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PHYSICAL REVIEW E 82, 016322 !2010"
Dynamics of an electrostatically modified Kuramoto–Sivashinsky–Korteweg–de Vries
equation arising in falling film flows
D. Tseluiko*
School of Mathematics, Loughborough University, Leicestershire LE11 3TU, United Kingdom
D. T. Papageorgiou†
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
!Received 13 January 2010; revised manuscript received 4 May 2010; published 30 July 2010"
The gravity-driven flow of a liquid film down a vertical flat plate in the presence of an electric field acting
in a direction perpendicular to the wall is investigated. The film is assumed to be a perfect conductor, and the
bounding region of air above the film is taken to be a perfect dielectric. A strongly nonlinear long-wave
evolution equation is developed for flow parameters near the critical instability conditions. The equation retains
terms up to second order in the slenderness parameter in order to incorporate the effects of shear-induced
growth, short-wave damping due to surface tension, electric stress effects, and dispersive effects. In the
additional asymptotic limit of small but finite interfacial perturbations the dynamics are shown to be governed
by a Kuramoto–Sivashinsky !KS" equation with Korteweg–de Vries dispersion, also known as the Kawahara or
the generalized Kuramoto–Sivashinsky !gKS" equation, which also includes a nonlocal energy growth term
that arises from the electrostatics. Extensive numerical experiments are carried out to characterize solutions to
this equation. Using perturbation theory and numerical solutions, it is shown that the electric field alters the
far-field decay characteristics of bound states of the gKS equation from exponential to algebraic behavior. In
addition, it is demonstrated numerically that chaotic solutions of the KS equation that are regularized into
traveling-wave pulses when sufficient dispersion is added can in turn become chaotic by applying a sufficiently
strong electric field. It is suggested, therefore, that electric fields can be utilized to enhance interfacial turbulence and in turn increase heat or mass transfer in applications. A physical example involving electrified falling
film flows of ethelene glycol fluids is furnished and shows that the theory is within reach of experiments.
DOI: 10.1103/PhysRevE.82.016322
PACS number!s": 47.15.gm, 47.15.Fe, 47.20.Ky
I. INTRODUCTION
Falling film flows of viscous liquids on vertical walls are
unstable at all nonzero Reynolds numbers and produce a rich
variety of complex two- and three-dimensional dynamics;
see the pioneering experiments of Kapitza and Kapitza #1$.
More recent experiments have established the presence of
spatiotemporal chaos and coherent structures; see Liu and
Gollub #2,3$, Vlachogiannis and Bontozoglou #4$, and Argyriadi et al. #5$. There are two modes of linear instability: a
long-wave surface mode at moderate Reynolds numbers associated with the moving free surface and a shorter shear
mode operating at high Reynolds numbers #6$. The stability
characteristics depend strongly on the angle of inclination of
the plate with the horizontal, and when the plate is vertical
the interfacial mode is unstable at all Reynolds numbers
while the shear mode enters at much higher values. For example, for an angle of inclination of 4° and a wide range of
liquids, Floryan et al. #6$ found that the interfacial mode
becomes unstable at a Reynolds number of 18, approximately, while the shear mode does not enter until a Reynolds
number of approximately 3 ! 103. The Kuramoto–
Sivashinsky !KS" equation !e.g., Ref. #7$" correctly captures
the weakly nonlinear interfacial dynamics in the vicinity of
the critical Reynolds number and is fully consistent with the
*[email protected]
†
[email protected]
1539-3755/2010/82!1"/016322!12"
linear stability characteristics. This equation has its roots in
the modeling proposed by Shkadov #8$ and has been demonstrated to hold at higher Reynolds numbers !see, for example, Ref. #9$ and numerous references therein". We also
refer to the monograph by Chang and Demekhin #10$ for an
extensive discussion of falling films and evolution equations
that describe such phenomena.
Here, we are mostly concerned with the effects of electric
fields on falling film flows down vertical plates. The experiments of Melcher #11$ !see also Refs. #12,13$" showed the
existence of electrically induced surface waves; typically the
electric field induces instability when the field is normal to
the free surface but introduces dispersive effects when the
field is in the plane of the undisturbed free surface. More
recent experimental and theoretical work by Griffing et al.
#14$ studied the effect of localized electric fields on nonuniform steady-state profiles in falling film flows. The present
theoretical study is mostly concerned with flow regimes that
support complex dynamics of electrified falling film flows, in
particular we derive and study numerically and analytically a
generalized Kuramoto–Sivashinsky !gKS" equation incorporating electrostatic effects. The gKS was derived by Topper
and Kawahara #15$ in the context of falling film flows, and
Kawahara #16$ first established numerically that a sufficient
amount of dispersion can regularize complex dynamical behavior into pulses of traveling waves. The equation was also
derived by Hooper and Grimshaw #17$ for two-fluid flows in
channels and in the context of core-annular flows by Papageorgiou et al. #18$. The dispersive term in the latter study is
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©2010 The American Physical Society
PHYSICAL REVIEW E 82, 016322 !2010"
D. TSELUIKO AND D. T. PAPAGEORGIOU
a pseudodifferential operator arising from the physics of the
problem and which localizes to the gKS for small wave numbers; numerical experiments in Ref. #18$ showed dispersive
regularization by the pseudodifferential operator also. Smyrlis and Papageorgiou #19$ showed numerically that solutions
to the nonlocal and local dispersion equations are very close
in the energy norm for several attractors including timeperiodic ones. Numerous studies have ensued including the
construction of traveling waves and their stability by Renardy #20$ and Chang et al. #21,22$, and the existence of
traveling waves study of Ercolani et al. #23$.
The presence of an electric field acting perpendicularly to
the undisturbed liquid-layer free surface enhances the instability with short waves growing the fastest. It provides a
mechanism which in the absence of other regularizing physical effects renders the problem ill posed with short waves
growing at an exponential rate proportional to the inverse
cube of their wavelength. The manifestation of this instability on the nonlinear spatiotemporal evolution of falling flows
was considered by Tseluiko and Papageorgiou #24,25$ who
studied an electrically modified Kuramoto–Sivashinsky
equation both numerically and analytically, in the latter instance proving the existence of an inertial manifold and finding improved estimates for its dimension. In a series of papers, Tseluiko et al. #26–29$ considered modifications of the
problem over two-dimensional topographically structured
substrates and provided considerable evidence !by comparing direct simulations with asymptotic solutions" of the accuracy of long-wave theories in describing low-Reynoldsnumber dynamics. Staying with flat substrates, it is possible
to retain shear-induced growth, surface tension stabilization,
electric field growth, and dispersive effects by developing a
long-wave evolution equation correct to second order in the
small slenderness parameter. The resulting equation is onedimensional but strongly nonlinear in the sense that the freesurface deformation scales with the film thickness #this is Eq.
