We study the relevance of various scalar equations, such as inviscid Burgers’, Korteweg-de Vries (KdV), extended KdV, and higher order equations (of Camassa-Holm type), as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified with two approaches. The first method consists in approximating the flow with two decoupled, cou...

Double cnoidal waves of the Korteweg-de Vries equation are studied by direct solution of the nonlinear boundary value problem. These double cnoidal waves, which are the spatially periodic generalization of the well-known double soliton, are exact solutions with two independent phase speeds. The equation is written in terms of two phase variables and expanded in two-dimensional Fourier series. T...

R esum e. La propagation unidirectionnelle d'ondes de faible amplitude et de grande longueur d'onde est d ecrite, dans de nombreux syst emes physiques, par l' equation de Korteweg-de Vries. L'objet de ce travail est de proposer un probl eme mixte bien pos e lorsque le do-maine spatial est born e. Plus pr ecis ement nous etablissons l'existence de solutions locales en temps pour des donn ees ini...

B E RT VA N E S , P E T E R S P R E I J 1 and HARRY VAN ZANTEN 2 Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. E-mail: [email protected]; [email protected] Division of Mathematics and Computer Science, Faculty of Sciences, Free University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands...

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P Es) is reviewed. The method assumes that the P Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for ...

Long's equation describes stationary flows to all orders of nonlinearity and dispersion. Dissipation is neglected. In this paper, Long's equation is used to attempt to model the propagation of a solibore -a train of internal waves in shallow water at the deepening phase of the internal tide. 1. The Solibore Phenomenon The internal tide in shallow water often has a sawtooth shape rather than a s...

In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg-de Vries (vcKdV) equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV e...

Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dyn...

The transformation of a solitary wave on an underwater step is studied analytically and numerically. The theoretical model includes the linear potential description of the wave transformation on a step and the weakly nonlinear theory of long waves based on the Korteweg–de Vries equation for reflected and transmitted waves far from a step. Numerical simulation of solitary wave transformation on ...

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristi...