorbing (3-~
media. The theory is based on an extended Fermat principle, (6) and the
transition to wave mechanics is straightforward. For linear absorbing media,
the wave mechanical analog has been recently examined, (7) showing the
significance of complex potentials and giving a meaning to space- and
time-independent probability density in dissipative systems. The similar
procedure of investigating nonlinear wave mechanical systems, with reference
to the electrodynamic model, pays even higher dividends, as shown below.
We start the next section with a formal representation of the weakly
nonlinear wave mechanical system. A periodic solution is then assumed,
facilitating the algebraization of the transformed equations. This yields a
dispersion equation, similar to linear systems; however, here the amplitudes
of the wave function are also involved in the dispersion equation.
The next section is concerned with the implications of the theory for
various branches of physics. The relevance to the theory of superconductivity
is pointed out. A somewhat tentative argument regarding "effective" and
"tired" photons in high-intensity laser beams is given. This may contribute
to the understanding of experiments cited by Panarella, (8,9) without invoking
his ad hoc modification of quantum theory.
Finally the problem of particulate self-focusing is considered, in analogy
with the electrodynamic case.
2. G E N E R A L F O R M A L I S M
A formalism is presented here for dealing with weakly nonlinear systems
as defined below. The formalism is quite general, and no attempt is made to
discuss special cases here. The method is the analog of the electromagnetic
case discussed previously. (1,2,5)
Consider a general wave mechanical system represented by
LiAbj + aiAbj = 0
(1)
where Lij is a square matrix of operators involving space and time derivatives,
alj is a square matrix whose entries may be space and time dependent, thus
describing potentials, and ~ = (¢1 ,.-., CJ ,---) is a vector of probability wave
functions. The dimensionality of q~ is determined by the quantum mechanical
model at hand, e.g., for the scalar SchrSdinger equation q~ = (¢1), and for
the Dirac wave equation (1°) t~ = (¢1 ,.:., ¢4).
Nonlinear Wave Mechanics and Particulate Self-Focusing
557
In the present discussion au are allowed to depend on de, i.e., nonlinear
wave equations are introduced by allowing self-interaction. The physical
implications of such a construct will be discussed below. It is therefore
appropriate to rewrite (t) in the general form
Lij~j + V~(q~) = 0
(2)
where Vi(q~) indicates that the potentials depend on the array q~ of wave
functions. Part of the motivation for the present study is the fact that wave
equations of the type (2) have found their use in physics, e.g., in the GinzburgLandau theory of superconductivity. (11)
Nonlinear field problems of physics are notoriously complicated, and
there exists no general mathematical tool for dealing with them. The present
approach is no exception, and cannot be expected to do better than to
illuminate a special aspect of the general problem. We are dealing here with
weakly nonlinear systems, for which V~ can be represented as a hierarchy
in powers of q~,
v~ = v~ 1> + ~ ! 2 ) + ... + v~> + ...
(3)
r
(1)
,-
(2).
,
(aij ; @} + ~ai~e, ~bj; 4/~.} ~
... - / { a i j ..... @ ..... (#}
where am,
~ a(~),
ijk etc., are tensors of increasing rank and the braces denote a
functional structure relating the field q~ to the tensors uij ... v. In the scalar
Schr6dinger equation V a) = a(X)~b and V (~) = 0 for n =~ 1, where a(X) is
the potential, depending on spacetime. The latter are compactly denoted by
the relativistic four-vector X = (x, ict). This simple form, on which a major
part of our knowledge of quantum physics is based, seems to be too specialized
for our present purposes. For reasons clarified below, we prefer to define
the first-order potential by means of a four-dimensional convolution integral
p~)
(~)
.
= {~i~ (x), ~j} =
(o~
d --co
(i)
d~xl a~ ( x , ) ~ ( x
- x0
(4)
=
f
~
(1)
d~Xl aij (X -- X 0 4,5(Xl)
The integral (4) is subject to causality, and no contribution is assumed from
(1)(X - X 1 ) = b~j(1) ( X 0 a ( X - - X 0 ,
outside the light cone. If in (4) we assume ai~
then V~a)= al~)(X)~bj(X) will be obtained, signifying a local interaction.
