Long-time dynamics of the modulational instability of deep water waves

Transcription

1 Physica D (2001) Long-time dynamics of the modulational instability of deep water waves M.J. Ablowitz a, J. Hammack b, D. Henderson b, C.M. Schober c, a Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, CO , USA b Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA c 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 Schrödinger (NLS) equation. The laboratory and numerical experiments 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 Schrödinger 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 experiments provide a physical realization of the chaotic behavior previously established analytically for perturbed NLS systems Elsevier Science B.V. All rights reserved. Keywords: Nonlinear Schrödinger equation; Modulated periodic waves; Long-time evolution 1. Introduction Historically, the study of water waves has provided researchers with a wide variety of interesting nonlinear 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 leading terms of a series expansion for the surface wave displacement, and how the frequency of the wave depended on the amplitude for small but finite values of the wave amplitude. In 1925, Levi-Civita [2] proved Corresponding author. addresses: markjab@newton.colorado.edu (M.J. Ablowitz), cschober@lions.odu.edu (C.M. Schober). rigorously that an infinite series describing periodic waves could be obtained and that it converged. However, 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 considering slowly modulated wave trains. It can be shown that for small amplitude, the leading-order complex amplitude of the surface displacement satisfies the nonlinear Schrödinger (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 /01/$ see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S (01)00183-X

2 M.J. Ablowitz et al. / Physica D (2001) 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, has an infinite number of conserved quantities and does not possess temporally chaotic orbits [6,7]. However, 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 generate 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 Schrödinger (HONLS) equation, which is obtained by retaining terms in the asymptotic expansion through fourth-order (see Eq. (3)), destroys the symmetry of the 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 instability and of the wave train. Earlier studies on near-integrable nonlinear wave equations demonstrated that the modulational instability can give rise to quite complicated dynamics. Using initial data for quasiperiodic solutions near to homoclinic 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 evolutions. 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. Similarly, the role of these structures in Hamiltonian perturbations of the NLS and sine-gordon equations have been examined in the context of computational chaos [14,15,17,27 29] 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 homoclinic crossings (see, e.g. [13,15,28]). Significantly, for a damped-driven perturbation of the NLS equation, 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 Mel nikov analysis adapted to the infinite-dimensional setting [24]. It should be noted that establishing chaos in the PDE framework is technically quite difficult and for the purely dispersive perturbation of the NLS considered in [15,17,28] (which is relevant to the water wave problem discussed here) the persistence of homoclinic orbits and chaotic dynamics has not yet been rigorously proven. When evenness is removed, little is known theoretically. In this case, numerical studies on computational chaos in nonlinear wave equations have provided considerable insight. For example, using semi-stable initial data for the NLS it was shown that if even symmetry was not preserved in the numerical codes then roundoff errors can excite an odd component 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 transition states develop near the homoclinic manifolds. The solutions are characterized by random switching across standing wave states into left- and right-going modulated traveling waves. The occurrence of chaos without homoclinic crossings in the noneven regime is a novel mechanism by which nonlinear dispersive 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 perturbations to the NLS equation are critical to the evolution. (2) What characterizes the wave train