!29" appearing in Sec. III$. In this study we specialize these
solutions to the weakly nonlinear regime and derive the following nonlocal generalized Kuramoto–Sivashinsky !ngKS"
equation:
ut + uux + uxx + "uxxx + #H#uxxx$ + uxxxx = 0,
!1"
where u!x , t" is the scaled interfacial shape, x is the coordinate along the vertical wall, t is time, and " , # $ 0. In addition to the nonlinear inertial term the destabilizing term uxx
originates from the shear at the free surface; the term "uxxx
represents dispersive effects !the coefficient " is defined
later"; the term uxxxx is a stabilizing term due to surface tension; and the nonlocal term #H#uxxx$, where H is the Hilbert
transform operator, is due to the electric field with # being a
dimensionless measure of the field’s strength. We study Eq.
!1" on periodic domains x ! #−L , L$. The nonlocal Hilbert
transform term arises from the electric field in the region
outside the liquid layer and enters to modify the gKS equation by inducing additional short-wave instabilities as mentioned earlier. There are three parameters in the problem, ",
#, and L, which measures the size of the system. In the
absence of dispersion and electric fields, complex dynamics
ensues as L increases with a transition to chaos through a
Z
X
U
κ<0
κ>0
h(X, T )
G
E
n
t
FIG. 1. Flow of a perfectly conducting liquid film down an
inclined plane in the presence of an electric field. The region above
the film is occupied by a passive dielectric gas.
Feigenbaum period-doubling bifurcation scenario !see Refs.
#30–32$". In the absence of an electric field but with dispersion present, it has been demonstrated that the small-time
dynamics can be described by the interaction of a discrete
number of bound-state solitons—see, for example, Ref. #33$.
When # ! 0, we compute the bound-state solitons of Eq. !1"
and analyze their far-field behavior showing that the decay is
algebraic rather than exponential, something that can have a
considerable influence on the soliton-soliton interaction dynamics. Finally, we compute the initial boundary value problem on periodic domains as ", #, and L vary in order to
classify the most dominant attractors and characterize the
long-time dynamics. We establish that, with all other parameters held fixed, sufficient dispersion regularizes the flow,
whereas sufficiently large electric field induces spatiotemporal chaos.
II. PROBLEM STATEMENT
We consider the two-dimensional gravity-driven flow of a
liquid film down a vertical flat plate when the film is subjected to an electric field applied perpendicularly to the plate
as illustrated in Fig. 1. We choose a rectangular coordinate
system !X , Z" such that the X axis points along the wall in the
direction of gravity and the Z axis is perpendicular to it. The
film surface is located at Z = h!X , T", where T denotes time.
The wall is an electrode held !without loss of generality" at
zero voltage potential and a uniform electric field of strength
E0 is imposed as Z → %. The liquid is assumed to be a perfect
conductor, so that there is no potential difference between the
wall and the free surface and, consequently, no electric field
in the film. The medium above the film is assumed to be a
perfect dielectric with permittivity & p. The electrostatic limit
is appropriate when the induced magnetic field can be ne-
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DYNAMICS OF AN ELECTROSTATICALLY MODIFIED…
glected. This is the case in the present problem since there
are no high-frequency ac fields imposed and no highfrequency free-surface oscillations. Charges reside on the
free surface alone and their distribution can be determined
from Gauss’s law a posteriori. For instance, a discussion of
such limits can be found in work of Melcher and Taylor #34$.
We nondimensionalize variables using the undisturbed
film thickness h0 as the length scale and the Nusselt surface
speed U0 = 'gh20 / 2# as the velocity scale, where ' and # are
the density and the viscosity of the fluid and g is the acceleration due to gravity. The time scale is h0 / U0, the pressure
is nondimensionalized by #U0 / h0, and the electric potential
is nondimensionalized using E0h0.
Standard dimensional analysis reveals the importance of
three dimensionless parameters: a Reynolds number R, a
Bond number B, and an electric Weber number We, which
are defined as
2
R=
gh30
2 ,
' h 0U 0 '
=
#
2#
B=
'gh20
2#U0
=
,
(
(
& ph0E20 & pE20
We =
=
,
2#U0 'gh0
!3"
where U = !U , V" is the fluid velocity, P is the pressure, and
G = !2 , 0" is the dimensionless gravitational force. The noslip and no-penetration conditions at the wall require that
U=0
!4"
at Z = 0, and the kinematic condition at the film surface, Z
= h!X , T", reads
V = hT + UhX .
!5"
The dynamic balance of stress at the film surface demands
that
" · n = − !2)/B + Pa"n + M · n,
where t is the unit tangent to the free surface depicted in
Fig. 1.
The electric potential * in the region Z + h!X , T" satisfies
Laplace’s equation,
"2* = 0.
!9"
Since the free surface is a null equipotential we demand that
*=0
!10"
!* → − k
!11"
&E&2 = !1 + hX2 "*Z2
!12"
at Z = h!X , T", and
as Z → %, where k is the unit vector pointing in the Z direction. We note that it can be easily verified that
!6"
Our goal is to derive a weakly nonlinear evolution equation that retains the fundamental physics of the electrified
film flow, namely, instability and energy production, stability
and energy dissipation, dispersion, and nonlinearity. We proceed with a long-wave analysis, that is, we assume that
streamwise variations occur over a much longer length scale,
say ,, than the typical film thickness h0, and introduce the
small parameter - = h0 / ,. Our first aim is to derive a longwave evolution equation including O!-2" terms. We proceed
as in Ref. #24$, where such an equation was derived up to
order O!-". We note that for a given free-surface shape the
electric field problem can be solved independently of the
fluid problem, and then the electric field contribution in the
normal stress balance condition !8" can be expressed in terms
of the free-surface shape !implicitly in general but explicitly
in the asymptotic analysis that follows", closing the fluid
problem.
