LONG-TIME DYNAMICS OF THE MODULATIONAL INSTABILITY OF DEEP WATER WAVES.pdf

8/11/2019 LONG-TIME DYNAMICS OF THE MODULATIONAL INSTABILITY OF DEEP WATER WAVES.pdf

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Physica D 152153 (2001) 416433

Long-time dynamics of the modulationalinstability of deep water waves

M.J. Ablowitz a, J. Hammackb, D. Henderson b, C.M. Schober c,a Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, CO 80309-0526, USA

b Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USAc Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA

Abstract

In this paper, we experimentally and theoretically examine the long-time evolution of modulated periodic 1D Stokes

waves which are described, to leading-order, by the nonlinear Schrdinger (NLS) equation. The laboratory and numericalexperiments indicate that under suitable conditions modulated periodic wave trains evolve chaotically. A Floquet spectral

decomposition of the laboratory data at sampled times shows that the waveform exhibits bifurcations across standing wave

states to left- and right-going modulated traveling waves. Numerical experiments using a higher-order nonlinear Schrdinger

equation (HONLS) are consistent with the laboratory experiments and support the conjecture that for periodic boundary

conditions the long-time evolution of modulated wave trains is chaotic. Further, the numerical experiments indicate that

the macroscopic features of the evolution can be described by the HONLS equation. Ultimately, these laboratory experi-

ments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems.

2001 Elsevier Science B.V. All rights reserved.

Keywords:Nonlinear Schrdinger equation; Modulated periodic waves; Long-time evolution

1. Introduction

Historically, the study of water waves has provided

researchers with a wide variety of interesting non-

linear phenomena. One of the classical examples of

nonlinear waves was the discovery by Stokes [1] in

1847 of traveling nonlinear periodic wave trains in

deep water. More precisely, Stokes found the lead-

ing terms of a series expansion for the surface wave

displacement, and how the frequency of the wave de-

pended on the amplitude for small but finite values of

the wave amplitude. In 1925, Levi-Civita [2] proved

Corresponding author.E-mail addresses:[email protected] (M.J. Ablowitz),

[email protected] (C.M. Schober).

rigorously that an infinite series describing periodic

waves could be obtained and that it converged. How-ever, the question of stability of these periodic water

waves remained open until 1967 when Benjamin and

Feir [3] established, experimentally and analytically,

that in sufficient deep water the Stokes water wave was

unstable.

The instability result can be obtained by consider-

ing slowly modulated wave trains. It can be shown that

for small amplitude, the leading-order complex ampli-

tude of the surface displacement satisfies the nonlin-

ear Schrdinger (NLS) equation [4,5,9]. The relevant

periodic solution of the NLS equation can be readily

shown by Fourier methods to be unstable in deep

water. Even though the modulational instability of

periodic waves has been established, the long-time

0167-2789/01/$ see front matter 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 7 - 2 7 8 9 ( 0 1 ) 0 0 1 8 3 - X


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M.J. Ablowitz et al. / Physica D 152153 (2001) 416433 417

behavior of the instability has not been completely

investigated. A natural question is whether the NLS

equation provides a satisfactory description of the

dynamics. Our results indicate that the answer to this

question is, in general, negative for periodic wave

trains.

The NLS equation is completely integrable, hasan infinite number of conserved quantities and does

not possess temporally chaotic orbits [6,7]. How-

ever, water wave dynamics are described only to

leading-order by the NLS equation. The results in

this paper indicate that the perturbations to the NLS

equation associated with periodic wave trains gener-

ate chaotic evolutions. Since there is no mathematical

proof of dynamical chaos available, we use this term

only in a broad sense. The higher-order nonlinear

Schrdinger (HONLS) equation, which is obtained by

retaining terms in the asymptotic expansion through

fourth-order (see Eq. (3)), destroys the symmetry ofthe NLS equation with respect to space translations

[18,20]. We show that the higher-order corrections

have a significant effect on the evolution of the insta-

bility and of the wave train.

Earlier studies on near-integrable nonlinear wave

equations demonstrated that the modulational instabil-

ity can give rise to quite complicated dynamics. Using

initial data for quasiperiodic solutions near to homo-

clinic orbits (which we will denote as semi-stable

initial data), certain damped-driven and Hamiltonian

perturbations of the sine-Gordon and NLS equations

have been shown to trigger temporally chaotic evolu-tions. In the seminal papers [11,12], the role of linear

instabilities and homoclinic orbits in the generation

of chaotic dynamics for damped-driven perturbations

of the sine-Gordon equation was established along

with the spectral criterion for these instabilities. Sim-

ilarly, the role of these structures in Hamiltonian

perturbations of the NLS and sine-Gordon equations

have been examined in the context of computational

chaos [14,15,17,2729] and chaotic energy transport

[25,26].

