Catherine Sulem

On asymptotic stability of solitary waves for nonlinear Schrödinger equations

Abstract We study the long-time behavior of solutions of the nonlinear Schrodinger equation in one space dimension for initial conditions in a small neighborhood of a stable solitary wave. Under some hypothesis on the structure of the spectrum of the linearized operator, we prove that, asymptotically in time, the solution decomposes into a solitary wave with slightly modified parameters and a dispersive part described by the free Schrodinger equation. We explicitly calculate the time behavior of the correction.

Nonlinear modulation of gravity waves: a rigorous approach

Weakly nonlinear gravity waves of given wavenumber in a horizontally unbounded two-dimensional domain are expected to undergo slow modulations in space and time. Together with an attendant analysis of the water wave equations, this work gives a mathematical justification of the modulation approximation. It proves that the resulting wavepacket, whose envelope is governed by the cubic nonlinear Schrodinger equation is a solution of the water wave equations to leading order. An upper bound of the remainder is also provided.

Solitary water wave interactions

This article concerns the pairwise nonlinear interaction of solitary waves in the free surface of a body of water lying over a horizontal bottom. Unlike solitary waves in many completely integrable model systems, solitary waves for the full Euler equations do not collide elastically; after interactions, there is a nonzero residual wave that trails the post-collision solitary waves. In this report on new numerical and experimental studies of such solitary wave interactions, we verify that this is the case, both in head-on collisions (the counterpropagating case) and overtaking collisions (the copropagating case), quantifying the degree to which interactions are inelastic. In the situation in which two identical solitary waves undergo a head-on collision, we compare the asymptotic predictions of Su and Mirie [J. Fluid Mech. 98, 509 (1980)] and Byatt-Smith [J. Fluid Mech. 49, 625 (1971)], the wavetank experiments of Maxworthy [J. Fluid Mech. 76, 177 (1976)], and the numerical results of Cooker, Weidman, and ...

The modulational regime of three-dimensional water waves and the Davey-Stewartson system

Abstract Nonlinear modulation of gravity-capillary waves travelling principally in one direction at the surface of a three-dimensional fluid leads to the Davey-Stewartson system for the wave amplitude and the induced mean flow. In this paper, we present a rigorous derivation of the system and show that the resulting wavepacket satisfies the water wave equations at leading order with precise bounds for the remainder. Key steps in the analysis are the analyticity of the Dirichlet-Neumann operator with respect to the surface elevation that defines the fluid domain, precise bounds for the Taylor remainders and the description of individual terms in the Taylor series as pseudo-differential operators and their estimates under multiple scale expansions.

Hamiltonian long–wave expansions for water waves over a rough bottom

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long–wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou (1983 Stud. Appl. Math. 68, 89–102) on periodic bottoms for two–dimensional flows.We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharovs Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990–1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length–scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length–scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two–dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg–de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long–scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three–dimensional long–wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev–Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple–scale operators. In the present case this involves the Dirichlet–Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

Stability of isotropic singularities for the nonlinear schro¨dinger equation

Abstract We describe a method by which the fully two- and three-dimensional cubic Schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities. In both cases, anisotropic initial data collapse very rapidly towards isotropic singularities. In three dimensions, the solutions become self-similar with a blowup rate ( t ∗ −t) -1 2 . In two dimensions, the self-similarity is weakly broken and the blowup rate is [( t ∗ −t)/ ln ln 1/(t ∗ −t)] -1 2 . The stability of the singular isotropic solutions is very firmly backed by the numerical results.

The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves

Abstract We analyze the self-focusing of gravity-capillary surface waves modeled by the Davey-Stewartson equations. This system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two-dimensional Schrodinger equation with a non local cubic nonlinearity. With accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical, as in the usual two-dimensional cubic Schrodinger equation, which is the deep water limit of the Davey-Stewartson equations. We also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation.

Interaction of lumps with a line soliton for the DSII equation

Abstract Exact solutions of the focussing Davey–Stewartson II equation are presented, which describe the interaction of N-lumps with a line soliton. These solutions are constructed by analysing the inverse spectral problem of the associated Lax pair. The dynamical properties of these solutions are also discussed.

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.

Global existence for the derivative nonlinear Schrödinger equation by the method of inverse scattering

ABSTRACT We develop inverse scattering for the derivative nonlinear Schrödinger equation (DNLS) on the line using its gauge equivalence with a related nonlinear dispersive equation. We prove Lipschitz continuity of the direct and inverse scattering maps from the weighted Sobolev spaces H2,2(ℝ) to itself. These results immediately imply global existence of solutions to the DNLS for initial data in a spectrally determined (open) subset of H2,2(ℝ) containing a neighborhood of 0. Our work draws ideas from the pioneering work of Lee and from more recent work of Deift and Zhou on the nonlinear Schrödinger equation.