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Numerical simulation of gravity waves
Abstract We present a new spectral method to simulate numerically the waterwave problem in a channel for a fluid of finite or infinite depth. It is formulated in terms of the free surface elevation ηExpand
The nonlinear Schrödinger equation : self-focusing and wave collapse
Basic Framework.- The Physical Context.- Structural Properties.- Rigorous Theory.- Existence and Long-Time Behavior.- Standing Wave Solutions.- Blowup Solutions.- Asymptotic Analysis near Collapse.-Expand
Nonlinear Schrödinger Equation
We now start with the last major part of the Internet Seminar which is devoted to the basic version of the nonlinear Schrödinger equation. In this lecture we first discuss a bit its derivation andExpand
On the continuous limit for a system of classical spins
The continuum limit of a cubic latice of classical spins processing in the magnetic field created by their closest neighbours is considered. Results concerning existence, uniqueness and (forExpand
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 analysisExpand
Tracing complex singularities with spectral methods
Abstract A numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems. With initial data analytic in the spaceExpand
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 integrableExpand
Longtime dynamics of a conductive fluid in the presence of a strong magnetic field
We prove existence in the large of localized solutions to the MHD equations for an ideal conducting fluid subject to a strong magnetic field. We show that, for large time, the dynamics may reduce toExpand
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 waveExpand
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 someExpand