However, many physical applications prescribe nonlocal interaction, and it
seems advantageous to adopt the form (4), which is mandatory for spatially
and temporally dispersive systems. (~2) A fourfold Fourier transform applied
to (4) yields
V?)(K) =
a")(U)
,~-, ~@(K)
(5)
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where for simplicity of notation the functions and their transforms are
denoted by the same symbols. The different functions are distinguished by
means of their argument, which is explicitly displayed where necessary. The
wavenumber vector and frequency are compactly denoted by the four-vector
K = (k, i~o/c). On the other hand, V/m = aij(1)(X)@. in general implies a
fourfold convolution integral in K space. For a general discussion (5) is
preferable, leading to an algebraic dispersion equation.
The definition of the higher order nonlinear terms VI") constitutes a
crucial step in our discussion. We introduce the forms used in nonlinear
optics by A k h a m a n o v and Khokhlov, <1~)which are equivalent to those used
by Schubert and Wilhelmi. <14) As done before, m a four-dimensional spacetime representation is used. Thus we define
=
f°-® daX1 ...
-~ d ' X , aa...v(Xa ..... X,) ~bj(X - - X 0 " " ~bv(X
X,)
(6)
This definition, although it seems to be quite arbitrary, will be shown to
provide a satisfacotry framework for our physical discussion. The reason for
that will be the conservation of m o m e n t u m and energy of quasiparticles
produced by nonlinear interaction. The Fourier transform of (6), with a
factor (27r)-4 for each integration, which is henceforth suppressed, is given by
=
-co d4K1 "" -~o d Kn_~ a,j...v(Kx .... , K,) ~bj(K1) "" ~bv(K~)
(7)
K = K1 + K2 + "'" + K .
Note that in (7) there are n - 1 fourfold integrations and an additional
constraint on K. The latter, standing for k = kl + k2 + "'" + k , and
oJ = o~1 + to z + ... + ~o,, provides for the m o m e n t u m and energy conservation of the quasiparticles created by the nonlinear interaction. To
choose a somewhat oversimplified example, (7) states that a (multiplication)
nonlinear interaction between two wave functions A1 exp(ikl • x - - i~olt ) and
A2 exp(ik2 -- i¢o2t) yields A exp(ik • x
i¢ot), such that k = kl + k2, and
¢o = to1 -t- o)2 • The intensity A 2 of the new quasiparticle depends on how
m a n y particles of species 1 and 2 interact. I f the plausibility of (7) is accepted,
then the nonlocal nature of (6) must also be accepted, since they form a
transform pair.
In order to consider applications of the model given above, we look for
a simple solution of the wave equation. It is clear that harmonic waves
cannot be considered as a solution for a nonlinear wave equation such as (2).
On the other hand, a periodic wave function can be used if it contains all the
harmonics which will be created due to the nonlinearity of the system. This is
Nonlinear Wave Mechanics and Particulate Self-Focusing
559
characteristic of a system of identical particles which are noninteracting in
the absence of nonlinearity, and this type of interaction due to nonlinear
effects is termed self-interaction. Therefore we choose a solution
~ =
~
~bj,~exp(imK" X)
(8)
where ~b~ is the amplitude of the nthe harmonic and K • X = k ' x -- oJt.
It is presently assumed that ~b;,, are constants, due to the homogeneity of
the system in space and time. Later, in order to discuss ray tracing and selffocusing, it will be necessary to replace (8) by an appropriate eikonal approximation. Substitution of (8) into (6) and exchange of order of integration and
summation yields
V
- i (n) =
co
oo
Z
@~exp(ic~K'X)"" Z
~,~exp(ic~K'X)
f
× -~ daX~ ... -~d4Xn
(~) CX1 ,..-, X,~) exp[--iK • (aX1 + ... +/3X~)]
X ai~...k
(9)
The integral in (9) defines a (4n)-fold Fourier transform, yielding a l j . . . ~ ( a K .....
ilK), depending on K, the summation indices ~ ..... fl, and the tensor indices
i, j , . . . , v. Since no new harmonics are produced in this process of selfinteraction, the series with a summation index 9,, such that y = a + ... + / 3 ,
can be defined
V~~) =
~
V~ ") exp(iyK" X)
(10)
St=--~
and compared to (9). By rearranging the series and noting that each term
picked out from (9) involves n wave amplitudes with indices going f r o m j to v,
it is possible to define new terms according to
V~,~)
-(~) ,,.
". : ai¢...vY'sv
"'" ~vv
(11)
where all the symbols in (11) depend on yK. Note that the new 8 are different
from a. At a first glance it might seem paradoxical that we succeeded in
transforming (7) into an algebraic form involving only one value yK.
However, it must be remembered, comparing (9) and (10), that this has been
achieved in the presence of all spacetime harmonics, hence if one term in (10)
is somehow filtered out, this will affect all other terms. This is the essential
distinction between linear and nonlinear systems; in the former all harmonics,
if present, are unrelated. Finally it is noted that the definition (11) is tanta-
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mount to a Taylor expansion of V~,
(")(q~,) near ~ ,
=
72>+...
+
Vi"
e~bj,
6~2 Vi"
@" + 2 0~bj,
0~bk,¢
Ilt~,=0
0, with a vanishing term
>
lit-/=0
@,~k, + ""
(12)
and by comparison of (11) and (12) the ~ tensors are identified.
In view of the above discussion it is now possible to derive an algebraic
dispersion equation for self-interacting, weakly nonlinear systems. One must
simply replace 0/0X operators in Lit, (2), by imK (i.e., O/Ox by imk, ~/Ot by
-- imm), and thus corresponding to (2) we now have
--(1)
--(n)
Gi~ = Li~(imK) ~mj + a~j(mU)
~b~j + ..- 4- a~;...,(mK)
~b,,j ... ~b,~ = 0 (13)
Let us consider now only m = 1, i.e., the fundamental harmonic, which is
the limiting case of the nonlinear case as it reduces to the linear case. All
other m 4 = 1 will be considered as equations on the amplitudes ~ j ..... 4J~.
Following the same argument as for linear systems of homogeneous algebraic
equations, the determinant of (13) must vanish in order to have nontrivial
solutions. Hence, if we rewrite (13) (suppressing m) in the form
=
= 0
(14)
noting that ~b are now the amplitudes ~le, then
F = det(F~e) : 0
(15)
We have therefore derived a dispersion relation F(K, q~) = 0, i.e., an algebraic
equation relating k and ~o, and involving the amplitudes of the fundamental
harmonic wave functions. It is a characteristic of nonlinear systems that
their dispersion equation involves the amplitudes of the fields. This appears
also in other models for discussing nonlinear systems. (la)
3. DISCUSSIONS AND APPLICATIONS
The distortion of harmonic signals in nonlinear systems, conducive to
harmonic generation, is an elementary phenomenon, whether we are dealing
with time signals in lumped electric circuits or with waves in a distributed
system. Still, it captures the imagination whenever it appears in a new and
somewhat unexpected context. This is the case in nonlinear optical systems,
Nonlinear Wave Mechanics and Particulate Self-Focusing
561
where harmonic generation gives rise to new colors in the light beams. The
emergence of probability wave harmonics in the presence of nonlinear selfinteraction of streams of otherwise identical particles is therefore not
surprising, but it requires physical interpretation. Thinking along classical
mechanical lines, it seems incomprehensible that new particles will be created
in a system of this kind. However, in wave mechanical terms it simply means
that pairs and higher order ensembles become correlated, or coherent, acting
as a new quasiparticle whose momentum and energy are n times those of a
single particle. Obviously the overall momentum and energy of the system
are conserved. According to this picture, harmonics in (8) describe the
motion of the "center of gravity" of the new quasiparticles, comprised of
several "real" particles. This bonding effect may involve particles situated
at different locations. At a first glance this is somewhat puzzling, but actually
this is a model accepted in the theory of superconductivity long ago. In fact,
all the ingredients appearing in the present theory have already been included
in the theory of superconductivity, albeit in a fragmentary form. The necessity
for a nonlocal field theory, as displayed by (6), is intimately connected with
the need for a nonlocal electrodynamic model as given by Pippard. (11) The
nonlinear wave mechanical equations of the Ginzburg-Landau theory m)
can be considered as a special case of (2). The celebrated BCS theory Im of
superconductivity is founded on the idea that pairs of electrons in a bound
state are producted due to mutual interaction mediated by phonons. This
leads to explanations of various effects such as Josephson tunneling. The
general ideas conform to the present model in which quasiparticles are
created due to nonlinear self-interaction, although the present model, being
very general, does not specify the nature and origin of this interaction. It is
of course unrealistic to assume that the present formalism, without further
specialization, could provide a general framework for the physics of superconductors. However, it is still interesting to observe how all the seemingly
disconnected aspects are consistently involved in the present theory. It is
hoped that this will contribute to a better formulation and deeper understanding of the subject in the future.
Due to the dual nature of light, the present model and the electrodynamic
analog m become overlapping when photons are considered. The present
discussion, based on a particulate wave mechanical model, may contribute
to our understanding of the redistribution of photon energy and momentum
in high-intensity laser pulses. Panarella ~s,9) discusses this effect and cites
earlier studies. He claims to verify experimentally that high-energy photon
are found in the presence of high-intensity light beams. His theory for
explaining this effect, admittedly an ad hoc one, attributes to photons an
intensity-dependent energy. Thus the fundamental quantum theoretical
relation e = hv (where e is the energy and v is the frequency) is questioned.
825/Io/7/8-4
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His discussion is very comprehensive, frankly pointing out the weak points
of this theory. Although his ad hoc theory can account for his data, the
modification of the fundamentals of quantum theory creates a bigger
difficulty. It is always with great reluctance, and understandably so, that
scientists are willing to modify fundamental physical theories. Witness the
special theory of relativity and the related Michelson-Morely experiment,
initiated by Michelson in 1881 and repeated in many ways by numerous
physicists over a period of about fifty years. The present model of nonlinear
quantum mechanics may contribute to the understanding of effects induced
by high-intensity optical pulses without affecting the foundation of quantum
mechanics. It is suggested that the appearance of "effective" and "tired"
photons, as termed by Panarella, is the result of association and dissociation
of quasiparticles produced in the presence of nonlinear self-interaction. A
detailed mathematical picture for these processes is not available presently,
and we have to take resort to a verbal argument which must be regarded
as a conjecture, at best.
Due to the nonlinearity inherent in the environment in which the light
travels, be it a crystal or a gas, particulate harmonic generation is induced,
manifested by the creation of quasiparticles, as described by the harmonics
in (8). These are created and annihilated in a random fashion and (8) describes
only the average density of the process. It is therefore conceivable that
because of the inhomogeneity of the system, e.g., due to the varying intensity
within the laser light pulse, pairs and higher order ensembles, upon dissociation, do not obtain the original energies of the single photons, although the
total energy is conserved. Although e = hv is preserved for individual
photons and bound aggregates, "effective" and "tired" photons of higher
and lower energy can be created as a result of unequal redistribution of
energy when dissociation occurs. The conjecture presented here can be
tested experimentally. According to Panarella, the effective photons are
created because of photon-photon interactions, while the present theory
presupposes a mediating medium which brings about the nonlinearity. If an
experiment is constructed showing the "effective" photons are produced in
vacuum, this will disqualify the present theory and add credibility to
Panarella's theory of intensity-dependent photon energy. Of course, negative
results will unequivocally invalidate his theory.