The base solution for the electric potential for a flat un¯ = −Z. We introduce a perturdisturbed film, i.e., h ' 1, is *
˜ by writing
bation potential *
where " = −PI + !!U + !U " is the Newtonian stress tensor
nondimensionalized by #U0 / h0, n is the unit normal to the
free surface pointing into the film, ) is the curvature of the
film surface taken to be positive when the surface is concave
downward as indicated in Fig. 1, and Pa is the dimensionless
air pressure. The Maxwell stress tensor M nondimensionalized by #U0 / h0 is given by !see Ref. #34$"
!13"
˜ +*
˜ = 0,
*
XX
ZZ
!14"
˜ = h − 1 at Z = h!X , T", and
with *
˜ → 0,
*
X
˜ →0
*
Z
!15"
1
X = .,
-
1
Z = /,
-
!16"
as Z → %. We introduce new independent variables . and /,
!7"
where E is the electric field and satisfies the relation
E = −!* with * denoting the electric potential. It may be
verified directly that the Maxwell stresses do not contribute
to the tangential stress balance. In fact, the normal and tangential stress balance conditions at the free surface are
¯ +*
˜,
*=*
so that
T
M = We!EE − I&E&2/2",
!8"
III. WEAKLY NONLINEAR EVOLUTION
!2"
! · U = 0,
n · " · n = − !2)/B + Pa" + We&E&2 ,
at the free surface and this result is used below.
where ( is the surface tension coefficient. Note that the Bond
number B is the square of the ratio of the film thickness to
the capillary length a = %( / 'g, and the electric Weber number We is the ratio of electric to gravitational forces.
The flow in the liquid film is governed by the incompressible Navier–Stokes equations,
R!UT + U · !U" = − !P + "2U + G,
t · " · n = 0,
and expand
˜ = * + -* + ¯ .
*
0
1
It can be verified that the problem at leading order is
016322-3
!17"
D. TSELUIKO AND D. T. PAPAGEORGIOU
*0.. + *0// = 0,
!18"
with *0 = h − 1 at / = 0, and
h1 +
+ ,
2We! 3
2 3
2 3
8R 6
h +h h... +
h H#h..$
h h. +
3
3
15
3B!
* 0/ → 0
!19"
+ -2
*0/!.,0" = − H#h.$
!20"
+
*0. → 0,
as / → %. It can be shown using complex variables that
!see Refs. #24,35$, for instance", where H is the Hilbert
transform operator defined by
H#g$!." =
1
P.V.
0
(
%
−%
g!.!"
d.! ,
. − .!
*1. → 0,
* 1/ → 0
as / → %. Solving as above, we obtain
!23"
*1/!.,0" = H#„h*0/!.,0"….$ = − H†!hH#h.$".‡.
!24"
˜ = − H#h $ − -)H†!hH#h $" ‡ + hh * + O!-2"
*
/
.
. .
..
!25"
Using Eqs. !20" and !24", we find
at the free surface whose position is / = -h in terms of the
outer variables. The value of &E&2 at the free surface is readily
determined using Eq. !12",
˜ "2
&E&2 = !1 + -2h.2"!− 1+ -*
/
= !1 + -2h.2"#1 + -E1 + -2E2 + O!-3"$2 ,
!26"
where for brevity we have introduced the notations
E1 = H#h.$,
E2 = H†!hH#h.$".‡ + hh.. .
!27"
Next we proceed with the long-wave analysis for the hydrodynamic problem as in Ref. #24$. We assume that R
= O!1", 1 / B = O!1 / -2", and We = O!1 / -" and introduce
B! =
B
,
-2
We! = -We .
.
2R 16 5 2
8
9
2
+ h6h.h...
h h. h.. + h6h..
5
4
B! 5
/
37 7
8
7 6
h h.H#h..$ −
h H#h...$ + h5H#2hh.2
30
210
15
/
2We! 3
!2h h.h.. + h4h... + h3H#h.$H#h..$
3
-0
= 0,
.
!29"
where a new slow time scale 1 = -T has been introduced. It
can be verified that the terms at the next order are at most of
order O!-3" times derivatives of h and Hilbert transforms of
the derivatives and therefore become of higher order when
the weakly nonlinear regime is considered.
The evolution equation !29" is strongly nonlinear in the
sense that it describes free-surface deflections that scale with
the undisturbed layer thickness. The existence of solutions
for such classes of nonlinear equations is not guaranteed as
the numerical results of Pumir et al. #36$ indicate in the
absence of electric fields. In the present work, we proceed by
deriving a canonical weakly nonlinear model that retains
long-wave destabilization, short-wave dissipation, dispersion, and electric field effects. Our analysis shows that the
consistent derivation of such a model from Eq. !29" is possible under the assumptions that R, 1 / B!, and We! are O!-".
Note that this is not in contradiction with the earlier assumptions that R, 1 / B!, and We! are O!1" since any parameter that
is O!-" is automatically O!1". Besides, as it will be shown
below, the terms that we neglected in Eq. !29" become accordingly of higher order and cannot appear in the weakly
nonlinear model. We introduce rescaled order-unity constants
!28"
These canonical scalings allow us to retain the competing
effects of nonlinearity, shear-induced instability, short-wave
stabilization due to surface tension, and instability due to the
electric field. It is important to note that any other scaling
that promotes the size of the electric field effects relative to
the other physical mechanisms leads to an evolution equation
that is ill posed in the sense that it is short-wave unstable—
the instability is similar to the Kelvin–Helmholtz one but
with much larger growth rates, namely, proportional to k3
rather than k, where k is the wave number of the perturbation. The mathematical model that emerges, then, facilitates
an extensive quantitative study of these competing physical
mechanisms. After a lot of algebra which involves the determination of the velocity and pressure in terms of h!X , T", the
kinematic condition provides the final equation correct to
order O!-2",
.
/
+ h3H†h..H#h.$ + 2h.H#h..$ + hH#h...$‡"
!22"
with *1 = −h*0/!. , 0" at / = 0, and
.
-
14 3 2
596 9 2 2 10
h h. + 2h4h.. + R2
h h + h h..
3
315 . 7
+ h2h..$ +
and P.V. denotes the principal value of the integral.
At the next order we have
*1.. + *1// = 0,
,
+ RWe!
!21"
PHYSICAL REVIEW E 82, 016322 !2010"
R̄ =
R
,
-
B̄ = -B! =
B
,
-
!30"
noting that also We = O!1" if We! = O!-". A balance of terms in
the weakly nonlinear limit is possible for amplitudes which
are of size -2. Therefore, we substitute h!. , 1" = 1 + -22!. , 1"
into Eq. !29" to obtain the following equation, correct to
order O!-4":
-221 + 2-22. + 4-422. + -4
+ -4
2We
H#2...$ = 0.
3
8R̄
-4
2.. + 2-42... +
2....
15
3C̄
!31"
Note that the terms that we neglected in Eq. !29" are
O!-3&h.&" or smaller, and they become O!-5" under the assumption that h = 1 + O!-2" and hence should be neglected in
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DYNAMICS OF AN ELECTROSTATICALLY MODIFIED…
the weakly nonlinear equation derived above. Utilizing the
change of variables,
. − 21 =
. /
5
1/2
4R̄B̄
x,
1=
. /
4 1 3
2=
R̄ B̄
15 5
75
2 2
32- R̄ B̄
t,
30
25
20
!32"
u0
10
1/2
5
!33"
u,
0
-5
-15
we arrive at the canonical equation
ut + uux + uxx + "uxxx + #H#uxxx$ + uxxxx = 0,
!34"
where " and # are order-unity constants defined by
"=3
./ ./
5B̄
4R̄
1/2
=3
5B
4R
μ
15
1/2
,
#=
We
".
3
!35"
Equation !34" is a ngKS equation and is considered on periodic domains of size 2L. The parameters " and # measure
the relative importance of dispersion and the electric field,
respectively. In the numerical work that follows, we treat "
and # as independent parameters and in particular consider
situations where " = 0 and # ! 0. In view of their definitions
!35", it appears that this is not permissible. However, a canonical equation which is identical to Eq. !34" but with "
= 0 and # ! 0 can be derived from the fully nonlinear Eq.
!29" by looking for weakly nonlinear solutions of larger amplitude than before, namely, by writing h = 1 + -2 and proceeding in the usual fashion. This derivation has been carried
out by Tseluiko and Papageorgiou #24$ who started with Eq.
!29" correct to order -. In the sequel, then, no confusion
should be caused by the independent treatment of " and
#—where we take " = 0 we implicitly assume that # ! 0 with
the understanding that this equation is canonical also. One of
our objectives is to study mathematically the dynamics in the
presence or absence of the different effects and this can be
done by studying Eq. !34" with the parameters varying
independently.
IV. NUMERICAL SOLUTIONS
Equation !34" is solved numerically on 2L-periodic domains with an initial condition of the form u!x , 0" = u0!x"
L
u0!x"dx = 0, i.e., zero-mean initial states. Note
with !1 / 2L"1−L
that in physical experiments one needs to consider the effects
of confining walls, but in many situations periodic wavy
structures emerge with minimal border effects, and these are
the types of wave regimes we focus on in this study. Since
L
u!x , t"dx is conserved, the solution has zero mean for all
1−L
subsequent times also, and from a Fourier decomposition
viewpoint this implies that the k = 0 mode of the spectrum is
zero. A pseudospectral method is used that treats the stiffness
of the linear operator accurately—for details, see Ref. #24$. It
is well established numerically that the KS equation #i.e., Eq.
!34" with " = # = 0$ exhibits large-time chaotic dynamics if
the period of the system is sufficiently large !see the Introduction". The addition of small amounts of dispersion does
not affect this complex behavior, but it has been shown that
sufficiently strong dispersion leads to a regularization of cha-
-10
-5
0
y
5
10
15
FIG. 2. Dependence of the pulse solution of the ngKS equation
on # for " = 0.5 and L = 60. Parameter # changes from 0 !the shape
with the smallest amplitude" to 2 !the shape with the largest amplitude" with the step of 0.4.
otic dynamics into traveling-wave pulses #18,19,33,37$. On
the other hand, in the absence of dispersion the addition of
electric field effects !# ! 0" enhances the chaotic behavior in
the sense that chaotic dynamics emerge for smaller values of
L #24,25$. The objective of the present study is in evaluating
the competing effects of dispersion and electric fields on the
spatiotemporal dynamics of the KS equation. We solve the
boundary value problem to construct nonlinear traveling
waves and also compute initial value problems to explore the
large-time dynamics including the stability of traveling-wave
pulses.
V. TRAVELING WAVES
As noted above, solutions of the KS equation !" = # = 0"
considered on sufficiently large periodic domains are chaotic
in both space and time. However, a sufficiently large dispersion parameter " arrests spatiotemporal chaos, and a solution
to the gKS equation looks like a superposition of weakly
interacting nonlinear pulses of approximately the same
shape, i.e., close to the shape of the solitary-pulse solution
!e.g., Ref. #16$". It is important, therefore, to study solitarypulse solutions of Eq. !34" and analyze the effect of the
nonlocal term on such pulses.
We rewrite Eq. !34" in a frame moving with the velocity
c = c!" , #" of a stationary pulse,
ut − cuy + uuy + uyy + "uyyy + #H#uyyy$ + uyyyy = 0,
!36"
where y = x − ct. A stationary pulse u0 = u0!y ; " , #" is a solution of the steady version of Eq. !36", i.e.,
− cu0y + u0u0y + u0yy + "u0yyy + #H#u0yyy$ + u0yyyy = 0.
!37"
Equation !37" is a nonlinear eigenvalue problem for the pulse
velocity c. We integrated Eq. !37" numerically on periodic
domains by using a spectral fast Fourier transform !FFT"
method and Newton iterations. The results for " = 0.5 and
various values of # on the periodic interval #−60, 60$ are
shown in Fig. 2. It can be seen that the pulse amplitude
increases with # with the shape with the smallest amplitude
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D. TSELUIKO AND D. T. PAPAGEORGIOU
-1.5
12
-2
log10(|u 0|)
10
-2.5
8
c
6
4
-3.5
2
0
0
-3
0.2 0.4 0.6 0.8
1
μ
1.2 1.4 1.6 1.8
-4
0.8
2
FIG. 3. Dependence of speed of the pulse of the ngKS equation
on # when " = 0.5 and L = 60.
corresponding to # = 0 and that with the largest amplitude
corresponding to # = 2. The speed of the pulse is also a
monotonically increasing function of #, as is seen in Fig. 3.
In addition, as # increases, the amplitude of the oscillations
downstream of the pulse grows and a depression forms upstream of the pulse. In contrast to the case when # = 0 for
which the tails of the pulse tend to zero exponentially as y
→ % #37$, for nonzero # the tails of the pulse tend to zero as
1 / &y&3 as &y& → %. This is confirmed numerically in Fig. 4
which depicts a logarithmic plot of &u0& with y for y 3 1. The
solid and the dashed lines correspond to the right and the left
tails of the pulse, respectively, and the thick straight line
included has a slope of −3 in agreement with the algebraic
behavior mentioned above !the analysis for small # below
also confirms this". As the period of the domain increases,
the pulse shape and the speed of the pulse tend to limiting
values. The dependence of c on the period of the domain is
shown in Fig. 5 and we can conclude that for sufficiently
large L the periodic solutions approximate well the solitary
pulses on an infinite domain.
Analysis for small #
In this section, we carry out a perturbation expansion to
analyze the effects of the electric field on the pulse solitons
of the gKS equation. It was found numerically !see Fig. 4"
that nonzero values of # alter the decay at infinity of the
solitary pulses from exponential to algebraic and we seek to
establish this analytically. Let us consider Eq. !37" on y !
!−% , %" and assume that # 4 1. We look for asymptotic
solutions of the form
u0 = f 0 + # f 1 + ¯ ,
c = c0 + #c1 + ¯ .
!38"
− c0 f 0y + f 0 f 0y + f 0yy + " f 0yyy + f 0yyyy = 0,
!39"
1
1.1
1.2
log10(|y|)
1.3
1.4
1.5
FIG. 4. The right !solid line" and the left !dashed line" tails of
the pulse solution of the ngKS equation for " = 0.5 and # = 2. Results are shown on a log-log plot, from which it is evident that the
tails of the pulse tend to zero as 1 / y 3 as y increases !the thick solid
line is a straight line of slope −3".
L#f 1$ = c1 f 0 − H#f 0yy$,
!41"
where L is a linear operator defined by
L#f$ ' − c0 f + f y + " f yy + f yyy + f 0 f .
!42"
The adjoint operator is defined by
L!#f$ ' − c0 f − f y + " f yy − f yyy + f 0 f .
!43"
By inspection we see that L#f 0y$ = 0 and hence f 0y is in the
null space of L—it has been verified numerically that the
null space is one dimensional and therefore spanned by f 0y
!this has been done by projecting the L2 space onto an
N-dimensional space and finding the matrix representation of
L in this space and verifying that the rank of the matrix is
N − 1 by computing its singular value decomposition". Similarly we verified numerically that the null space of the adjoint operator L! is also one dimensional. Let 5 be in the null
space of L! normalized, so that 2f 0y , 53 = 1, where 2· , ·3 denotes the usual inner product in LC2 . We can compute 5 numerically and the result for the typical case " = 0.5 is shown
in Fig. 6. The solvability condition for Eq. !41" requires that
the right-hand side of this equation is orthogonal to 5, which
gives the following expression for c1:
2.5179
2.51785
2.5178
At leading order we find
c
i.e., f 0 is a pulse solution of the gKS equation for a given ".
At first order, we obtain
2.51775
2.5177
2.51765
2.5176
− c0 f 1y + f 1yy + " f 1yyy + f 1yyyy + !f 0 f 1"y = c1 f 0y − H#f 0yyy$.
2.51755
16 20 24 28 32 36 40 44 48 52 56 60
!40"
Integrating once and noting that the constant of integration is
zero due to the decay to zero of the solution at infinity, we
find
0.9
L
FIG. 5. The variation of speed of traveling-wave pulse solutions
of the ngKS equation on the half-period L when " = 0.5 and #
= 0.4.
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PHYSICAL REVIEW E 82, 016322 !2010"
DYNAMICS OF AN ELECTROSTATICALLY MODIFIED…
2.97
-2.97
500
0.3
u
0.2
0.1
-0.1
1
u0
300
t
ψ -0.2
-0.3
-0.4
-1
200
-2
-30 -20 -10 0
100
-0.5
0
-20
150
-0.6
-0.7
-20
2
400
0
-15
-10
-5
0
y
5
10
15
20
F†H#f 0$‡!k" = − i sgn!k"F#f 0$!k",
H#f 0$ 5
1
y
50
0
0
200
t
300
1
y3
H#f 0yy$ = H#f 0$yy 5
400
500
as y → 6 %,
!47"
and, therefore, from Eq. !41" it follows that
f1 5
1
y3
!48"
as y → 6 %.
Thus, the asymptotic result for small # confirms the above
observation that the tails of the ngKS pulse should tend to
zero as 1 / y 3 as y → 6 %, in contrast to the gKS case. This
observation is important for the development of a weak5.70
-2.77
500
u
5
4
3
2
u1
0
-1
-2
-30 -20 -10 0 10 20 30
400
!46"
as y → 6 %.
100
FIG. 8. Solution of the KS equation #Eq. !34" with " = # = 0$
when L = 30. Top left: evolution of u!x , t". Top right: profile at the
last computed time t = 500. Bottom: evolution of the energy norm.
!45"
%
p!y"e−ikydy is the usual Fourier transwhere F#p!y"$!k" = 1−%
form operation. By integrating Eq. !39" twice we obtain
%
% 2
%
f 0dy = !1 / 2"1−%
f 0dy, which shows that 1−%
f 0dy ! 0.
c01−%
Therefore, F[H#f 0$]!k" has a jump discontinuity at zero
!note that the zero Fourier mode is given by F#f 0!y"$!0"
%
f 0!y"dy" implying that !see Ref. #38$, Theorem 19, p.
= 1−%
52"
20
||u||2
!44"
For example, when " = 0.5 we find c1 4 1.58. Next, we can
solve Eq. !41" numerically with an additional requirement
that 2f 1 , 53 = 0 !this is imposed to obtain uniqueness since
any term proportional to f 0y can be added to f 1"; a numerical
solution of Eq. !41" for " = 0.5 is shown in Fig. 7.
It is well known that the tails of a solitary pulse of the
gKS equation tend to zero exponentially as y → 6 %. Moreover, the left tail tends to zero monotonically as y → −%,
while the right tail tends to zero either in an oscillatory manner or monotonically as y → % depending on whether " is
below or above a threshold value, "! 4 3.71 #37$. The Fourier
transform of the Hilbert transform H#f 0!y"$ is given by
x
10 20 30
100
FIG. 6. The null space eigenfunction 5 of the adjoint operator L
given by Eq. !43", when " = 0.5.
c1 = 2H#f 0yy$, 53.
