Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings

+-I@

cJ%&L

PHYSICA hi

I--

EISEVIER

Physica A 228 (1996) 212-235

Computational chaos in the nonlinear Schr6dinger equation without homoclinic crossings M.J. Ablowitz, B.M. Herbst ‘, CM. Schober Programin Applied Mathemutics, Chiuersi/y of Colorado, Boulder CO 80309.

USA

Abstract A Hamiltonian difference scheme associated with the integrable nonlinear Schrodinger equation with periodic boundary values is used as a prototype to demonstrate that perturbations due to truncation effects can result in a novel type of chaotic evolution. The chaotic solution is characterized by random bifurcations across standing wave states into left and right going traveling waves. In this class of problems where the solutions are not subject to even constraints, the traditional mechanism of crossings of the unperturbed homoclinic orbits/manifolds is not observed.

1. Introduction In recent

years there has been considerable

interest

in the study of perturbed

integrable

nonlinear wave equations with periodic boundary values. An important feature associated with this class of problems is that the integrable problem has large classes of exact solutions, which can be written in terms of Riemann theta functions of arbitrary genus and are quasi-periodic in time [ 21. Unfortunately these solutions are rather complicated and it is difficult to analyze qualitative behavior in all but the simplest cases. For many integrable problems of interest (e.g. the nonlinear Schrodinger, sine-Gordon and modified Korteweg-deVries equations) there is a special class of solutions (they are limits of the Riemann theta function solutions) referred to as homoclinic solutions which limit to N-phase solutions for long times. Homoclinic solutions are especially interesting since it is well known that they can bc the “seeds” for the chaotic evolution in a perturbed problem [9,12]. Since the ’ Permanent address: Department of Applied Mathematics, University of the Orange Free State, Bloemfontein 9300, South Africa.

Elnevier Science B.V. .~.~DI0~78-4~71(9S)00434-3

homoclinic

solutions

the perturbation

M.J. Ablowirz

et al./Physica

are unstable,

frequent

A 228 (1996) 212-235

and continual

213

homoclinic

crossings

due to

can occur, and this gives rise to what is usually referred to as homoclinic

chaos. In fact, for the focusing Nonlinear iu, + IA,, + 2(4u

= 0,

Schriidinger

equation

(NLS), (1)

under periodic boundary conditions, u(x + L, t) = u(x, t), we have demonstrated that, given appropriate initial data, perturbations due to either truncation or even roundoff effects in numerical schemes can lead to numerically induced chaos [ 931. This chaos as a result of homoclinic crossings is conceptually analogous to the way chaos is generated in many lower-dimensional dynamical systems, though in this case far richer and more complicated homoclinic structures are available [ 10,6]. However, in all of the previous studies a symmetry condition, i.e. evenness u( x+ L, t) = u( X, t), was imposed. Generic solutions do not have this even symmetry. In physical applications, evenness is difficult to impose or maintain. For example, experimental noise can lead to growth of asymmetries and it is therefore relevant to examine the behavior of solutions in the full NLS phase space. In this paper we study the effects of relaxing the symmetry condition on the near integrable dynamics. In particular, we ask the following questions: (i) Is chaotic dynamics readily observed (if at all) when considering the noneven situation? (ii) In general, do crossings of the unperturbed homoclinic manifolds occur‘? (iii) If chaotic evolutions occur, what is the underlying mechanism responsible for this behavior? Using as a prototype a Hamiltonian difference scheme (which by itself is a physically significant equation) associated with the NLS equation, we observe that (i) the evolution appears to be chaotic and is as readily observable as in the even case; (ii) homoclinic crossings are neither necessary nor generic; (iii) a class of standing wave solutions are the underlying entities which provide the bifurcating solutions across which the evolution obtains. The various “sides” of these standing wave solutions, which are left/right going traveling waves, assume the role of what formerly was the “inside/outside” states on the various sides of the homoclinic manifold. These results therefore indicate that integrable Hamiltonian nonlinear wave equations such as the nonlinear Schrodinger equation can be very sensitive to perturbations, leading to irregular and apparent chaotic dynamics without unperturbed homoclinic crossings playing a role in the dynamics. It is easy to specify initial data in which a chaotic evolution is observed without the occurrence of homoclinic crossings and to enhance the complexity of the chaotic evolution. In the next section we show, using perturbation analysis on typical initial data, that when we do not restrict consideration to the subspace of even solutions, the phase space is not separated by homoclinic manifolds into disjoint invariant regions. Consequently the mechanism of homoclinic chaos due to homoclinic crossings is initially lost. In the subsequent evolution the numerical solutions show no indication of homoclinic crossings. Moreover, there are no restrictions on the spectral data to force these homoclinic crossings. In the last section we examine in detail the effect of relaxing the evenness on numerical solutions of the NLS equation. Using initial data which are (noneven) harmonic perturbations of a constant state, we compute the solution and

214

M.J. Ahlowit~

the relevant associated do not find homoclinic

et al./Physica

A 228 (1996) 212-235

spectral information. As mentioned above, in the dynamics WC crossings, but in its place we observe intermittent crossings of

standing wave states. Thus we have observed and we discuss a scenario which extends the mechanism by which integrable nonlinear wave equations can be chaotically excited by Hamiltonian or damped driven perturbations and one which WC believe should be experimentally

observable

2. Perturbation

(e.g water waves, nonlinear

optics).

analysis

The NLS equation

arises as the compatibility

condition

between

two linear operators

[41, CC.” =

C(r)

-14 d/dx + iA f?/d.x 1/*

(

--

iA

a/l% - i( lIljZ ~ 2h2) -itr:

+ 2A11*

(2)

> ’ -it4,, - 2h1i d/at 4 i( 1,112- 2A2) >

(3)

The operators CC.‘.‘) depend on u,u,, where 11(x, I) is a solution of the NLS equation, and a complex parameter A. The solutions of the NLS equation are characterized in terms of the spectrum of fZ(.‘) dchned by a( c’-“)

:= {A t @lC (‘)u = 0, JuJ bounded

VX}

(4)

Since the potential u(x.,~) solves the NLS equation and is of spatial period L, the spectrum is obtained using the Floquet theory. Starting with the fundamental solution matrix, M(x; N, A) defined by the conditions P)(14,

A)M

=

0.