3 418 M.J. Ablowitz et al. / Physica D (2001) evolution? (3) Are the dominant features of the evolution captured by the HONLS model? We show experimentally and analytically that periodically modulated nonlinear Stokes water waves are, in fact, temporally 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 are reproducible. The Floquet spectral theory of the NLS equation is used to determine the characteristics of the evolution 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 Fermi Pasta Ulam chain [10] and later to perturbations 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 homoclinic regimes which in turn indicates bifurcations in the waveform between different physical states. The spectral decomposition of the laboratory data demonstrates that the left right switching mechanism for chaotic excitations occurs in the water wave problem. In numerical experiments with the HONLS equation we also find numerous left right homoclinic transitions. The correlation between the results of the laboratory 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 by 2 φ = 0, for z η, φ z 0asz, where φ(x,z,t),η(x,t) are the velocity potential and free surface displacement, respectively, and by the boundary conditions on the free surface z = η(x, t), η t + η x φ x = φ z, φ t + gη ( φ)2 = 0. (1) In the small amplitude approximation, the velocity potential is expanded about z = 0. For slowly modulated waves one assumes the ansatz φ = ɛ(ae iϑ+ k z + ) +ɛ 2 ( φ + A 2 e 2(iϑ+ k z) + ) +, η = ɛ(b e iϑ + ) + ɛ 2 ( η + B 2 e 2iϑ + ) +, (2) where θ = kx ωt, denotes complex conjugate and the deep water dispersion relation is used: ω 2 = g k with k 0 = 0.44 rad/cm (neither damping or surface tension is taken into account in the theory). The variables A, φ,a 2 are assumed to be functions of X = ɛx,z = ɛz,t = ɛt, and B, η, B 2 are functions of X and T only. ɛ is a dimensionless number, i.e. a measure of small amplitude and balances slow modulation; ɛ = ka, where a is the size of the initial surface displacement. 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) ( 2iω A T + ω ) 2k A X ɛ ( ( ω ) 2 AXX + 4k 4 A A) 2 2k = ɛ 2 ( iω 2 8k 3 A XXX + 2k 3 ia 2 A X 12k3 i A 2 A X +2ωk φ X A ). Note that φ satisfies 2 φ = 0 with the boundary conditions: φ Z = (2ωk/g)( A 2 ) X on z = 0 and φ 0asz. The free surface amplitude is obtained from η = (ɛω/g)((ia + (ɛ/2k)a x )e iθ (k 2 ɛ/ω)a 2 e 2iθ ) + ( ). We introduce the following dimensionless and translating variables: T = ωt, X = kx, Z = kz, η = kη, φ = (2k 2 /ω) φ, u = (2 2k 2 /ω)a, τ = 1 8 ɛt,χ = X 1 2 T. Solving Laplaces equation on Z 0 by Fourier methods for φ in terms of A yields the following perturbed nonlocal NLS equation (HONLS): iu τ +u χχ + 2 u 2 u + ε( 1 2 iu χχχ 6i u 2 u χ +iu 2 u χ + 2u[H( u 2 )] χ ) = 0, (3) where H(f) represents the Hilbert transform of the function f. The Fourier transform of the Hilbert transform yields H(f) ˆ = i sign(k) f(k). ˆ In the periodic case (0, L), k is replaced by k n = 2nπ/L. These Fourier relations are readily implemented numerically

4 M.J. Ablowitz et al. / Physica D (2001) 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 transform is H(f) = (1/π)PV (f (ξ)/ξ x)dξ, where PV denotes the Cauchy principal value integral Integrable theory of the NLS Setting ɛ = 0 in (3) with χ x and τ t, we obtain the focusing cubic NLS equation which can be written in Hamiltonian form ( ) ū i t = J u with ( ) 0 1 J =, 1 0 and Hamiltonian H (u, ū) = L 0 δh δu δh δū (4) ( u x 2 u 4 ) dx. (5) Zakharov and Shabat [8] established the complete integrability of the NLS equation by discovering its Lax pair: φ x = L (x) φ, φ t = L (t) φ, (6) where ( ) iλ iu L (x) =, iū iλ ( i[2λ 2 ) L (t) uū] 2iλu + u x =. (7) 2iλū ū x i[2λ 2 uū] The compatibility condition for the pair of linear systems (7) is satisfied for all values of the complex spectral parameter λ if the coefficient u(x, t) satisfies the NLS equation (3) at ɛ = 0. If one imposes periodicity 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 σ(l (x) (u)) := {λ C L (x) v = 0, v bounded x}. (8) Given a fundamental solution matrix M(x; λ) (such that M(0; λ) = I) for the Lax pair (6) with potential u(x), one defines the Floquet discriminant to be the trace of the transfer matrix M(L; λ) across one period: (u; λ) := Tr[M(L; λ)]. (9) Since det[m(x; λ)] = 1, the Floquet spectrum can be characterized as follows: σ(l (x) (u)) := {λ C (u; λ) is real and 2 (u; λ) 2}. (10) The properties of the Floquet discriminant that we will use in the following sections are: 1. (u; λ) is constant under the NLS evolution. 2. Floquet discriminants at different values of the spectral parameter Poisson commute, for λ λ, { (u; λ), (u; λ )}, where the Poisson bracket is defined as L ( δf δg {F,G}=i δu δū δf ) δg dx. δū δu 0 These properties show that (u; λ) encodes the infinite family of NLS constants of motion (in fact, parameterized by λ C). Within the discrete spectrum, periodic/antiperiodic eigenvalues are the roots of (u; λ) =±2. We also distinguish the following points of the spectrum: 1. The simple periodic/antiperiodic spectrum { } σ s = λ s d j (λ, u) =±2, dλ 0. (11) 2. Critical points of spectrum λ c j, specified by the condition d (u; λ)/dλ λ=λ c = Double points of the periodic/antiperiodic spectrum { } σ d = λ d d j (λ, u) =±2, dλ = 0, d2 dλ 2 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