4. PARTICULATE SELF-FOCUSING
Since it was first discussed by Askaryan] TM the self-focusing effect has
been the subject of intensive investigation. Recently a ray-theoretic model
for nonlinear media m has been discussed in connection with self-focusing
Nonlinear Wave Mechanics and Particulate Self-Focusing
563
lossless c2~and dissipative (5) media. The present formalism for solving a system
of equations of the type (2) is analogous, hence a similar effect of wave
mechanical, or particulate, self-focusing effect is expected.
In order to take into account spatially and temporally varying systems,
the phase K • X in (8) is replaced by f K • dX, a line integral in four-space,
between the limits X1 and X. The slowly varying amplitudes Cj,,(X) now
replace the constants @,~ in (8), taking into account slow variations in space
and time. Accordingly we have to modify (14) and (15),
Gi(K, ,-k, X) = 0
(16)
F(K, g,, X) = 0
(17)
where X denotes the dependence of fi (13) (the analogs of the constitutive
parameters in electrodynamics) on space and time, and da = ~bj~ are the
amplitudes of the fundamental harmonic. The problem of ray tracing in
nonlinear media has been discussed previously, (1,2,5J including real spacetime
ray tracing in absorbing media. (3,4,7) It has been realized recently (~7) that
F, (17), for the electromagnetic case must be augmented by the self-consistent
requirement that the group velocity v be in the direction of the Poynting
vector. Incorporating into F with a suitable Lagrange multiplier yields a new
F = 0. Similarly, in the wave mechanical analog we require that the group
velocity v be in the direction of the particle probability current (in the
generalized sense, see Messiah, (~.s) p. 888), i.e., g = 9, where ~ and 9 are unit
vectors in the direction of current density and group velocity, respectively.
This again modifies F into F.
By retracing the electromagnetic argument we derive the Hamilton
equations pertinent to the present case of lossless wave mechanical systems:
dx
v--
dt
--
Fk
F~ '
dk
F,
dt
--f~'
do)
dt
Ft
--
(18)
F,~
where
F,-
0F
OF [ 0 G j ] -1 0Gj
09)
for any varible I. For the linear case we replace F by F and ~ F / ~ ¢ i = 0
identically, since the dispersion equation is not dependent on field amplitudes.
These are the familiar Hamilton equations of geometrical optics or geometrical
mechanics. (6) For dissipative systems cv) we add the constraint Im v = 0 at
the departure point of the ray and ensure that the ray stays in real spacetime
by adding i[3 and iv. [~ to the second and third eqs. of (18), respectively,
where
.
[Re(v~
. + v~v)]-I
.
. Jm (v~ L
U~
F, T vt + v~ • v)
v=~
F~
(20)
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Censor
Equations (18)-(20) are supplemented by
r°6 1-!
L-~-~J
dt - -
r°a
I Ok
dk
" d--t- +
dco
oc,
,
Oco dt @ TX-x" v ~-
(21)
describing the evolution of the fields + along the ray.
5. D I S C U S S I O N AND A SIMPLE EXAMPLE
The author is not aware of self-focusing phenomena that have been
discussed in the context ot particulate systems. However, as particle beams,
considered to date for fusion reaction, will achieve larger densities, the
relevance may arise. The simple example considered here shows, as is
expected, aG> that self-focusing depends on the gradients of the intensity in a
beam. Consequently, if for some stochastic reasons ("noise") the intensity
profile becomes a little corrugated, instability wilt set in, leading to filamentation. This might be of interest for high-energy dense particulate beams.