M(O;n,A)

=

;

:’

( we

,

(5)

>

introduce the Floquet discriminant A ( II, h) := trM( L; 14, A). The fundamental property of A is that it is entire in both its arguments.

Ihr a fixed /\, A is invariant $A(ll(f).

along solutions

Morcovcr.

of the NLS equation,

(6)

A) = 0.

Since A is invariant and the functionals A(A, u), A(A’,lc) arc pairwisc in involution, A provides an infinite number of commuting invariants for NLS. The level set M,, on which 11resides is implicitly defined by M,, = {r,tF’lA(r:,A)

In determining the spectrum of C(“‘(14, yields the following characterization: ,(C“‘)

(7)

=A(14,A)b”A}. A),

:= {A E @~A(u, A) is real and

condition

- 2 5

(4) for bounded eigenfunctions

A(14, A)

<

2).

(Xl

M.J.

We distinguish (i)

Ahlowitz

the following

et al./Physica

(iii)

212-235

21.5

points of the spectrum:

The simple periodic spectrum os = {A:J:lA(h,u) = l 2, dA/dA

(ii)

A 228 (1996)

Critical points of spectrum

# 0).

(9)

h’, specified by the condition

dA(q; A)/dhjh=Ar = 0.

Double points of the periodic spectrum ad = {A;lA( A, 14) = 12, dA/dA

= 0, d2A/dh2

+ 0).

( 10)

The nonlinear spectral transform can be used to interpret any solution in terms of a set of nonlinear modes. Each periodic eigenvalue corresponds to a nonlinear mode whose structure and dynamical stability is completely determined by the location and order of the periodic spectrum. The location of the double points plays a particularly important role in the geometry of the phase space. Real double points label inactive nonlinear modes whereas complex double points are associated with linearized instabilities of the NLS equation. The complex double points also label the orbits homoclinic to the unstable solutions [ 11. As a concrete example, consider the plane wave solution, u(x, t) = ae2i]a]21. For the plane wave, the Floquet discriminant is given by A=2cos(&*+A2L) and ~r(fZ(~)) =!RU[-

(11) ia, ia]. In this case, the periodic spectrum

is

2

/+

( > y

-

(12)

a*.

The simple periodic spectrum is &tilu( (n = 0) and the remaining periodic spectrum consists only of double points (see Fig. 1). As mentioned earlier, it is important to distinguish between the complex double points (i.e. n = 1,. . , [d/n-], where [ 91 = max integer< 4) and the real double points, (i.e. n = [uL/n-] + 1,. . .). In the numerical study we consider initial data that are small perturbations of modulationally unstable plane waves. The initial conditions are chosen to be in the “one or two complex double point regime”, i.e. when [d/r] = We are interested in the fate of bations since they characterize the when symmetry conditions are not

1, 2. the complex double points under non-even perturhomoclinic manifold. In near integrable dynamics, imposed, we ask: do chaotic evolutions develop as

in the even case? If so, what is the underlying mechanism? Can it be characterized by homoclinic crossings? To begin to address these questions, we determine the spectral configurations of an E neighborhood of the plane wave at t = 0. Initial data of the following form is representative of states near the plane wave: I4 = 14(O)

+

EU(

=u + E [exp(ioi) where a, r,,, ical solution

131

(13)

‘)

cos(p,x)

+ v,exp(i&)

sin(p,,x)]

,

and 192are real, and ,u,, = 2rn/L. Under small perturbations evolves to superpositions of these representative states.

(14) the numer-

216

M.J. Ahlowitz et al. /Physicu

-

= =

x Fig.

I.

of

band double

A 228 (1996) 212-235

spectrum

point

Spectrum of the plane wave u = ur’2/a12r

The discriminant A (A, U) and the eigenfunctions arguments. Therefore, to determine the spectrum perturbation expansions:

arc entire functions of each of their for ( 13) we assume the following

/\ = A’()’ + CA(‘) + . .

(15)

“(0) + E”(I) +

(161

and L’

=

..

This type of perturbation analysis was also used to compute the splitting of double points for even solutions (see e.g. [3] ). Substituting these expansions into the spatial operator, (2), and collecting terms at the various orders of E, we obtain at 0( E’), p.~)y(o) = (),

(17)

where

and at O(E), p)“(

_iA(‘)ujO’

I) = {

-i/p’u;o’

+ u(l)L,:o), (19)

+ nm*L,/o).

At a double point, A = h,, the eigenspace is two dimensional. the leading order problem (17) is given by u(O) = A+$+ + A-$-.

(

of the plane wave given by 1

(i/n)(ik+A)

and where k,f = A: + a2 = jv/L.

of

(20)

where q%* are the eigenfunctions $* = exp( fikx)

The general solution

)

M.J. Ablowitz et al./Physica

Proceeding anti-periodic

to O(E),

A 228 (1996) 212-235

we note that the solvability

eigenfunctions,

condition,

217

assuming

L-periodic

or

for the system

CCx)u = G,

(21)

with

is given by the orthogonality

condition

L

s

(G,wr

+G2w;)dx=0

0

for all w in the null space of the Hermitian

operator, CH, where

(22) It follows that the null space of LH is spanned by

<(cl,“>* (G&9* )’

(23)

and the general solvability

condition

assumes the form

(24)

, i2-

The solvability

condition

(k+A’O’P+ 2%

applied to ( 19), yields the following

system of equations:

2A(O) -A(I)

Lp

2 A+

*(,,

, (-k+h)2 2

a

a2

p*++*

(

A-

=fJ >

(25) ’

1

where cz = (exp(iOt) /3= (exp( -iOt)

+ ir,exp(i&))

,

f ir, exp( -292))

(26) ,

(27)

and k = i,u,, (for k # ip,,, CY= p = 0). This condition on k means that a specific double point is selected by the perturbation. After some algebraic manipulation one finds in the case of the imaginary double points, that nontrivial solutions, A*, exist provided

218

M.J. Ablowitz

-$Lsin(HI (A”‘)2

=

+

t-4)

et trl. /Phy.Gctr A 228 (1996) 212-235

sin(4

- 0,)

sin(f% f 4) sin(q5 -- 0,) + ir,, sin(& - 8, ) sin(24)] rz

if k = ,u,,/2, if k f

0

(28)

~,,/2.

where tan q5 = Im( A(‘)) /k. Note 1. Since the real axis is always

the spectrum and (r(C) has the following symmetry. A E C(C) then h* g c(C), only the spectrum in the upper half A plant will hc displayed in the spectral plots throughout the remainder of the paper. Note 2. The above analysis

is valid when the spectrum of the plane wave contains an arbitrary number of complex double points, however we restrict our consideration to the relevant “one or two complex double point regimes”.

Remarks. _ If 81 = 192or 81 = 82 5 7r , i.e. II is even, then this is the situation

studied before 131. WC summarize these earlier results in order to compare them with the asymmetric case. The right hand side of (28) is real and can be either negative, zero or positive. If A”’ = 0 the double point does not split and we ohtain a solution homoclinic to the plane wave which therefore belongs to the isospectral set of the plant wave (see Fig. 2a). The homoclinic solution is characterized by a single mode which limits to the unstable plane wave as t -+ &co. When the complex double point splits into two simple points, it can only splits in one of two directions: along the imaginary axis (gap configuration, see Fig. 2b), or symmetrically about the imaginary axis (cross configuration, see Fig. 2~). This is a realization of the saddle structure of the real part of A when A(“’ is complex. Solutions 2b and 2c can be thought of as “inside” and “outsidc” ohit

the homoclinic

separates

orbit, respectively.

the symetric

subspace

In the synmetric

case, the hnmclinir~

into disjoint invariant .sul~rnanifolds. For a

rigorous discussion, see [ I 1. Due to the analyticity of the discriminant A, under small perturbations it is possible to cvolvc from the one configuration to the other, while maintaining the even symmetry, only by passing through the complex double point, i.e. by crossing the homoclinic manifold. _ If 01 # 02 and 01 f 02 f ?r, i.e. u is asymmetric and r,! is nonzero, then the right hand side of (28) is complex. In this case, setting A’ ‘,*I = ir’/*e’J’/‘, where p can take on any value between 0 and 277-,the double point can now split in any direction. Examining the behavior of A( 14, [) in a neighborhood of 14~“) we show (see the appendix) that the band of continuous spectrum breaks at the double points which are resonant with the perturbation, i.e. A”) is no longer an clement of spectrum (unless a special subclass of even solutions, allowing translations, is being considered). Thus. when II(‘) resonates with a particular mode, the corresponding double point splits in such a way that an asymmetric version of the “cross” state can not occur. The spectral

M.J. Ablowitz

et al. /Physicn

219

A 228 (1996) 212-235

h--plane

;;ii -0.4

PO.2

0.0

0.2

0.4

0 2

0.4

0 2

0.4

h-plane

,jii PO.4

-0.2

0.0

h-plane

,i”ri PO.4

Fig. 2.

(a)

NLS, A”)

The surface lu(x, r)

/ and

0.0

the nonlinear spectrum with one double point for a homoclinic solution of

1and the nonlinear I + ,I cospx). (c) The

= 0. (b) The surface lu(x, t)

wave solution of NLS, UC,= O.S(

PO.2

spectrum with one imaginary “gap” for a standing surface lu(x, t)I and the nonlinear spectrum with

one “cross” for a standing wave solution of NLS locked in the center and wings, ~0 = 0.5(

configuration

is determined

by

the

location

I + I ices

px).

of A+, where A(*) E A(‘) + ,A(‘,*).

At

also determines the speed and direction of the associated phase. In the one complex double point regime there are only two basic spectral configurations associated with non-even perturbations: For 0 < p < r, Re A+ > 0 and Re A- < 0. The resulting upper band of spectrum lies in the first quadrant and the lower band lies in the second quadrant. The wave form is characterized by a single mode traveling to the right (“right state”). See Fig. 3a. For 7r < p < 25-, the situation is reversed. Re A+ < 0 and Re A- > 0. The resulting upper band of spectrum lies in the second quadrant and the lower band lies in the first quadrant. The wave form is characterized by a single mode traveling to the left (“left state”). See Fig. 3b. Since the spectrum can evolve from the one configuration to the other without passing through a double point, it appears that the homoclinic orbit does not separate the full NLS phase space.

220 -

M.J. Ahlowi~z et ul./Physicu

Using

the same methodology

A 228 (1996) 2/2-235

as [ 31, which treats the even situation,

the above

analysis can be carried to higher order. This generalization is significant when there is more than one complex double point present initially. In the two complex double point regime (the initial data used in Fig. 4, At and A2 split into Ai*’ and A:*‘. We obtain the following spectral configurations in the noneven case: If A:, Ai arc in the same quadrant, the two modes in the multi-phase solution travel in the same direction (with different speeds). In Fig. 4a, Re A: < 0 and Re A: < 0; the two modes travel to the left. In Fig. 4b, RCA: > 0 and Re Al > 0; the two modes travel to the right. If A:. At arc in different quadrants, the two modes in the multi-phase solution travel in opposite directions. In Fig. 4e, the Re A: < 0 and the Re AZ > 0; the first mode travels to the left and the second mode travels to the right. In Fig. 4f, the Re A; > 0 and the Re Ai < 0; the first mode travels to the right and the second mode travels to the left. It is possible to split only A2. In this case, we obtain a solution homoclinic to a traveling wave. In Fig. 4c, the RCA: > 0 and the wave travels to the right. In Fig. 4d, the RCA; < 0 and the wave travels to the left. Notice the similarity with the solutions in Figs. 3a.b. For a discussion of the spectral configurations and their topological description in the two double point regime in the even case, see [ 6 1. A description of the topology of the level sets in the one and two double point regime in the nonevcn case is in progress. In this paper our goal is to describe the new phenomena introduced by asymmetric perturbations and at this stage a topological classification is not yet available. The main result of this paper is that for near integrable dynamics in the noneven situation the evolution of the spectrum between the two distinct configurations for left and right traveling waves generically does not involve complex double point crossings. In the one double point regime, generically the spectral configuration evolves from the “right” state to the “left” state by crossing a nearby cross state (e.g. see Fig. Sb) or gap state (e.g. see Fig. 6c), rrat by moving through a double point. Similarly, in the two double point regime, the evolution of the spectrum from one configuration to another (see Figs. 4a-d) can be accomplished by executing an appropriate sequence of crossings of gap and cross states and does not entail homoclinic crossings. Consequently, if chaotic evolutions develop in the noneven case, generically they will no longer be characterized by homoclinic crossings. In the next section we show that a chaotic evolution can develop in nearby systems and is produced by a new mechanism involving random bifurcations through nearby standing waves.

M.J. Ahlowirz

et d./Physica

A 228 (1996) 212-235

0.6/

221

h-plane

h-plane

Fig. 3. (a)

“Right” state; the surface lu(x, I)

1and

the nonlinear spectrum for a right traveling wave solution

of NLS, u,, = O.S( I + O.OS(rwi cos /AX + eni sin fix) ) (b) “Left” state; the surface ju( x. t)

( and

spectrum for a left traveling wave solution of NLS, ug = O.S( I + O.OS( e’“cos p.r + e3” sin ,u)

3. Numerical

the nonlinear ).

experiments

We numerically integrate the NLS equation using the following conservative scheme (DDNLS) : iti,, + (u,+, + U,_I - 24

/h* + 214J2u,, = 0,

difference

(29)

where h = L/N, u,j+N = u,i. This is a simple scheme which can be used to study the shortmoderate time scale effects of Hamiltonian perturbations on NLS dynamics. Moreover, (29) has physical relevance. For the time implementation of this discretization, we use a fourth order adaptive Runge-Kutta algorithm in the NAG library (D02BBF). The DDNLS system may be considered as an O(S = h*/12) perturbation of the integrable NLS equation. Thus the near-integrable dynamics of (29) can be interpreted in terms of the known dynamical properties of the NLS equation through an “integrable nonlinear diagnostic”: i.e., we examine the evolution of the nonlinear spectrum under the perturbation induced by DDNLS. At each time t, we take the numerically generated solution {z*,,(t) In = I,. . . , N} and perform the direct spectral transform by numerically determining v(A). The initial data used in the numerical study is close to a homoclinic manifold represented by either one or two complex double points as explained in the previous section.

222

M.J.

Ahlowitz

et ul. /Physicu

A 228

(1996)

212-235

h--plane 0.6

Fig. 4. The surface double

point

Irc(.r, I) 1 and the nonlinear

regime”

(i.e.

la =4&n).

11,) = O.S(

(c)

u,, = O.S( ltO.OSicos2p.r):

(e)

II,, = O.S( l+O.Os(2e~“’

(d)

that

points

and that the manner

under

small

inary

we monitor

and present

DI(A+,A-)

2p.r)

=

equation

in the “two

complex

by

110 = O.S(l+0.2(e”“cos~.~+sin~..r)): ); (f)

data, when about

point

-A_1

u(b = 0.51 l+0.0S(rcos~x+2r~‘~‘sin

double

depends splits

point

splits

a “gap”

of this splitting -AL)

distance distance, =O,

if Im( A+ - A_ ) = 0.

into two simple

the perturbation

is even

- A- ) is either

(A,

state along

axis into a “cross”

of the splitting

ifRe(A,

splits

into either

2p.r)).

point

on whether

the double

the imaginary

measure

+lA+ - A-1 -)A,

of the NLS

are given

each complex

the evolution

a signed

for solutions

conditions

of the splitting

i.e. the double

axis or symmetrically

in time,

(b)

perturbations

For even initial

or pure imaginary; cxperimcnts

initial

UCI = O.S( l+0.1e3”‘sin2~.r);

sm pcLx+icos

Recall

or noneven.

spectrum

The

I +0.2(cos~,Y+r”“’smpx));

(a)

v-1

state.

D( A,, A-)

real

the imagIn the

even

= /A, ~ A- /

i.e.

(30)

M.J.

Ablnwitz

et cd. /Physicu

A 228 (I 996)

212-235

223

(f1 Fig. 4 -continued

Positive and negative values represent gap and cross states, respectively, and a sign change represents a homoclinic crossing. In the noneven case, ( A+ - A_ ) has, in general, both a real and an imaginary part. The spectral measurements which are relevant in the experiments using noneven initial data are: the splitting distance D( A+, AL) and the “vertical” splitting, denoted V( A+, A-). The vertical splitting is a signed measure of the splitting distance Im( A+ - A-) where we correlate the sign with left and right traveling wave states, i.e. V(A+,A_)

=

+Im( A+ - A_) -Im( A+ - A-)

ifReA+ ifReA+

> 0,
(31)

For V > 0 the wave is right going, for V = 0 the wave is standing and for V < 0 the wave is left going. For a homoclinic crossing or double point crossing to occur, A+ and A_ must coalesce and D(A+, A-) = 0. For D(A+, A-) # 0, when the vertical

224

M.J.

Ahlowitz

et al. /Phy.sim

A 228 (1996)

splitting passes through zero the spectral configuration

212-235

evolves through a cross state and

the waveform bifurcates across a standing wave solution into (either left or right going) traveling waves. In the noneven case, the spectrum can also change configuration by evolving through a gap state. This can be detected by examining the Re( A+ -A_ ). However, here we will not present the evolution of the splitting of Re( A+ - A-), since in the experiments the crossing of gap states did not have a significant effect on the dynamics of the solution.

The me cmplex double point regime for a left traveling wave solution,

We begin by considering

110= 0.5( I + 0.01(e”~90’cos~ux- + e6”sinpux)),

asymmetric

initial data

(32)

with L = 2\/27~,~ = ~z-/L. Fig. 5a shows the surface (0 < t < 500) obtained with the discretization defined in Eq. (29) for N = 24. Initially the waveform travels to the Icft; as time evolves the perturbation induced by the discretization causes the waveform to jump between left and right traveling waves. We observe that this bifurcation occurs randomly and intermittently throughout the entire time series for 0 < t < IO, 000 CFig. 5a also shows the surface for 5000 < t < 5500). As alluded to previously, the bifurcation between left and right traveling waves occurs when the spectrum evolves through a nearby cross state (corresponding to a standing wave solution) and not by executing homoclinic crossings. In the surface plot (Fig. 5a) the first bifurcation from a left to a right traveling wave occurs at t = 131.8. To illustrate how the spectrum changes configuration when the waveform bifurcates, in Fig. 5b the evolution of the spectrum due to the perturbation induced by DDNLS is shown at three successive time slices in this transition region when the waveform bifurcates from a “left” state (t = I3 I .5) to a “right” state (t = 132. I ) by evolving through a “cross” state (t = 1.31.X). The evolution of the spectrum for 0 < t < 500 under the DDNLS llow is shown in Fig. 5c. From the evolution of D( A,., A-) it is clear that the spectrum is bounded away from complex double points and that homoclinic crossings do not occur. On the other hand, the evolution of V( A,, A-) shows that the solution is characterized by random intermittent bursts in which the solution bifurcates between left and right traveling waves. The intermittent random bifurcations between left and right traveling waves occur during long time intervals. The evolution of the spectrum for (5000 < t < 5500) for initial data (32) is given in Fig. 5d. Further numerical experiments, in which the perturbation strength is varied, indicate that as the strength is increased the intermittent bursts are replaced by more frequent random bifurcations. Note. Numerical solutions in the noneven regime arc sensitive to perturbations of the initial data. In additional experiments, if the initial data are varied slightly or even if the time step in the integrator is modified slightly, the solution bifurcates at different times which are apparently totally uncorrelated to the previous bifurcation times.

M.J.

Ahlowirz

ef nl./Physica

A 228 (1996)

212-235

22.5

i_t,LL_ h-plane

J--7

-0.0s

0.0

t=

, ,.. t _._.__,_ ,,,.,.,.1

/.,I OD

-.i

&,

,111. _-Do

_JD_

T/.___

111.1..

----.z

e_yy

il

0.05

132.1

---+---

IJC,

cloy _1.__-

Fig. 5. (a) Surface for ~(1 = 0.5( I + 0.01 (c”.~’ cospx + e”‘j sm fix)) obtained with the difference scheme (29) for 0 < f < 500 and 5000 < t < 5500. Note that the first bifurcation from a left to a right traveling wave occurs at t = 131.8; (b) The nonlinear spectrum at three time slices showing that the transition from a “left” state (r = 131.5) to a “right” state (t = 131.8)occurs when the spectrum evolves through a “cross” state (f = 132. I ). (c-d) The evolution of the vertical splitting V( A +,A_)andofD(A+,A_)forO
When the numerical solution for initial data (32) is obtained with discretization (29) for N = 20, the spectrum evolves through cross states and occasionally through gap states. Fig. 6a illustrates how the spectrum can change configuration by evolving through a gap state. However, in these experiments, U( 1) bifurcations were always observed to occur when the spectrum evolves through a cross state. For spectral configurations very close to a gap state, the speed of the corresponding traveling wave is very small. Since

h-plane

Fig. 6. (a) “left”

the nonlinear

h-plane

spectrum

at three time slices showing

state can also occur when the spectrum evolves through

II,) = O.S( I + 0.01 cospx).