5 420 M.J. Ablowitz et al. / Physica D (2001) the location of the corresponding element of the periodic/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 double points are in general associated with linearized instabilities of the NLS equation and label the orbits homoclinic to the unstable solution [12] 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 u 0 (x, t) = a e 2i a 2t, where for convenience, a is assumed to be real. After dimensionalizing and transforming to the surface displacement η, the exponent corresponds to the nonlinear frequency shift found by Stokes that includes the term 4k 2 A 2 in Eq. (2). Its stability can be examined by considering perturbations of the form u(x, t) = u 0 (1 + ɛ(x,t)) and linearizing for small ɛ. Assuming that ɛ(x,t) = ɛ n (0) e iµ nx+iσ n t + ɛ n (0) e iµ nx iσ n t,µ n = 2πn/L, it is found that the growth rate σ n is given by σn 2 = µ2 n (µ2 n 4a2 ). Thus, the plane wave is unstable to long wavelength perturbations, i.e. provided 0 < (πn/l) 2 < a 2 and the number of unstable modes is the largest integer M satisfying 0 <M< a L/π. This stability criterion is applicable to the modulated periodic wave train produced via η p (t) described in the laboratory experiments. Another approach to examining instabilities is using the associated Floquet theory. The discriminant for the plane wave u 0 (x, t) is readily computed to be (a, λ) = 2 cos( a 2 + λ 2 L). (13) Then, the associated Floquet spectrum consists of continuous bands R [ ia,ia], and a discrete part containing the simple periodic/antiperiodic eigenvalues ±i a, and the infinite number of double points ( nπ ) 2 λ 2 n = a 2, n Z. (14) L The complex double points, 0 < (πn/l) 2 < a 2 (this is the same condition as for linear instability), are associated with critical saddle-like level sets and can be used to label their homoclinic orbits. The remaining λ n s for n >Mare real double points. If [al/π] = M, there are 2M complex (pure imaginary) double points. Fig. 1a shows the spectrum of the plane wave for the case M = 1. Although homoclinic orbits of the modulationally unstable plane wave can be explicitly constructed, here we simply provide the waveform of the homoclinic solution along with its associated spectrum (see Fig. 1a). The homoclinic solution is characterized by a single mode which limits to the plane wave as t ±. The isospectral set of the plane wave includes the homoclinic orbit. In the numerical experiments, we use initial data which are small perturbations of unstable plane waves with M = 1, 3, 5 complex double points or unstable modes. The experimental data specifically corresponds to perturbations of the Stokes wave in the three unstable mode regime. Note that since the real axis is always spectrum and σ(l) has the symmetry, λ σ(l) then λ σ(l), only the spectrum in the upper-half λ-plane is displayed in the spectral plots throughout the paper. Numerical experiments involving perturbed NLS and sine-gordon equations have shown that the existence of instabilities and their associated homoclinic orbits can generate a variety of interesting phenomena, including temporally chaotic evolutions. We summarize the results of [17] which are relevant in interpreting the laboratory and numerical experiments Nearby states We give a brief description of the multi-phase solutions whose Floquet spectra are O(ɛ) close to one of the modulationally unstable plane wave with one complex double point. The spectral configurations are computed at t = 0. In [17], initial data of the form u(x, 0) = u 0 + ɛu 1 = a + ɛ[e iθ 1 cos(µx) + r e iθ 2 sin(µx)] are considered, where 0 <ɛ 1, r and θ i s, for i = 1, 2, are real parameters and µ a real frequency to be selected so that the perturbation affects one or more specific complex double points. The selection criteria are found to be: for a = mπ/l,