For spatially symmetric data, the mechanism for

chaotic behavior involves random crossings of the

critical level sets of the constants of motion or homo-

clinic crossings (see, e.g. [13,15,28]). Significantly,

for a damped-driven perturbation of the NLS equa-

tion, analytical arguments have been found when even

symmetry is imposed to explain the onset of chaotic

dynamics [24]. The persistence of hyperbolic solutions

and the transversal intersection of their homoclinic

manifolds has been rigorously proven using singular

perturbation theory and Melnikov analysis adaptedto the infinite-dimensional setting [24]. It should be

noted that establishing chaos in the PDE framework

is technically quite difficult and for the purely disper-

sive perturbation of the NLS considered in [15,17,28]

(which is relevant to the water wave problem dis-

cussed here) the persistence of homoclinic orbits and

chaotic dynamics has not yet been rigorously proven.

When evenness is removed, little is known the-

oretically. In this case, numerical studies on com-

putational chaos in nonlinear wave equations have

provided considerable insight. For example, using

semi-stable initial data for the NLS it was shown thatif even symmetry was not preserved in the numerical

codes then roundoff errors can excite an odd com-

ponent which subsequently evolves chaotically [16].

More recently, the evolution of asymmetric initial

data has been studied numerically for a Hamiltonian

perturbation of the NLS equation as well as for a

simple symmetry-breaking perturbation of the NLS

[17,23]. The work in [17] identified for the first time a

distinctive new mechanism whereby homoclinic tran-

sition states develop near the homoclinic manifolds.

The solutions are characterized by random switching

across standing wave states into left- and right-goingmodulated traveling waves. The occurrence of chaos

without homoclinic crossings in the noneven regime

is a novel mechanism by which nonlinear disper-

sive wave systems can be chaotically excited and as

discussed in this paper is observable in laboratory

experiments.

In this paper (in [22], some of these results were

announced), we examine in detail the water wave

approximation to the NLS equation and consider

the following questions: (1) Are chaotic evolutions

in deep water physically observable? If so, then the

NLS is not an adequate description, and the per-

turbations to the NLS equation are critical to the

evolution. (2) What characterizes the wave train


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418 M.J. Ablowitz et al. / Physica D 152153 (2001) 416433

evolution? (3) Are the dominant features of the evolu-

tion captured by the HONLS model? We show exper-

imentally and analytically that periodically modulated

nonlinear Stokes water waves are, in fact, tempo-

rally chaotic, non-symmetric and not reproducible. In

contrast, when the envelope of the slow modulation

is taken to be that of a soliton, the experiments arereproducible.

The Floquet spectral theory of the NLS equation

is used to determine the characteristics of the evo-

lution of the wave train. This type of normal-mode

analysis, obtained by projecting numerical data

onto integrable nonlinear modes, was first applied

to certain perturbations of the Toda chain such as

the FermiPastaUlam chain [10] and later to per-

turbations of the sine-Gordon and NLS equations,

e.g. [11,17] amongst others. Here we compute the

spectrum, in particular the discrete eigenvalues, and

observe when they evolve into sensitive homoclinicregimes which in turn indicates bifurcations in the

waveform between different physical states. The

spectral decomposition of the laboratory data demon-

strates that the leftright switching mechanism for

chaotic excitations occurs in the water wave problem.

In numerical experiments with the HONLS equation

we also find numerous leftright homoclinic transi-

tions. The correlation between the results of the lab-

oratory and numerical experiments indicates that the

macroscopic or gross features of the wave evolution

are indeed modeled by the HONLS equation.

2. Analytical background

2.1. Governing equations

The equations governing the surface waves are

given by2= 0, for z, z 0 as z ,where(x,z,t),(x, t)are the velocity potential and

free surface displacement, respectively, and by the

boundary conditions on the free surface z=(x, t),

t+ x x= z, t+ g + 1

2 ()2

=0. (1)In the small amplitude approximation, the velocity po-

tential is expanded about z=0. For slowly modulated

waves one assumes the ansatz

= (A ei+|k|z + )+2(+ A2e2(i+|k|z) + ) + ,

= (Bei + ) + 2( + B2e2i + ) + , (2)where

=kx

t ,

denotes complex conjugate and

the deep water dispersion relation is used: 2 = g |k|with k0= 0.44 rad/cm (neither damping or surfacetension is taken into account in the theory). The vari-

ables A,, A2 are assumed to be functions of X= x ,Z=z, T= t, and B ,, B2 are functions ofXand T only. is a dimensionless number, i.e. a mea-

sure of small amplitude and balances slow modula-

tion;=ka, where a is the size of the initial surfacedisplacement. Substituting the above ansatz into the

expanded form of the free surface equation (1) leads

to the following perturbed NLS equation on z= 0([19], corrected for misprints)

2i

AT+ 2k

AX

2k

2AXX+ 4k4|A|2A

= 2

i2

8k3AXXX+ 2k3iA2AX 12k3i|A|2AX

+2kXA

.