N = 24. obtained

with difference

that the transition

n “gap”

state. (b)

scheme (29)

from :I “right”

state to n

Surfncc for even initial

for 0 <

data

I c- 500.

the observed transitions through gap states are small and not long enough in duration (given the small speeds under consideration) there was no significant effect on the waveform. In all the experiments we considcrcd, crossings of gap states occurred less often than crossings of cross states and they did not have a significant effect on the solution. Let us, for the moment, restrict ourselves to the subspace of even solutions in the one double point regime. Consider initial data for a standing wave solution whose spectral confguration is a gap state, 110= O.S( I + O.ol(cos~ux)),

(33)

with L = 2&7r,p = 277/L. Fig. 6a shows the surface of the standing wave obtained with DDNLS for N = 24 for the time slice (0 < r < 500). The solution displays no irregularity and persists as a standing wave locked in the center for the duration ol the experiment (0 < t < 5000). Although homoclinic crossings occur, they arc not significant (large enough in amplitude or long enough in duration) to have an effect on the solution. In the even case, the nonlinear instability associated with one complex double point evidently is insufficient to produce chaotic motions in conservative near integrable dynamics. Although homoclinic crossings were not observed in any of the experiments using noneven initial conditions, the proximity to the unstable homoclinic structures appears to play a role in the near integrable dynamics. For example, if the initial data for a traveling wave is chosen farther from the homoclinic orbit,

M.J. Ahlowitz

Fig.7.

(a),(b)

scheme (29)

et nl./Phy.sicu

221

A 228 (I 996) 212-235

Surfaceforinitialdatau~~=0.5(1+0.1(e0~”’cos~ux+e

6~‘~sin ~CLX) ) obtained with difference

for 0 < I < 500 and the corresponding evolution of the spectrum for 0 < t < SOO. Note that

the solution is a right traveling wave and no bifurcations occur. Correspondingly, the vertical splitting distance has no zeros and the spectrum is locked in a “right” configuration.

uo=0.5(1

+0.1(e.9icosq+e60i

sin w))

,

(34)

with L = 2&r, ,U = 2r/L, and the same perturbation strength is applied, i.e. N = 24, then the solution obtained with (29) displays no irregularity and persists in a right traveling wave state for the entire duration of the experiment (0 < t < 5000). Figs. 7a,b show the surface of the persistent traveling wave state and the corresponding spectral

diagnostics for the solution obtained with DDNLS for 0 < t < 500. Fig. 7b shows that D( A+, A- ) is bounded away from zero and crossings of standing wave states do not occur in the evolution of V( A+, A_). From this last example it seems reasonable to pursue an analysis to establish persistence of traveling wave solutions that are sufficiently far away from homoclinic structures under the DDNLS perturbation.

The two complex double points regime Next we consider multi-phase solution with two excited modes, 110= O.S( I + 0.2( eO.‘)‘cospx + e60i sin P-X) ),

noneven

initial

data for a

(35)

218

Mel. Ahlowirr

et trl. /Phwicu

A 228 (I 996) 212-235

where L = 4&r, p = 2r/L. Using the discrete scheme (29) and N = 64, the solution is observed to be quite irregular. Fig. 8a shows the surface for (35) obtained with the difference scheme (29) for 0 < t < 100 and 200 < t < 300. An intcrcsting feature of the two complex double point regime is that although the solution is a multiphase solution, in long time simulations the solution appears to degenerate into a single coherent

structure

which bifurcates

(Fig. Xh) shows that the dominant

from left to right. The surface contour

feature of the waveform

is a coherent

structure

plot that

begins by traveling to the left and then reverses it’s direction of motion between left and right traveling states. The spectral diagnostics, the splitting distances V( A+, A- ) and D(A, , A_) are shown for each complex double point for 0 < t < 1000 in Fig. 8c. As in the one double point regime, we do not observe crossings of the underlying homoclinic manifold. Therefore, the absence of homoclinic crossings in the one double point regime is not an isolated example. Intermittent bursts of crossings of underlying gap and cross states occur. The evolution of V( A_+,A_ ) for the two modes shows that the bifurcation activity is dominated by the first mode for this period of time. One can view the speed of the traveling waves as the significant paramctcr in the asymmetric case and that the zero speed wave (corresponding to standing waves) is the bifurcation state. In contrast with even solutions in the one double point regime, even solutions obtained with (29) in the two double point regime are characterized by temporal chaotic behavior [9.12]. Consider the following even initial data for a multi-phase standing wave: 110= O.S( I + 0.2COS~.X),

(36)

where L = 4&n-, p = 27r/L. Using discretization (29) with N = 64, an extremely irrcgular waveform develops. In this numerical experiment the even symmetry is preserved by working on half the s-interval, 0 < x < L/2, and imposing a symmetry condition at .Y= L/2. Fig. 9a shows the waveform given the initial data (36) for the time slices 0 < t < 100 and 900 < t < 1000. Fig. 9b,c show the corresponding evolution ol Dt (A,, A- 1 for the first and second modes for 0 < t < 500 and 500 < t < 1000 respectively. The temporal chaotic solution obtained in this regime is characterized by random intermittent homoclinic crossings which, as in the noneven case, become more frequent as the perturbation strength is increased. Note. The frequency of the bifurcations across standing waves in the noneven cast is comparable to the frequency of homoclinic crossings in the cvcn cast when the experimental parameters are the same, i.e. the number of lattice points N is fixed and the initial data is the same distance from the underlying homoclinic structures. While in the class of cvcn solutions, the chaotic dynamics appears to be related to the hrcakup of the underlying homoclinic structures, in the non-even situation scvcral novel fcaturcs appear. The role of the separatrix (the unperturbed homoclinic manifold in the even case) is now played by a class of standing wave solutions and there is a continuum of standing wave states available. The random flipping of the wave form carries informatton about the speed and the direction of travel, producing a richer structure in the chaotic Ilipping.