6 M.J. Ablowitz et al. / Physica D (2001) Fig. 1. The surface u(x, t) and the nonlinear spectrum with (a) one double point for a homoclinic solution of NLS, λ (1) = 0; (b) one imaginary gap for a standing wave solution of NLS, u 0 = 0.5( cos µx); (c) one cross for a standing wave solution of NLS locked in the center and wings, u 0 = 0.5( i cos µx); (d) right state: a right-traveling wave solution of NLS, u 0 = 0.5(1+0.05(e 90i cos µx+e 0i sin µx)); (e) left state: a left-traveling wave solution of NLS, u 0 = 0.5(1+0.05(e 0i cos µx+e 30i sin µx)). if µ = µ j = 2πj/L, 1 µ m 1, then the jth complex double point splits into two simple points and a complex band of spectrum or a gap is created. The parameters θ 1 and θ 2 govern the symmetry of the solution: if θ 1 = θ 2 +nπ, then the perturbed spectrum exhibits the symmetry λ λ (this corresponds to the solution being even in the spatial variable, i.e. q(x) = q( x), a constraint commonly imposed in the 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

7 422 M.J. Ablowitz et al. / Physica D (2001) the symmetric case, the homoclinic orbit separates the symmetric subspace into disjoint invariant submanifolds. 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 θ 1 and θ 2, the selected complex double point splits asymmetrically in the complex plane. (For a complete analysis of the O(ɛ) 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 second quadrant. 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 main result obtained for near-integrable dynamics in the noneven regime is that the spectrum can evolve between the two distinct configurations for left- and right-traveling waves without passing through a complex double point; the homoclinic orbit does not separate the full NLS phase space Temporal chaos in the noneven regime 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 or gap state, not by moving through a double point. Similarly, when more complex double points are involved, the evolution of the spectrum from one configuration to another can be accomplished by executing an appropriate sequence of crossings of gap and cross states, and again does not entail homoclinic crossings. So do chaotic evolutions develop in the noneven regime in nearby systems? The answer is an emphatic yes, but they are no longer characterized by homoclinic crossings; instead they are produced by a new mechanism involving random bifurcations through nearby standing waves (see Fig. 2a). As an example, consider the initial data for a left-traveling modulated 3-phase Fig. 2. (a) Surface for u 0 = 0.5( (e 0.9i cos µx + e 60i sin µx)) obtained with the difference scheme DDNLS for 0 <t<500 and 5000 <t<5500; (b) The nonlinear spectrum at three time slices.