0
x
300
t
From this, we conclude that
200
100
5
0
-20
300
4
0
x
x
20
3
f1
200
2
||u||2
1
100
0
-1
-2
-20
0
0
-15
-10
-5
0
y
5
10
15
20
FIG. 7. The unique solution f 1 of the first-order perturbation
problem !41", when " = 0.5. Uniqueness is fixed by imposing
2f 1 , 53 = 0 as described in the text.
100
200
t
300
400
500
FIG. 9. Solution of the gKS equation #Eq. !34" with " = 1, #
= 0$ when L = 30. Top left: evolution of u!x , t". Top right: profile at
the last computed time t = 500. Bottom: evolution of the energy
norm.
016322-7
PHYSICAL REVIEW E 82, 016322 !2010"
D. TSELUIKO AND D. T. PAPAGEORGIOU
3.00
-2.58
500
u
140
400
300
t
u
100
||u||2
60
200
0
x
interaction theory of the pulses and this will be pursued elsewhere.
VI. SPATIOTEMPORAL DYNAMICS
In this section we solve the initial value problem !34" on
periodic domains as the half-size of the domain L and the
parameters " and # vary. The initial condition for most simulations presented is taken to be u!x , 0" = cos!0x / L"; random
initial conditions have been used also where noted in the
text. If the dispersion is sufficiently strong, then travelingwave pulses result and it is important to have an accurate
way of computing the speed of steady-state traveling waves.
A spectrally accurate method is possible by noting the formula
c=
(
−L
(
L
−L
(
L
−L
2
uxx
dx
!49"
,
u2x dx
100
80
60
||u||
40
20
0
1
7
8
9
3
4
5
6
Dispersion δ
-20
0
20
20
10
0
-10
-10
-20
0
20
x
x
2
1
0
0
10
1
2
3
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
Dispersion δ
60
||u||
2
20
x
spectral accuracy using trapezoidal integration, for instance.
We begin by exhibiting the regularizing effect of dispersion on the KS dynamics. Taking # = 0 !no electric field", L
= 30, " = 0, and an initial condition u!x , 0" = cos!0x / L", we
observe chaotic long-time dynamics !this has been noted by
many authors and in fact the route to chaos has been shown
to follow a Feigenbaum period-doubling scenario #30–32$".
Results up to 500 time units are shown in Fig. 8 which depicts the evolution of u, the final computed profile u!x , 500",
and the evolution of the energy 6u!· , t"62 of the solution.
Analogous solutions starting from the same initial condition
but having a dispersion parameter " = 1.0 are given in Fig. 9.
The dispersion is sufficient to regularize the chaotic dynamics and produce traveling-wave pulses with computed speed
c 4 0.79. It can be shown #39$ that the traveling wave-train
is composed of the solitary pulses computed from Eq. !37"
plus a small correction function. In the results of Fig. 9
the solution is almost exactly a traveling wave composed of
four solitary pulses. If " is too small, then the large-time
dynamics remains chaotic—an example is given in Fig. 10
where " = 0.1. The variation with " of the traveling-wave
speed and the corresponding L2 norm 6u6!defined by 6u6
L
u!x , t"2dx$1/2" for this branch of three-pulse waves are
= #1−L
given in Fig. 11 from which we observe a monotonic increase in both the speed and the norm, both scaling linearly
2
6
0
FIG. 12. Computed traveling-wave profiles for the three-pulse
branch in Fig. 11 corresponding to the values " = 7 !top left", " = 8
!top right", " = 9 !bottom left", and " = 10 !bottom right".
4
5
-20
40
0
3
Dispersion δ
x
u
6
4
-10
20
10
4
3
10
30
8
2
0
20
Speed c
Speed c
which follows from multiplication of Eq. !37" by u0y and
integration by parts over the domain. Note that we used
u!x , t" in Eq. !49" above, so that if a traveling-wave state
exists then the quantity c!t" will tend to a constant value
which is the speed of the pulse. Given a periodic function
u!x , t", the right-hand side of Eq. !49" can be computed with
0
1
20
0
-20
30
u
uu2x dx − "
u
10
40
t
FIG. 10. Solution of the gKS equation #Eq. !34" with " = 0.1,
# = 0$ when L = 30. Left: evolution of u!x , t". Right: evolution of the
energy norm.
L
30
20
-10
20
0
-20
40
0
20
0 100 200 300 400 500
100
40
30
7
8
9
40
20
0
0
10
FIG. 11. Variation of the speed c !top" and norm 6u6 !bottom"
with " for traveling-wave solutions of the gKS equation #Eq. !34"
with # = 0$ when L = 30. Three-pulse branch.
Dispersion δ
FIG. 13. Variation of the speed c !top" and norm 6u6 !bottom"
with " for traveling-wave solutions of the gKS equation #Eq. !34"
with # = 0$ when L = 30. Two-pulse branch.
016322-8
PHYSICAL REVIEW E 82, 016322 !2010"
DYNAMICS OF AN ELECTROSTATICALLY MODIFIED…
24
Speed c
0
20
-2
-3
0
18
1
2
3
150
4
5
6
Dispersion δ
7
8
9
10
||u||
16
14
12
100
||u||
50
0
0
Bimodal
Trimodal
22
-1
10
8
1
2
3
4
5
6
Dispersion δ
7
8
9
6
10
0
0.2
0.4
0.6
μ
0.8
1
1.2
1.4
FIG. 14. Variation of the speed c !top" and norm 6u6 !bottom"
with " for traveling-wave solutions of the gKS equation #Eq. !34"
with # = 0$ when L = 30. Heptamodal branch.
FIG. 16. Bifurcation diagram: L2 norm of steady states as #
increases. L = 7, " = 0. Diamonds: bimodal steady states. Circles:
trimodal steady states.
with " as first noted in Ref. #16$. Traveling-wave profiles
from this branch at " = 7 , 8 , 9 , 10 are depicted in Fig. 12. We
emphasize, however, that the solutions are not unique; there
exist other branches with different modal behavior and we
have followed the characteristics of two-pulse traveling
waves and the results are presented in Fig. 13. Another
branch can be constructed by using as initial condition the
most unstable wave provided by linear theory. In this case
the resulting traveling waves are heptamodal, i.e., they have
period 2L / 7 and have negative speeds, relative to the
traveling-wave frame used to derive the gKS equation, for all
nonzero values of ". This branch emanates from a heptamodal steady-state solution of the KS equation for L = 30
with the speed being zero when " = 0. Characteristics of the
traveling waves for this branch are given in Fig. 14 and
typical profiles for " = 1 , 4 , 7 , 10 are depicted in Fig. 15. We
note that in all time-dependent computations, steady-state
traveling waves with equally separated pulses must emerge
when " is sufficiently large and the time of integration is also
large. It has been shown by Tseluiko et al. #39$ that times of
integration of 104 or larger may be required to obtain convergence with equal pulse separation. In the present numerical results our typical integration times are 100–500 and in
all cases the L2 norm reaches to within less than 0.1% of its
steady-state value. An interesting feature of the dynamics is
that the relaxation to equally separated pulses is very slow
and also different initial conditions result in different pulse
separation distances at moderate times and hence require different integration times to converge to the ultimate equally
separated pulse solutions—see Ref. #39$.