Note that satisfies2 = 0 with the boundaryconditions: Z = (2k/g)(|A|2)X on z = 0 and 0 as z . The free surface amplitude isobtained from = (/g)((iA+ (/2k)Ax )ei (k2/)A2 e2i)

+(

). We introduce the following di-

mensionless and translating variables:T=T , X=kX, Z = kZ, = k, = (2k2/), u =(2

2k2/)A, = 18 T , = X 12 T. Solving

Laplaces equation on Z0 by Fourier meth-ods for in terms ofA yields the following perturbednonlocal NLS equation (HONLS):

iu+u + 2|u|2u + ( 12 iu 6i|u|2u+iu2u+ 2u[H (|u|2)] )=0, (3)

where H( f ) represents the Hilbert transform of the

functionf. The Fourier transform of the Hilbert trans-

form yields H( f )= i sign(k)f(k). In the periodiccase (0, L), k is replaced by kn = 2n/L. TheseFourier relations are readily implemented numerically


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when using a pseudo-spectral method, which is the

method we employ. We also note that on the infinite

line the physical space realization of the Hilbert trans-

form isH( f )=(1/ )PV(f( )/x) d, wherePVdenotes the Cauchy principal value integral.

2.2. Integrable theory of the NLS

Setting = 0 in (3) with x and t, weobtain the focusing cubic NLS equation which can be

written in Hamiltonian form

it

u

u

=J

H

u

H

u

(4)

with

J=

0 1

1 0

,

and Hamiltonian

H(u,u)= L

0(|ux |2 |u|4) dx. (5)

Zakharov and Shabat [8] established the complete in-

tegrability of the NLS equation by discovering its Lax

pair:

x= L(x) , t= L(t) , (6)where

L(x)

= i iu

iu i ,L

(t) =

i[22 uu] 2iu + ux2iu ux i[22 uu]

. (7)

The compatibility condition for the pair of linear

systems (7) is satisfied for all values of the complex

spectral parameter if the coefficient u(x,t)satisfies

the NLS equation (3) at=0. If one imposes period-icity by requiring the potential u(x,t) to be periodic

in the spatial variable x with period L, then one can

characterize u (for any fixed time t) in terms of its

Floquet spectrum

(L(x) (u)):= {C|L(x)vvv=0, |vvv| bounded x}.(8)

Given a fundamental solution matrix M(x ; ) (suchthat M (0; )=I) for the Lax pair (6) with potentialu(x), one defines the Floquet discriminant to be the

trace of the transfer matrix M(L; ) across one period:(u; ):=Tr[M(L; )]. (9)

Since det[M(x ; )]=1, the Floquet spectrum can becharacterized as follows:

(L(x) (u)) := {C|(u; ) isrealand2(u; )2}. (10)

The properties of the Floquet discriminant that we

will use in the following sections are:

1. (u; ) is constant under the NLS evolution.2. Floquet discriminants at different values of the

spectral parameter Poisson commute, for =,

{(u; ), (u; )},where the Poisson bracket is defined as

{F, G} =i L

0

F

u

G

u F

u

G

u

dx.

These properties show that (u; ) encodes the in-finite family of NLS constants of motion (in fact,

parameterized by C).Within the discrete spectrum, periodic/antiperiodic

eigenvalues are the roots of(u; )= 2. We alsodistinguish the following points of the spectrum:

1. The simple periodic/antiperiodic spectrum

s =

sj|(, u)= 2,d

d=0

. (11)

2. Critical points of spectrum cj, specified by the con-

dition d(u; )/d|=c= 0.3. Double points of the periodic/antiperiodic spectrum

d =

dj|(, u)= 2,d

d=0, d

2

d2=0

.

(12)

The nonlinear spectral transform is used to represent

solutions in terms of a set of nonlinear modes whose

structure and dynamical stability is determined by


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the location of the corresponding element of the pe-

riodic/antiperiodic spectrum [11]. Generic multiple

points have multiplicity 2 and 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 dou-

ble points are in general associated with linearizedinstabilities of the NLS equation and label the orbits

homoclinic to the unstable solution [12].