8 M.J. Ablowitz et al. / Physica D (2001) solution of the NLS equation u 0 = 0.5(1+0.01(e 0.90i cos µx + e 60i sin µx)) (15) with L = 2 2π, µ = 2π/L. Fig. 2a shows the surface (0 < t < 500) obtained with the following Hamiltonian discretization (DDNLS): i u n = (u n+1 + u n 1 2u n )/h u n 2 u n = 0 for N = 24. Initially, the waveform travels to the left; 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< 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. 2a), the first bifurcation from a left- to right-traveling wave occurs at t = The evolution of the spectrum due to the perturbation induced by DDNLS is shown in Fig. 2b at three successive time slices in this transition region when the waveform bifurcates from a left state (t = 131.5) to a right state (t = 132.1) by evolving through a cross state (t = 131.8). The evolution of spectrum was computed to 5000 and it showed random intermittent bursts in which the spectral configuration bifurcates between the left state and the right state. (For a complete presentation of the spectral results and description of the behavior in the noneven regime, see [17].) Although it is still an open question as to what analytical object under perturbation gives rise to the chaos in the noneven regime and how to rigorously prove the mechanism, it seems intuitively clear that the random switching between left- and right-running waves is similar to the switching between standing wave states observed in the even regime. In the noneven regime the signature of the chaos is different as there are no homoclinic crossings (observed both in discrete problems and now in water waves, as shown in Section 4). Even so, a persisting homoclinic structure seems a likely candidate to play a role in the ensuing chaos. For the results of a Mel nikov analysis of a simpler symmetry-breaking perturbation of the NLS carried out to explain the chaotic evolutions in terms of a transversal intersection of the homoclinic manifolds of persisting hyperbolic sets, see [23]. In the asymptotic theory, the HONLS equation is a more accurate approximation to the water wave dynamics and since it destroys even symmetry, homoclinic crossings are not allowed. However, the left right switching mechanism for chaotic excitations is available and is in fact observed in the laboratory experiments. The question at hand is: In the description of deep water waves, when is the NLS equation adequate and when is the HONLS equation critical to the description? 3. Laboratory results 3.1. Experimental apparatus and procedures The experimental apparatus comprised a wavetank with motion-controlled instrumentation carriage, water supply, wavepaddle, wavegages, and computer system. The glass wavetank is 14.3 m long by 25.4 cm wide and 20 cm in depth with a sanded, glass beach at the downstream end for energy absorption. At this depth it is sufficient to use the deep water limit. Mounted on the top are siderails that support the instrumentation carriage that is attached to a belt. The belt is driven by a servo-controlled motor that allows the instrumentation carriage to translate downstream of the wavemaker at the group velocity of the underlying wave train. The tank is cleaned with alcohol before it is filled with water to a depth greater than the desired 20 cm. The water is distilled in-house, then run through a filtering system that removes ions, organics and particles. A film of baby powder is spread on the surface (following Scott, 1979). The carriage is fitted with a brass rod and then translated down the tank to scrape the surface film toward the end of the tank. This film moves as a rigid body in front of the carriage, until it reaches the end of the tank where it is vacuumed with a wet-vac. This procedure provides for a reproducible water surface, for which the linear damping rate of the 3.33 Hz wave train is cm 1. This value is more than the standard boundary layer

9 424 M.J. Ablowitz et al. / Physica D (2001) calculation of clean surface damping, but less than that of a contaminated surface calculation. The water is allowed to rest for 15 min before experiments are conducted; the surface is cleaned at least every 2 h. Waves are generated using a vertically oscillated, anodized wedge with both position and velocity control. The programmed velocity for both sets of 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 measured the maximum amplitude of the envelope soliton at 23 positions down the tank. This amplitude decayed exponentially at a rate of cm 1 /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 1 4 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 surface displacement near the wavemaker to insure that the surface displacement there is reproducible. A mechanical 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 correlation coefficients 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 series) 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 coefficients among experiments of 0.99 or better, except for two experiments (among 40) for which the correlation coefficients were 0.96 or 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 1 4 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 second are 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 s to allow for transient motions to pass and then set in motion at the desired speed. In the temporal measurements we compare time series obtained from different experiments with identical initial conditions. These measurements are graphed against each other to produce a phase plane diagnostic 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 near the wavemaker produces a 45 line with a very slight width, indicating a 1 2% noise level. This indicates what should be expected from time series showing the wavefield evolution Envelope solitons The control experiment is performed with the soliton solution of NLS since theoretically it should evolve reproducibly (see below). The wavemaker was programmed to oscillate as η s (t) = a sin(ω 0 t)sech(aω 0 t/ 2) where ω 0 = rad/s, a = 0.2cm,k 0 = 0.44 rad/cm, ω (k 0 ) = 24.4cm/s (group velocity), g = 980 cm/s 2 and T = 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 reproducible, 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 wavemaker. Fig. 5a graphs the water surface displacements