M.J. Ahlowitz

et al. /Physics

A 228 (1996)

212-235

229

(4 500 400

2 F

300 200 100 0 0

10

20

30 40 SPACE

50

60

(b)

Fig. 8. (a)

Surface for u() = O.S(

I+

0.2(e”~“‘cos~~

+ ehoi smj~~)) obtained with difference scheme (29)

for 0 < f < 100 and 200 < t < 300; (b) Contour plot for initial data (3

I)

for 0 < t < SO0 (c) the evolution

ofV(A+.A_)andD(A+,A-)formodelandmode2forO
If evenness is not enforced in the numerical computation, then a loss of symmetry occurs and the chaotic solution is no longer characterized by homoclinic crossings. Our earlier studies in effectively chaotic regimes indicate that even round off error is capable of triggering the growth of asymmetry [5]. The surface for the even initial data (36) obtained with the difference scheme (29), N = 64, is shown in Fig. 9d for

M.J.

Ahlowitz

et (11. /Plrysiccr

A 228

(1996)

212-235

‘1 (4 Fig.

9. (a)

Surface

scheme (29) D, (A,.

for even initial

t < 100and

data IQ, = 0.5( 900 < t <

1000.

1 + O.~(COS~.~~)). N = Even symmetry

A- ) for the first and second modes for 0 < t i

for even initial

In this

for 0 c

data ~(1 = O.S( I + 0.2coq q).

500 and 500 c;

t<

N = 64, obtained with diffcrcnce

experiment. since evenness is not enforced. function develops two components.

truncation

64, obtained

is preserved.

(b).(c)

with

1000. respectively. schcmc (29)

effects lend to loss of symmetry

difference

The evolution (d)

for 0 <

ol

Surface

t < 250.

and the splitting

M.J. Ablowitz

et al. /Physica

A 228 (1996) 212-235

231

0 < t < 250.In this case the use of Dt is not appropriate for 0 < t < 250 since there is a qualitative change in the phenomena during the experiment. In Fig. 9d we present the evolution of DI until the asymmetry reaches 0( IO-‘) (the accuracy of the nonlinear spectral solver) and (A+ - A_) has a real and an imaginary part. Thereafter we indicate that the splitting function has developed two components by presenting IRe( A+ - A_) 1 and -IIm(A+ - A_)1 simultaneously. The solution is now characterized by bifurcations across nearby standing wave states. In the even case, when the NLS is damped and driven, chaotic behavior develops in the one double point regime. In this case, an interpretation of this behavior using symbol dynamics has been proposed [ I I 1. The expression “symbol dynamics” refers to the cxistcnce of an invariant set in the phase space which is topologically equivalent to a set of all symbol valued sequences. Possibly, a symbol dynamics interpretation may be feasible in the noneven situation. In the noneven case, the sequence might take on values of (R) for right going waves and (L) for left going waves. The random bifurcations in the waveform between left and right traveling waves might then be represented as a shift on this sequence space. These examples highlight the essential role symmetries play in near integrable dynamics. Perhaps the most important feature is that it is as easy to observe chaotic dynamics in the full NLS phase space as in the restricted subspace of even solutions. Homoclinic crossings do not appear to be associated with any of the irregular waveforms examined in the numerical experiments in the noneven case as they were in earlier studies when even initial data was considered. When restricting consideration to the invariant subspace of even solutions, a Mel’nikov analysis indicates that there is no persistent transversal homoclinic orbit in the one complex double point regime in agreement with the numerical observations of regular waveforms [ 81. In the two complex double point regime, an analogous Mel’nikov analysis predicts transversal intersections of the perturbed homoclinic orbits [6]. However, in the noneven case the lack of homoclinic crossings is a truly distinctive behavior. When symmetry conditions are relaxed, as we have seen, a new phenomenon involving traveling waves becomes significant. It is not clear what the entire class of solutions is that can act as bifurcating states in the one and two complex double point regimes. The geometry of the problem and the underlying mechanisms arc currently being investigated.

Acknowledgements

This work is partially supported by the NSF, Grants No. DMS-9404265, the Office of Naval Research, Grant No. N00014-94-0194 and the Air Force Office of Scientific Research. Grant No. F49620-94-0120.

Appendix

A

To establish that A’“) does not persist as an element of spectrum under noneven perturbations, we examine the behavior of A( II, A(“)) in a neighborhood of U(O). F( 10 = A( A”‘)(U), u) can be expanded F(U) = F(P))

in a Taylor series about 11= U(O) # 0.

+ SF(U’“‘)

+ @F(rP’)

+

)

(A.1 )

where 6F and tT2F are computed using variation of parameters. The first variation of Fi has the following representation in terms of the gradient

ot

F, (see [ 1I]): I>

tw,(h,rr)

=

0

h

SF-

---Q%*(x) + L%(x)

&SLi(.Y) dx, >

(A.2)

where (A.?) Since 6Fj/6u

( uc) = 0.

F,(M)=f2+6*Fj(l40)

(A.4)

+“‘.

Therefore the second variation 6*F identifies the spectral configurations that are permissible under perturbation. For Fj = F,! + iF,!, the real and imaginary parts of the Hessian are given by [ I I ] “‘F;

= [(62’d2

S’F;

=2 [S&S&

+ (8X2)‘]

eigenspace,

62‘1 = (i@(+.*‘,

[@a&)

+ (SZd2]

,

(A.5)

- 62’,6,ci], of 8~ (i.e. II(I) in 14) onto the members

where 62’j is the projection linearized

-

of the basis of the

@(*,“‘) (y, A), at A = A,;“,

SU).