2.3. Modulational instability and homoclinic

solutions

An issue of main importance is the stability of

the periodic Stokes wave. For the nondimensionalized

NLS equation the Stokes wave, or plane wave, is given

by u0(x,t)= a e2i|a|2t, where for convenience, a isassumed to be real. After dimensionalizing and trans-

forming to the surface displacement , the exponent

corresponds to the nonlinear frequency shift found by

Stokes that includes the term 4k2|A|2 in Eq. (2). Itsstability can be examined by considering perturbations

of the form u(x,t)=u0(1 + (x,t ))and linearizingfor small . Assuming that (x,t)=n(0) einx+int+n(0) einxint, n=2n/L, it is found that thegrowth raten is given by

2n= 2n(2n 4a2). Thus,

the plane wave is unstable to long wavelength pertur-

bations, i.e. provided 0 < (n/L)2


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Fig. 1. The surface |u(x,t)| and the nonlinear spectrum with (a) one double point for a homoclinic solution of NLS, (1) = 0;(b) one imaginary gap for a standing wave solution of NLS, u0= 0.5(1+0.1 cos x); (c) one cross for a standing wave so-lution of NLS locked in the center and wings, u0= 0.5(1+0.1i cosx); (d) right state: a right-traveling wave solution of NLS,u0=0.5(1+0.05(e90i cos x+e0i sin x)); (e) left state: a left-traveling wave solution of NLS, u0=0.5(1+0.05(e0i cos x+e30i sin x)).

if = j= 2j/L,1 m1, then the jthcomplex double point splits into two simple points

and a complex band of spectrum or a gap is created.

The parameters1and 2 govern the symmetry of the

solution: if1=2 +n , then the perturbed spectrumexhibits the symmetry (this corresponds to

the solution being even in the spatial variable, i.e.

q(x)= q(x), a constraint commonly imposed inthe study of perturbations of the NLS equation). In

this case the double point splits along the imaginary

axis (gap configuration, see Fig. 1b) or symmetrically

about the imaginary axis (cross state, see Fig. 1c). In


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the symmetric case, the homoclinic orbit separates the

symmetric subspace into disjoint invariant subman-

ifolds. Due to the analyticity of the discriminant ,

under small perturbations it is possible to evolve from

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.This even symmetry constraint, first considered in

[11], has been used almost exclusively in the literature

so that homoclinic crossings can be easily identified.

On the other hand, for generic 1 and 2, the se-

lected complex double point splits asymmetrically in

the complex plane. (For a complete analysis of the

O() splitting of the complex double points in the

noneven regime, see [17].) When one complex double

point is present the two basic spectral configurations

associated with noneven perturbations are as follows:

(1) The resulting upper band of spectrum lies in the

first quadrant and the lower band lies in the secondquadrant. The wave form is characterized by a single

mode traveling to the right (right state, see Fig. 1d).

(2) 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. 1e). The

Fig. 2. (a) Surface for u0= 0.5(1 + 0.01(e0.9i cos x+ e60i sin x)) obtained with the difference scheme DDNLS for 0 < t < 500 and5000 < t


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solution of the NLS equation

u0=0.5(1+0.01(e0.90i cos x+ e60i sin x)) (15)with L = 2

2, = 2/L. Fig. 2a shows the

surface (0 < t < 500) obtained with the follow-

ing Hamiltonian discretization (DDNLS): iun =(un+1 + un1 2un)/ h

2

+ 2|un|2

un=0 forN= 24.Initially, the waveform travels to the left; as time

evolves the perturbation induced by the discretiza-

tion causes the waveform to jump between left- and

right-traveling waves. We observe that this bifurca-

tion occurs randomly and intermittently throughout

the entire time series for 0 < t


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calculation of clean surface damping, but less than that

of a contaminated surface calculation.

The water is allowed to rest for 15 min before ex-

periments are conducted; the surface is cleaned at

least every 2 h. Waves are generated using a vertically

oscillated, anodized wedge with both position and ve-

locity control. The programmed velocity for both setsof experiments is the linear, fluid particle velocity;

the programmed position is the desired free-surface

displacement.

The theoretical formulation of Section 2 does not

include the effects of viscous damping. To obtain a

measure of viscous decay in the experiments, we mea-

sured the maximum amplitude of the envelope soliton

at 23 positions down the tank. This amplitude decayed

exponentially at a rate of 6 104 cm1/s.The surface displacement is measured using five

capacitance-type wave gages. Gage 0 is fixed at 40 cm

downstream from the wavemaker. It is an intrusive,capacitance-type wavegage. Gages 14 were mounted

40 cm apart on the carriage; they are non-intrusive,

capacitance wave gages that span the width of the

tank, and thus average out any surface motion in the

direction perpendicular to wave propagation.

The purpose of gage 0 is to measure the water sur-

face displacement near the wavemaker to insure that

the surface displacement there is reproducible. A me-

chanical cam is used to close a switch that begins the

data collection; it has a few milliseconds of slop. To

adjust for that, we shift the total time series of all five

gages by a few milliseconds based on the correlationcoefficients between the time series obtained by gage

0. That is, we compare the time series from gage 0

from every experiment with the same initial conditions

to that obtained for the first such experiment. We shift

the time series of all the gages by the small amount

necessary to give the maximum correlation coefficient

between the measurements obtained at gage 0. This

small shift (a few milliseconds out of a 65 s time se-

ries) insures that the starting point of each set of time

series is the same from experiment to experiment.

The time series from gage 0 have correlation coeffi-

cients among experiments of 0.99 or better, except for

two experiments (among 40) for which the correlation

coefficients were 0.96 or 0.97. This high correlation

indicates that, indeed, the initial conditions, i.e. the

water surface displacement near the wavemaker, was

reproducible within a small noise level.

Gages 14 are mounted on the carriage so that two

types of time series are obtained from them. The first

are fixed measurements in which the gages are located

at a particular position along the tank. The secondare measurements obtained in a reference frame that

translates at the linear group speed of the underlying

wave train. The carriage that supports the gages is at

rest for 21.28 s to allow for transient motions to pass

and then set in motion at the desired speed.

In the temporal measurements we compare time se-

ries obtained from different experiments with identi-

cal initial conditions. These measurements are graphed

against each other to produce a phase plane diag-

nostic for reproducibility. If the results of the two

experiments are identical the graph will be the 45

line. In particular, the time series from gage 0 nearthe wavemaker produces a 45 line with a very slightwidth, indicating a 12% noise level. This indicates

what should be expected from time series showing the

wavefield evolution.

3.2. Envelope solitons

The control experiment is performed with the

soliton solution of NLS since theoretically it

should evolve reproducibly (see below). The wave-

maker was programmed to oscillate as s(t)

=a sin(0t)sech(a0t/2) where 0= 20.94 rad/s,a= 0.2 cm, k0= 0.44 rad/cm, (k0)= 24.4 cm/s(group velocity),g=980 cm/s2 andT= 71.9 dynes/cm (surface tension).

Fig. 3 shows time series obtained by the five wave

gages when the carriage is near the wavemaker. There

is evidence of some dispersion in the envelope soliton.

The effects of viscous decay are observed as well, e.g.

in Fig. 4, which shows an envelope soliton near the

wavemaker and 800 cm downstream of it.

To determine if the soliton experiment is repro-

ducible, the experiment was run every 15 min over a

2.5 h period. Time series are obtained at gage 4 when

the carriage was 800 cm downstream of the wave-

maker. Fig. 5a graphs the water surface displacements


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Fig. 3. Time series from gages (a) 0, (b) 1, (c) 2, (d) 3, and (e) 4 obtained when the carriage was fixed so that gage 1 was 40 cm from gage 0.

measured in the first and last experiments against each

other. There was a slight phase shift between the twodata sets. Shifting each data point in one of the data

sets the same amount (0.00921 s), produces the nearly

perfect 45 line (Fig. 5a). The experiments using soli-tons indicate that we are able to conduct a repro-

ducible experiment when the initial conditions are

such that the wavefield evolution is predicted to be

reproducible. The soliton experiment reflects a stable

nonchaotic evolution which the NLS equation ade-

quately describes.

3.3. Modulated periodic wave trains

For modulated wave trains the position of

the wavemaker is programmed to be p(t) =

a sin(0t)(1

+Esin pt ) where a

= 0.5 cm, p

=1.047rad/s, E=0.1, the values 0, g , k0, T are thesame as in the soliton case and the corresponding

unperturbed periodic wavelength of the modulation is

L=147 cm.Fig. 6 shows output from the five wavegages for

the modulated wave train. The output from gage

0, shown in Fig. 6a, shows a periodic modulated

wave train. The output from gages 14 are shown

in Figs. 6be. For times less than 21.3 s the carriage

supporting these gages is at rest and so the periodicity

is the same as that near the wavemaker. At 21.3 s, the

carriage moves with the group speed of the waves

as evidenced by the Doppler shift that occurs at this

time. One would expect that when the gage starts to

move at the group speed, the amplitude at the time


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Fig. 4. Time series from gages (a) 0, (b) 1, (c) 2, (d) 3, and (e) 4 obtained when the carriage was fixed so that gage 1 was 840cm from

gage 0.

the carriage begins motion would remain constant,

except for some viscous decay, as happens in Fig. 6e.