10 M.J. Ablowitz et al. / Physica D (2001) 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 two data sets. Shifting each data point in one of the data sets the same amount ( s), produces the nearly perfect 45 line (Fig. 5a). The experiments using solitons indicate that we are able to conduct a reproducible 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 adequately describes Modulated periodic wave trains For modulated wave trains the position of the wavemaker is programmed to be η p (t) = a sin(ω 0 t)(1 + δ E sin ω p t) where a = 0.5cm,ω p = rad/s, δ E = 0.1, the values ω 0,g,k 0,T are the same 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 1 4 are shown in Figs. 6b e. 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

11 426 M.J. Ablowitz et al. / Physica D (2001) 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 840 cm 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. 6b d, presumably because of a mismatch in linear and actual group speed. We note that evidence of reflections 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 develops between the waves of two experiments and cannot be simply removed. Indeed subsequent experiments show that different Lissajous figures, with diverging 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 is reconstructed 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 successively 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 conducting 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 wavemaker, before the wave has reached the wavegages. This profile provides a benchmark for the level of

12 M.J. Ablowitz et al. / Physica D (2001) 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 = 15 s, 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 modulated Stokes wave trains evolve chaotically for certain parameter regimes. 4. Numerical experiments We have shown experimentally that periodically modulated nonlinear Stokes waves can evolve chaotically. 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 following: (1) we do some additional post-processing of the data from the laboratory experiments. We calculate the Floquet spectral decomposition of the laboratory data at sampled times. This establishes that the water wave dynamics is characterized by left right 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(ɛ) 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 as initial 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 parameters used in the laboratory experiments. We use a fourth-order pseudo-spectral code for integrating the HONLS equation (3) with N = 512 Fourier modes in 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 surface displacement η obtained using initial data u(x, 0) is graphed against the evolution of the surface displacement η obtained using u (x, 0), where u (x, 0)=u(x, 0)(1 + δu r (x)), δ is on the order of experimental error (1%) and u r (x) is taken to be a random field. The formula for the reconstruction of η (the phase plane plot is the one diagnostic where dimensional coordinates are presented) in terms of u

13 428 M.J. Ablowitz et al. / Physica D (2001) 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 1 4 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 both the physical and numerical experiments is projected onto the nonlinear spectrum of the NLS and we follow 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 only shown 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 soliton with zero velocity. The space time evolution of the waveform obtained using the HONLS equation with ɛ = 0.14, for 0 <t<5, is given in Fig. 8a. The soliton develops an O(ɛ) velocity and sheds a small amount of radiation off the front of the soliton, but the dynamics is regular. The phase plane plot, for 0 <t<10, remains close to the 45 line with little spread (Fig. 8b), as was observed in the laboratory experiments. Although the spectrum is not invariant, the spectral plots show that the spectrum does not change configuration and that there are only small O(ɛ) changes in the amplitude and speed of the soliton. This indicates that we obtain a reproducible experiment 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 corresponded to modulationally unstable periodic wave trains and is of the form u(x, 0) = a(1 + δ T cos µ n x) (16) with δ T = 0.1. We varied the amplitude a in the

14 M.J. Ablowitz et al. / Physica D (2001) Fig. 7. Spatial profiles corresponding to times of (a) 1.00 s, (b) s, (c) s, (d) s, (e) s, and (f) s 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 ɛ = 0.14 for 0 <t<5; (b) Phase plane plot of the surface amplitude (mm) η vs. η for model soliton initial data for 0 <t<10.