62’2 E (-@I,‘-~“&), 6Z3 2 (~@‘+~“,&), S&E

(A.61

(i@‘-.2’,6u).

Each of the projections ly = (qt,Pz), where PC*’ = (4//T)‘* I

8Z.j is real. The following

(l&Y,

“squared

Bloch eigenfunctions”

(A.7)

satisfy the linearized NLS equation; therefore they provide a basis for the linearized cigenspaces. Since the projections need to be real quantities, we consider the following decompositions:

M.J. Ahlnwitz et ol./Phy.k~

A 228 (1996) 212-235

233 (A.8)

where

(A.9) The vectors @C*,.i) form a basis for the linearized eigenspaces. When expanding 614 in terms of the linearized basis, the expansion coefficients are computed by projecting onto this basis. For the plane wave potential under consideration, using an appropriate of the eigenfunctions I+?(20)) the squared eigenfunctions are

normalization

2n2 cos 2k.x

WI =P

2(/\2 + k2)

P- =p

2a2i sin 2kx 2(A2 + k’)isin2kx - 4Akcos2kx

cos 2k.x - 4hki sin 2kx

’ (A.lO)

> ’

where cy2sin2 kL

P=-

k2

The basis for the linearized @(+,I)

[“,1

-p

eigenspaces

is

where t = 2v2 cos 2kx - 2vksin 2kx, where s = 2k2 sin 2kx - 2vk cos 2kx,

@(+.2) = p

w [

@(+.I)

=p

W

z [

The projections

-Z

1 1

where w = 2k2 cos 2kx + 2vk sin 2kx, where z = 2v2 sin 2kx + 2vk cos 2kx.

of 6~ onto the linearized

(A.1 I)

basis are

L 8x3

= -p

2Rdth)

dx = 2Lp(

2WZ6u)

dx = -2Lp(

-v2 cos 8, + vkr, cos 0,) ,

.I 0

L

824

= -p .I 0

v2r, cos O2 + vkcos 8, )

,

1.

6‘,

= -/3

2Re( i~6m)

.

dx = 2L4?( k2 sin 81 + vkr,, sin 82)

Thcrcforc. 8”;;’

= 8/3’L’

[(k’r,,

_ (k’sin If 6’F: h(“)

f

cannot

where

@F,

Im(G’F(~r”“))

The

01) (Y’Y,, cos82 + vkcos@l

vksin

81 + vkr,, sin 02) ( -v2

of spectrum.

= 0 is shown in Fig. A.1 #

The

surface

for a typical

= 0 iff 81 = 02 or 81 = 01 i 0. The

configurations

hand of spectrum

band of spectrum

arc equivalent only

remains

)

cos 0, + vkr,, cm Hz)]

0 for the plane wave then, since A(rr,A) bc an clement

one sees that K’F,’

possible

sin 82 -

unbroken

must he real for A E rr(C”)), of 6’F~(Q1.

Hz) and the level sets

value of T,,, i.e. r,, = 0.7. z-. Thus,

on the imaginary

to the spectral

(A.12)

for nonevcn axis breaks

configurations

for a special

shown

subclass

From

this

perturbations, and the two in Figs. 3a.h.

of cvcn solutions

M.J. Ahlowitz et ul./Physica

A 228 (1996) 212-235

2.35

(allowing translations). These results can be extended to an arbitrary number of complex double points: the perturbation ( 14) will affect the mth harmonic of /.L,,at E”‘.

References I I ] N.M. Ercolani and D.W. McLaughlin, Toward a Topological Classification of Integrable PDE’s, preprint (1992). [ 2 ] A.R. Its and VP Kotlyarov. Dokl. Akad. Nauk Ukain. SSR Ser. A I I ( 1976) 965. 131 M.J. Ablowitz and C.M. Schober, Effective chaos in the nonlinear Schrodingerequation, Contemporary Mathematics 172 ( 1994) 253. 14 1 M.J. Ahlowitz and H. Segur. Solitons and the Inverse Scattering Transform (SIAM, Phyladelphia. PA, 1981). 1S1 M.J. Ablowitz, C.M. Schober and B.M. Herb& Phys. Rev. Lett. 71 ( 1993) 268.1. 161 A. Calini, N. Ercolani, D.W. McLaughlin and C.M. Schober, Melnikov analysis of numerically induced chaos in the nonlinear Schrodinger equation, Physica D 89 ( 1996) 227. 17 1 N. Ercolani. M.G. Forest and D.W. McLaughlin, Geometry of the Modulational Instability. Part I: Local Analysis; Part II: Global Analysis, Memoirs of the AMS ( 1991). preprint. 181 A. Calini and CM. Schober, Mel’nikov analysis of a Hamiltonian perturbation of the nonlinear Schrodinger equation. in: Proc. NATO Workshop: 3D Hamiltonian Systems ( 1995). to appear. 19 1 B.M. Herbst and M.J. Ablowitz, Phys. Rev. Lett. 62 ( 1989) 206.5. I 101 M.J. Ablowitz and B.M. Herbst, On homoclinic structure and numerically induced chaos for the nonlinear Schriidinger equation, SIAM J. Appl. Math. SO ( 1990) 339. I I I I Y.D.W. Li McLaughlin, Comm. Math. Phys. 162 (1994) 349. I I2 I D.W. McLaughlin and CM. Schober, Physica D 57 (1992) 447.