However, this does not occur in the traces of Figs.

6bd, presumably because of a mismatch in linear

and actual group speed. We note that evidence ofreflections occur after about 61 s.

For the modulated wave train initial data, phase

plane plots show that the wave is reproducible near

the wavemaker. However, as the wave travels down

the tank we obtain Lissajous-type figures (see, e.g.

Fig. 5b) indicating that a complicated phase shift de-

velops between the waves of two experiments and

cannot be simply removed. Indeed subsequent exper-

iments show that different Lissajous figures, with di-

verging phase trajectories are obtained, each of which

correspond to different complicated phase shifts. The

phase shifts are sensitive to small changes in initial

data. Unlike the soliton, the two time series start to

diverge indicating the experiment is irreproducible.

Spatial data associated with modulated periodic

wave trains was also obtained yielding a different

perspective of the evolution. Some of the spatial data

were used in the numerical experiments, discussed

below, as initial conditions. The spatial envelope isreconstructed by concatenating 40 sets of data for

each of the four gages. In the 40 experiments, the

initial location of the carriage differs by 1 cm suc-

cessively for each experiment. The result is 160 time

series of the water surface spaced 1 cm apart which

are used to reconstruct the spatial profile of the water

surface, 160 cm long, by concatenating the data sets.

Our ability to measure a spatial envelope by conduct-

ing 40 experiments requires the experiments to be

reproducible.

Fig. 7 shows six spatial profiles obtained in this

way. Fig. 7a shows the spatial profile near the wave-

maker, before the wave has reached the wavegages.

This profile provides a benchmark for the level of


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Fig. 5. Phase plane plots for: (a) experimental soliton data; (b)

experimental modulated wave train data.

noise as indicated by the miniscule blips in what

should actually be a flat surface level. At t= 15s,the waveform is somewhat close to the wavemaker

and the blips in the data are not significant (Fig. 7c).

Further down the tank additional crests start to form.

The blips become significant and no longer represent

a simple non-smoothness in the wave profile. By 40 s,the periodicity of the underlying wave train is lost

(Fig. 7f). The degeneration of the spatial coherence

of the wavefield indicates that the experiments are not

reproducible. This experimental irreproducibility and

the theory presented below are evidence that modu-

lated Stokes wave trains evolve chaotically for certain

parameter regimes.

4. Numerical experiments

We have shown experimentally that periodically

modulated nonlinear Stokes waves can evolve chaoti-

cally. Armed with this information and the analytical

background, in this section we turn to the issue of

determining the qualitative features of the evolution

and whether it can be successfully modeled with the

HONLS equation. This is accomplished by the fol-

lowing: (1) we do some additional post-processing of

the data from the laboratory experiments. We calcu-

late the Floquet spectral decomposition of the labora-tory data at sampled times. This establishes that the

water wave dynamics is characterized by leftright

homoclinic transitions. (2) We numerically examine

the long-time dynamics of the HONLS equation. We

first establish the parameter regimes for which the

HONLS exhibits chaotic behavior (if at all) using

model initial data. When solutions to the HONLS

equation are regular and O() close to NLS solutions

for the same initial data we consider the NLS equation

to adequately describe the dynamics in that regime.

Once the basic HONLS dynamics is understood,

we examine the HONLS using experimental data asinitial data, to allow a closer comparison with the

laboratory experiments. The diagnostics indicate that

the numerical experiments compare very well with

the laboratory experiments. For certain regimes (i.e.

when a higher number of unstable modes are present)

the higher-order nonlinear terms become critical and

for these cases the dominant features of the chaotic

behavior is captured by the HONLS equation.

In the numerical experiments the parameter values

are specified in the nondimensionalized framework

and have been carefully matched with the parame-

ters used in the laboratory experiments. We use afourth-order pseudo-spectral code for integrating the

HONLS equation (3) with N= 512 Fourier modesin space and a fourth-order adaptive Runge-Kutta

scheme in time. As in the laboratory experiments,

reproducibility is studied using phase plane plots.

