A class of doubly periodic waves for nonlinear evolution equations

Abstract

A class of doubly periodic waves for several nonlinear evolution equations is studied by the Hirota bilinear method. Analytically these waves can be expressed as rational functions of elliptic functions with different moduli, and may correspond to standing as well as propagating waves. The two moduli are related by a condition determined as part of the solution process, and the condition translates into constraints on the wavenumbers allowed. Such solutions for the nonlinear Schrödinger equation agree with results derived earlier in the literature by a different method. The present method of combining the Hirota method, elliptic and theta functions is applicable to a wider class of equations, e.g., the Davey-Stewartson, the sinh-Poisson and the higher dimensional sine-Gordon equations. A long wave limit is studied for these special doubly periodic solutions of the Davey-Stewartson and Kadomtsev-Petviashvili equations, and the results are the generation of new solutions and the emergence of the component solitons as the fundamental building blocks, respectively. The validity of these doubly periodic solutions is verified by MATHEMATICA. © 2002 Elsevier Science B.V. All rights reserved.