15 430 M.J. Ablowitz et al. / Physica D (2001) instability criterion to have M = 1, 3, 5 unstable modes nearby (and we denote this as the M unstable mode regime). In [21], we have shown that for data T there are 2n + 1 simple imaginary points in the spectrum: λ 0 and {λ n+,λ n },n= 1, 2,...,M. When δ T is asymptotically small, the simple points λ n+,λ n of the spectrum of L get successively closer to each other with their distance from a double point being O(δT n ), n = 1, 2...,M. The distance from a double 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, δ T = 0.1 and L = 2 2π), we found that the solution of the HONLS equation with ɛ = 0.14, 0 <t<2.5, is well approximated by the solution obtained using the NLS equation. The waveform does not display any temporal irregularities and homoclinic transitions do not occur; stable nonchaotic dynamics ensue. The phase plane plot is very close to the 45 line. The spectral configuration is the least complicated 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, the upper band of spectrum has moved slightly to the right of the imaginary axis which is reflected by the waveform developing a very small speed. Temporal irregularities or homoclinic transitions do not occur. As in the soliton case, for the one unstable mode case the NLS equation yields a satisfactory description of the dynamics as the HONLS terms do not significantly alter the dynamics. As the number of unstable modes is increased, the NLS equation loses this ability. Fig. 9a shows the surface obtained using the HONLS equation with ɛ = 0.28, 0 < t < 10 for initial data (16) with a = 0.7,δ T = 0.1 and L = 4 2π. In this higher unstable mode regime (M = 3), the behavior of HONLS solutions is considerably more complicated than what was previously observed for the discrete systems with noneven initial data (cf. Section 2). Here, each of the unstable modes is exhibiting left right switching leading to a very complex waveform. In addition to displaying temporal chaos, the solution appears to be spatially irregular. The phase plane plot is spread significantly away from the 45 line (see Fig. 9b). Additionally, numerous homoclinic transitions are observed in the spectrum. These transitions are similar to those observed for the experimental initial data case (also in the three unstable mode regime). Fig. 9. (a) Surface for initial data (16) with a = 0.7 and L = 4 2π obtained using the HONLS equation with ɛ = 0.28 for 0 <t<10; (b) Phase plane plot of the surface amplitude (mm) η vs. η for model soliton initial data for 0 <t<10.

16 M.J. Ablowitz et al. / Physica D (2001) Fig. 10. (a) Phase plane plot of η vs. η for experimental initial data for 0 <t<5. Notice the similarity to Fig. 5b. Spectral plots corresponding 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 experimental data (Figs. 10b f) 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

17 432 M.J. Ablowitz et al. / Physica D (2001) modes are not significant in the description of the chaotic state. For the case M = 5, we found that the phase plane plot diverges from the 45 line even more strongly than for M = 3. The spectrum evolved significantly and we found more numerous homoclinic transitions than with M = 3. The cases M = 3, 5 yield strong temporal irregularities and, for the time scales under consideration, chaotic dynamics. The examination of the model data demonstrates that when there are a higher number of unstable modes present initially, the NLS is inadequate and the HONLS perturbations 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 results. For the experimental data, denoted as E we use u(x, 0) = the spatial envelope of the modulated wave 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 the wavefield evolves, the experiment is rendered irreproducible. 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 of spectrum. 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. 10b g 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. 10e g). In this sequence of plots the second and third curves in the spectral configuration switch orientation and back again indicating left right flipping in the second mode. For the timeframe examined, 0 <t<5, there are frequent random left right homoclinic transitions in all of the unstable nonlinear modes. Each of these transitions corresponds to a change in the characteristics of the nonlinear modes and leads to physical changes in the wave field. Finally, we remark upon the spectral decomposition of 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. Significantly, the evolution is unmistakably characterized by left right 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 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.

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