In the phase plane plots the evolution of the sur-

face displacement obtained using initial data

u(x, 0) is graphed against the evolution of the sur-

face displacement obtained using u(x, 0), whereu(x, 0)=u(x, 0)(1+ ur (x)), is on the order ofexperimental error (1%) and ur (x) is taken to be a

random field. The formula for the reconstruction of

(the phase plane plot is the one diagnostic where

dimensional coordinates are presented) in terms ofu


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Fig. 6. Time series from gages (a) 0, (b) 1, (c) 2, (d) 3, (e) 4. Gage 0 is at rest, 40 cm from the wavemaker. Gages 14 are mounted on a

carriage that translates downstream at the linear group velocity starting at 21.3 s.

can be found in Section 2. The associated nonlinear

spectral theory of the NLS equation is also used to

investigate the dynamics. The data provided by boththe physical and numerical experiments is projected

onto the nonlinear spectrum of the NLS and we fol-

low its evolution in time to determine changes in the

nonlinear mode content. Although in the experiments

the spectrum is computed every dt= 0.1, it is onlyshown at sampled times.

The benchmark case is the soliton case using model

initial data of the form u(x, 0)=sech(x), i.e. a soli-ton with zero velocity. The spacetime evolution of

the waveform obtained using the HONLS equation

with

= 0.14, for 0 < t < 5, is given in Fig. 8a.

The soliton develops an O() velocity and sheds a

small amount of radiation off the front of the soli-

ton, but the dynamics is regular. The phase plane plot,

for 0 < t < 10, remains close to the 45 line withlittle spread (Fig. 8b), as was observed in the labo-

ratory experiments. Although the spectrum is not in-variant, the spectral plots show that the spectrum does

not change configuration and that there are only small

O() changes in the amplitude and speed of the soli-

ton. This indicates that we obtain a reproducible ex-

periment when the initial conditions are solitons and

supports the notion that, for envelope solitons for the

time scales under consideration, the unperturbed NLS

equation adequately describes the long-time dynamics.

The other model initial data we considered cor-

responded to modulationally unstable periodic wave

trains and is of the form

u(x, 0)=a(1 + T cos nx) (16)with T = 0.1. We varied the amplitude a in the


14/18


Fig. 7. Spatial profiles corresponding to times of (a) 1.00s, (b) 12.00 s, (c) 15.00s, (d) 29.00 s, (e) 33.00s, and (f) 40.00s after the start

of data collection. The dots represent the experimental data.

Fig. 8. (a) Surface for u(x, 0)= sech x obtained using the HONLS equation with = 0.14 for 0 < t < 5; (b) Phase plane plot of thesurface amplitude (mm) vs. for model soliton initial data for 0 < t


15/18


instability criterion to have M = 1, 3, 5 unstablemodes nearby (and we denote this as the M unsta-

ble mode regime). In [21], we have shown that for

data T there are 2n+1 simple imaginary points inthe spectrum: 0 and{n+, n}, n= 1, 2, . . . , M .When T is asymptotically small, the simple points

n+, n of the spectrum ofL

get successively closerto each other with their distance from a double point

being O(nT), n= 1, 2 . . . , M . The distance from adouble point can be made arbitrarily small by taking

M sufficiently large. Thus, the evolution can exhibit

homoclinic transitions in which case the dynamics is

irregular and chaotic.

For initial data in the one unstable mode regime (e.g.

(16) with a= 0.5, T= 0.1 and L= 2

2 ), we

found that the solution of the HONLS equation with

= 0.14, 0 < t < 2.5, is well approximated by thesolution obtained using the NLS equation. The wave-

form does not display any temporal irregularities andhomoclinic transitions do not occur; stable nonchaotic

dynamics ensue. The phase plane plot is very close

to the 45 line. The spectral configuration is the leastcomplicated of all the modulated wave train cases. The

initial data is for a standing wave gap state. As time

evolves, the discrete eigenvalues remain well separated

and do not evolve into sensitive regions. By t= 2.5,

Fig. 9. (a) Surface for initial data (16) with a=0.7 and L=4

2 obtained using the HONLS equation with =0.28 for 0 < t


16/18


Fig. 10. (a) Phase plane plot of vs. for experimental initial data for 0 < t < 5. Notice the similarity to Fig. 5b. Spectral plotscorresponding to times of (b) 0.0, (c) 0.5, (d) 1.0, (e) 2.0, (f) 2.1, and (g) 2.6. Solid darkened curves are curves of spectrum and we have

included some of the curves of real (the dashed curves) to give an indication of the topological changes in the spectrum.