The next set of numerical experiments considers the effect
of the electric field on the nonlinear dynamics. We fix L = 7
which in the case of " = # = 0 supports a bimodal steady state.
Introducing an electric field !but no dispersion for the moment" modifies the flow to produce larger energy steady
states, initially, that maintain the bimodal spatial structure. In
the vicinity of # = 1 there is a competition between the bimodal steady state, a much higher energy trimodal steady
state, and a time oscillatory attractor !for example, when #
= 1.03 computations with 256 modes and time step 7t
= 20/ 1282 up to 500 time units indicate nonperiodic oscillatory behavior in time; this is supported on a very small interval of values of #". We emphasize that the results given
here are large-time solutions of the initial value problem with
the same cosine initial condition and constitute a description
of the most attracting states for this set of initial conditions
!typical maximum integration times are 100–200 time units".
Figure 16 is a bifurcation diagram showing the variation of
6u6 with # when steady states emerge from the large-time
computations. The disconnected branches represent steady
30
30
10
20
20
8
10
10
6
u
u
0
-10
-20
-20
0
x
20
-20
-20
30
20
0
x
2
20
u0
-2
20
10
-4
10
0
u
0
-10
-20
4
-10
30
u
0
-20
0
x
20
μ =0
μ =1.0
μ =1.2
-6
-10
-8
-20
-10
-20
0
x
20
FIG. 15. Computed traveling-wave profiles for the heptamodal
branch in Fig. 14 corresponding to the values " = 1 !top left", " = 4
!top right", " = 7 !bottom left", and " = 10 !bottom right".
-6
-4
-2
0
x
2
4
6
FIG. 17. Representative steady-state profiles as # increases. L
= 7, " = 0. Solid line: # = 0 !bimodal"; dashed line: # = 1.0 !bimodal";
dotted line: # = 1.2 !trimodal".
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D. TSELUIKO AND D. T. PAPAGEORGIOU
108.1
-107.9
100
u
150
100
50
u 0
-50
-100
-150
80
60
t
40
20
0
0
-5
x
4
7 x 10
-6 -4 -2 0 2 4 6
x
5
6
5
||u||
4
2
3
2
1
0
0
20
40
t
60
80
100
FIG. 18. Solution of the electrified KS equation #Eq. !34" with
" = 0, # = 4.0$ when L = 7. Top left: evolution of u!x , t". Top right:
profile at the last computed time t = 100. Bottom: evolution of the
energy norm.
states with decreasing periods as # increases. The branches
are labeled 2M and 3M to denote bimodal and trimodal solutions, respectively. Typical profiles from the two branches
are presented collectively in Fig. 17 for # = 0 , 1.0, 1.2. The
modal nature of the solutions can be seen clearly in addition
to the higher energy state of the trimodal branch. For values
of # between 1.3 and 1.4, approximately, the large-time dynamics gets attracted to time oscillatory and chaotic states.
Results for # = 4 are depicted in Fig. 18 which shows the
evolution of the energy, the spatiotemporal evolution of the
solution between 0 8 t 8 100, and the final computed profile
at t = 100.
The effect of introducing dispersion into the electrified
KS equation is considered next. To fix matters we consider
4
4 x 10
4
6 x 10
3.5
5
3
4
2.5
||u||2 2
||u||2 3
1.5
2
1
1
0.5
0
0
10
5
t
15
20
0
0
25
4
7 x 10
20
25 30
35 40
45 50
10 15
20
25 30
35 40
45 50
t
8
6
4
||u||
3
2
5
4
3
2
2
1
0
0
10 15
7
5
||u||
5
4
9 x 10
6
2
three cases # = 0.5, # = 1.1, and # = 4.0 which, in the absence
of dispersion, produce large-time states which are bimodal
steady states, trimodal steady states, and chaotic states, respectively. For the dispersionless steady-state branches corresponding to # = 0.5 and # = 1.1, the effect of " is to produce
traveling waves with negative speeds that maintain the bimodal and trimodal nature of the solutions; as " increases both
the speed and norm 6u6 of the solutions behave linearly as
established in earlier analogous numerical experiments.
Uniqueness is not guaranteed and in fact when # = 1.1 both
bimodal and trimodal traveling waves are found as " varies
depending on the initial conditions. The trimodal branch for
# = 1.1 was constructed by continuation, whereas the bimodal branch results for # = 0.5 emerged as the large-time solutions of a cos!0x / L" initial condition. In fact, using this
initial condition for the case # = 1.1, " ! 0 can produce bimodal traveling waves which are connected to the bimodal
branch in Fig. 16. We do not discuss these characteristics
further because of their analogs with the results shown in
Fig. 14, for example.
We finally turn to the case having # = 4 which produce
chaotic dynamics in the absence of dispersion as indicated in
Fig. 18. It is found numerically that if " is larger than 1.2,
approximately, the dynamics is regularized into a traveling
wave while chaotic or quasiperiodic dynamics are found for
weaker dispersion. To illustrate this we provide in Fig. 19 the
evolution of the energy norm 6u62 for " = 0.5, 1.0, 1.5, 2.0; the
results show that for " = 0.5 and 1.0 the dynamics is still
oscillatory in time but with the amplitude of the oscillations
decreasing with increasing " !for comparison see also Fig. 18
in the absence of dispersion". As " increases to 1.5 and beyond, the energy tends to a constant value reflecting the fact
that the solution evolves to become a traveling wave. In addition, the energy increases with " as established previously.
More detailed characteristics of the traveling-wave branch
for the range 1.28 " 8 5 are given in Fig. 20 which shows
the variation of the speed and the norm 6u6 with ". The speed
is positive at the lowest " for which a traveling wave exists
1
5
10
15
20
25 30
t
35 40
45 50
0
0
5
t
016322-10
FIG. 19. Solution of the ngKS equation #Eq.
!34" with # = 4.0 and increasing values of "$
when L = 7. Top: left, " = 0.5; right, " = 1.0. Bottom: left, " = 1.5; right, " = 2.0.
PHYSICAL REVIEW E 82, 016322 !2010"
DYNAMICS OF AN ELECTROSTATICALLY MODIFIED…
150
Speed c
5
0
100
-5
-10
-15
1
1.5
2
2.5
3
3.5
Dispersion δ
4
4.5
50
5
u
500
||u||
0
400
-50
300
200
1
1.5
2
2.5
3
3.5
Dispersion δ
4
4.5
-100
5
FIG. 20. Variation of the speed c !top" and norm 6u6 !bottom"
with " for traveling-wave solutions of the ngKS equation #Eq. !34"
with # = 4$ when L = 7. Hexamodal branch.
but then decreases monotonically to negative values with a
linear dependence for large "; at the same time the norm
increases monotonically and again follows a linear dependence as expected by scaling arguments #16$. The solutions
are hexamodal traveling waves #i.e., they have period
!2L" / 6 = 7 / 3$ and the solution for " = 2 is shown in Fig.
21—as " increases the modal property is preserved but the
amplitude increases linearly with ".
VII. PHYSICAL CONTEXT
The perfect-conductor-film model would be appropriate
for liquid metals, e.g., mercury. Also, salted water acts as a
near-perfect conductor. For our model to be valid we would
need the Reynolds number R and the Bond number B to be
small and of the same order. This is hardly achievable for
liquid metals and water since for film thicknesses that would
evolve dynamically over reasonable time scales the Reynolds
number turns out to be significantly larger than the Bond
number. This can be fixed by considering more viscous liquids. Appropriate choices would be, for example, solutions
of ethylene glycols and NaCl. Ethylene glycols play a significant role in a wide range of industrial applications; for
example, they are used in heat transfer fluids, in natural gas
hydration, and treating applications to remove water and impurities, in production of a variety of products, e.g., antifreeze, coolants, coatings, emulsifiers, and lubricants, among
others. The conductivity of a solution of an ethylene glycol
and NaCl depends on the concentration of NaCl. For example, the conductivity of monoethylene glycol and NaCl
can be 18.9 S m−1, which is approximately four times the
conductivity of sea water !see Ref. #40$".
As a putative experimental setup, let us consider tetraethylene glycol, the most viscous of ethylene glycols, at room
temperature. The density, viscosity, and surface tension of
tetraethylene glycol are
' = 1.1 ! 103 kg m−3,
# = 5.8 ! 10−2 Pa s,
( = 4.5 ! 10−2 N m−1 .
!50"
For a film of thickness h0 = 0.55 mm, we obtain the following values of the Reynolds number, the Bond number, and
the dispersion parameter:
-6
-4
-2
0
x
2
4
6
FIG. 21. Typical hexamodal traveling-wave profile from the
branch in Fig. 20, at " = 2 !# = 4 , L = 7". The hexamodal solution
has period !2L" / 6 = 7 / 3.
R 4 0.29,
B 4 0.074,
" 4 1.69,
!51"
which are reasonably suitable for the validity of the ngKS
equation. For the validity of the model, we also need the
electric Weber number We of order unity. For example, to get
the value We = 1, which gives # 4 0.57, we would require the
electric field E0 4 8.18! 105 V m−1. Although this value is
large, it is significantly less than the critical value for the
dielectric breakdown of air, 3 ! 106 V m−1, and can be
achieved in experiments !note that in their experiments on
electrified liquid films, Griffing et al. #14$ used a vacuum
system and were able to achieve very strong electric fields in
excess of 1.2! 107 V m−1". The velocity scale U0 turns out
to be 0.028 m s−1. Finally, we note that the dimensional time
t̃ is related to the time t in the ngKS equation through the
formula t̃ = At, where the time scale A is defined by
A=
75h0
.
64U0R2C
!52"
For a 0.55-mm-thick film of tetraethylene glycol, A turns out
to be 7.38 s.
VIII. CONCLUSIONS
A weakly nonlinear equation has been derived and studied
numerically and analytically to describe falling film flows at
small Reynolds numbers in the presence of both dispersion
and electrostatic effects. The model extends the known dispersive !or generalized" Kuramoto–Sivashinsky equation
!denoted by gKS" to a nonlocal equation with an additional
linear pseudodifferential part due to electric field effects. The
electric field is energy supplying and causes linear short
waves to grow the fastest. Surface tension dominates at large
wave numbers and damps them, thus keeping the nonlinear
problem well posed. We have carried out extensive numerical experiments in order to characterize the effect of the electric field on the gKS equation. The results can be summarized as follows: in the absence of dispersion and electric
fields, chaotic dynamics result as long-time solutions of the
model if the length L of the domain is sufficiently large
!these are the familiar phenomena associated with the
Kuramoto–Sivashinsky equation". Starting with such a cha-
016322-11
PHYSICAL REVIEW E 82, 016322 !2010"
D. TSELUIKO AND D. T. PAPAGEORGIOU
otic state of the KS equation, an increase in the dispersion
coefficient beyond a certain threshold value that depends on
the length L regularizes the dynamics into a traveling wave
composed of pulse solitons of the gKS. On the other hand,
steady-state solutions of the KS equation that are supported
on sufficiently small lengths L can become chaotic if the
electric field parameter is increased sufficiently. These chaotic states can in turn be regularized into a train of traveling
waves by increasing the value of the dispersion parameter.
We have illustrated numerically that the physical parameters
representing dispersion and electric field strength can be
tuned to guide the dynamics from complex to regular behavior, and vice versa.
We have also analyzed pulse soliton solutions of the gKS
in the presence of the pseudodifferential operator represent-
The work of D.T.P. was supported in part by Grant No.
DMS-0707339 from the National Science Foundation.
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ing the electric field. We have constructed such solitons numerically and established that the behavior at infinity is altered from exponential decay to an algebraic one. A
perturbation theory valid for small electric fields has also
been carried out and analytical grounds for the algebraic decay have been given. The change in far-field behavior can be
significant in theories using pulse soliton interactions to describe the dynamics with few degrees of freedom, and further
investigations are left for future work.
ACKNOWLEDGMENTS
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