As a simple example, consider the scalar Schr6dinger equation in which
a nonlinear term is included. By assuming an instantaneous local plane wave
(essentially the W K B approximation), the present form of (17) is
(22)
G = F~/J~ = [k 2 -- co + a + b~b~2]~b~= 0
where k and co are proportional to the m o m e n t u m and energy, respectively,
~b1 is the amplitude of the fundamental harmoni c, and a and b are real
constants. Since there is no explicit sPace a n d time dependence in (22),
(18) prescribes that dco/dt = 0 and d k / d t = 0. Consequently (21) vanishes,
prescribing ~bl = const along a ray path. Let us assume a steady state
situation such that in
dco
Oco
Oco
dt - - 8t + ~
dk
"v
O,
Ok
cqk
dt -- 0t + - ~ - x ' v
0
(23)
we have &o/Ot = 0 and Ok/Ot
0. The latter prescribes &o/0x = 0 because
of the uniqueness condition <19> Ok/Ot + &o/0x = 0. This leaves in (23)
(0k/0x) • v = 0, i.e., k can change in s p a c e such that the changes are everywhere perpendicular to the group Velocity, i.e., to the ray path. Since
~bl = const, all that is left of (18) is
v
=
-
Fk/F,o =
2k=
2(,.o
a -- b~,.2)1/~
(24)
and g = ¢z = [~ is satisfied automatically, F r o m (24) it is clear that for those
parts of the beam where the amplitude is larger, the group velocity is smaller.
Nonlinear Wave Mechanics and Particulate Serf-Focusing
565
If we assume, according to (23), that the changes in k (or v) are perpendicular
to the ray path, then a family of orthogonal trajections is defined, with the
rays perpendicular to the wavefronts. Since v depends on the intensity, the
rays will converge toward :regions of locally higher intensity, which is in
essence the manifestation of self-focusing. This also explains the instability
with respect to undulations on the intensity profile, finally leading to
filamentation. The electromagnetic analog of the present situation has been
considered in greater detail previously. <~7~
6. C O N C L U S I O N S
A formalism for analyzing weakly nonlinear wave mechanical systems
is presented. The analog to nonlinear optics facilitates the discussion o f
similar phenomena.
The concept of wave mechanical harmonic generation is discusseed and
its relevance to the theory of superconductive materials is pointed out.
The present theory is used to explain, in a somewhat speculative manner,
the possible existence of "effective" and "tired" photons in high-intensity
light beams.
Finally, it is pointed out that the theory implies particulate self-focusing,
and a simple example is discussed.
REFERENCES
1. D. Censor, or. Plasma Phys. 16, 415 (1976).
2. D. Censor, Phys. Rev. A 16, 1673 (1977).
3. D. Censor and K. Suchy, Kleinheubacher Berichte 19, 617 (1976) (Proc. Conf. URSI
Nat. Committee, Fed. Rep. Germany, October 1975).
4. D. Censor, J. Phys. A 10, 1781 (1977).
5. D. Censor, Phys. Rev. A 18, 2614 (1978).
6. J. L. Synge, Geometrical Mechanic's and de Broglie Waves (Cambridge University
Press, 1954).
7. D. Censor, Phys. Rev. D 19, 1108 (1979).
8. E. Panarella, Found. Phys. 4, 227 (1974).
9. E. Panaretla, Found. Phys. 7, 405 (1977).
I0. L. I. Schiff, Quantum Mechanics (McGraw-Hill, 1955).
11. M. Tinkham, Introduction to Superconductivity (McGraw-Hill, 1975).
12. V. L. Ginzburg, The Propagation of Electromagnetic Waves in Plasmas, 2rid ed.
(Pergamon Press, 1970).
13. S. A. Akhamanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon and
Breach, 1972).
14. M. Schubert and B. Wilhetmi, Einj'dhrung in die nichttineare Optik (Teubner, 1971),
Vol. 1.
15. G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1974).
566
Censor
16. G. A. Askaryan, Zh. Eksp. Teor. Fiz. 42, 1567 (1962) [Soy. Phys.--JETP 15, 1088
(1962)].
17. D. Censor, Wave packets and solitary waves--a dual approach. II: nonlinear systems,
in preparation.
18. A. Messiah, Quantum Mechanics (North-Holland, 1970).
19. H. Poeverlein, Phys. Rev. 128, 956 (1962).