We refer the reader to the spectral plots for the ex-

perimental data (Figs. 10bf) for a sample of the

changes in the spectral configuration that occur in

this regime. This is a generalization of the basic

mechanism for chaos in the noneven regime that we

observed for one unstable mode (cf. Section 2). In

the spectral plots only the spectrum related to the

dominant low modes is shown. The amplitude of

the higher modes is very small and the spectrum of

these radiative states is not depicted as the radiation


17/18


modes are not significant in the description of the

chaotic state. For the case M= 5, we found that thephase plane plot diverges from the 45line even morestrongly than for M= 3. The spectrum evolved sig-nificantly and we found more numerous homoclinic

transitions than with M= 3. The cases M= 3, 5yield strong temporal irregularities and, for the timescales under consideration, chaotic dynamics. The ex-

amination of the model data demonstrates that when

there are a higher number of unstable modes present

initially, the NLS is inadequate and the HONLS per-

turbations make a significant difference to the final

evolution.

The numerical results for the HONLS using the

experimental data as initial data provide evidence of

chaotic evolutions consistent with the laboratory re-

sults. For the experimental data, denoted as E we

use u(x, 0)= the spatial envelope of the modulatedwave train obtained near the wavemaker (see Fig. 7b).This data corresponds to a multi-phase solution in the

three unstable mode regime. Using initial data E , for

short times the experiment is reproducible and the

phase plane plot stays close to the 45 line. As thewavefield evolves, the experiment is rendered irrepro-

ducible. A complicated phase shift develops between

the experiments that changes with time resulting in a

phase plane plot (Fig. 10a) that is remarkably similar

to that of the laboratory data (Fig. 5b).

The spectral results are striking. The spectrum at

t

= 0 (Fig. 10b) depicts four curves of spectrum

with seven simple eigenvalues as the end points ofspectrum. The three nearby complex double points

(labeling the three unstable modes) correspond to

the center locations between the simple eigenvalues.

We number them according to their distance from

the origin with the first mode being farthest. The

spectrum evolves significantly in time and a number

of homoclinic transitions between modes occur. As

mentioned previously, only the spectrum related to

the dominant low modes is shown in Fig. 10 as the

transitions occur within a fixed, low-dimensional set

of nonlinear modes. Figs. 10bg give an overview of

the changes in the spectral configuration and show the

spectrum at six time slices, at (b) t=0.0, (c)t=0.5,(d) t= 1.0, (e) t= 2.0, (f) t= 2.1, and (g) t= 2.6.

In Figs. 10b and c the transition in the orientation of

the third and fourth curves of spectrum indicates a

bifurcation in the third nonlinear mode of the solution

between left- and right-traveling. Similarly, between

Figs. 10c and d, there is a change in the orientation

of the first and second curves indicating a homoclinic

transition for the first mode. Later in the evolution,the eigenvalues and homoclinic double points move

away from the imaginary axis (see Figs. 10eg). In

this sequence of plots the second and third curves

in the spectral configuration switch orientation and

back again indicating leftright flipping in the second

mode. For the timeframe examined, 0 < t < 5, there

are frequent random leftright homoclinic transitions

in all of the unstable nonlinear modes. Each of these

transitions corresponds to a change in the character-

istics of the nonlinear modes and leads to physical

changes in the wave field.

Finally, we remark upon the spectral decompositionof the laboratory data. Using the actual physical data

(not evolving with the HONLS equation) we compute

the spectrum. Fig. 10b and Figs. 11a and b provide

the spectrum at t= 12, 29 and 32 s, respectively. Sig-nificantly, the evolution is unmistakably characterized

by leftright homoclinic transitions! These three time

slices demonstrate a switching in the orientation of

the first and second bands of spectrum indicating that

the first mode is switching from left- to right-traveling.

We also note that at later times (see Figs. 11a and b)

there are only two nonreal eigenvalues, as opposed to

three earlier, as indicated in Fig. 10b. We believe the

Fig. 11. Spectral plots for the laboratory data corresponding to

times of (a) 29.0 s, and (b) 32.0 s. Solid darkened curves are curvesof spectrum and we have included some of the curves of real

(the dashed curves) to give an indication of the topological

changes in the spectrum.


18/18


reason for the loss of an eigenvalue, i.e. reduction

from three unstable modes to two unstable modes, is

due to viscous damping which has not been incorpo-

rated into the theory. Nevertheless, it is remarkable that

the leftright switching scenario is still apparent. Thus,

the dissipation does not interact significantly with the

chaotic mechanism. Based upon the results presentedin this paper, the long-time evolution of the modu-

lational instability and subsequent chaotic dynamics

is adequately described by the HONLS equation (3).

However, we note that (linear) damping should be

added to better describe amplitude changes. We shall

investigate this effect in the future.

Acknowledgements

This work was partially supported by the AFOSR

USAF, Grant No. F49620-00-1-0031 and the NSF,

Grant Nos. DMS-0070772, DMS-9803567 and

DMS-9972210.

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