Quasi-periodic wave solutions and asymptotic behavior for an extended ( 2 + 1 ) $(2+1)$ -dimensional shallow water wave equation

Wenjuan Rui

doi:10.1186/s13662-016-0832-4

Abstract

Based on Riemann theta function and bilinear Backlund transformation, quasi-periodic wave solutions are constructed for an extended $(2+1)$ -dimensional shallow water wave equation. A detail asymptotic analysis procedure to the one- and two-periodic wave solutions are presented, and the asymptotic properties of this type of solutions are proved. It is shown that the quasi-periodic wave solutions converge to the soliton solutions under small amplitude limits.

Highlights

1 Introduction Nonlinear evolution equations (NLEEs) have attracted much interest in the past few decades since they appear in many areas of scientific fields such as fluid mechanics, plasma physics, solid-state physics, and mathematical biology [ – ]
The investigation of solutions for NLEEs plays an important role in the study of nonlinear physical phenomena, and many effective methods have been discovered
Based on bilinear forms, Nakamura proposed a straightforward way to construct a kind of quasi-periodic solutions of nonlinear equations [, ], where the quasi-periodic wave solutions of the Korteweg-de Vries equation (KdV equation) and the Boussinesq equation were obtained by using the Riemann theta function

Summary

Introduction

Nonlinear evolution equations (NLEEs) have attracted much interest in the past few decades since they appear in many areas of scientific fields such as fluid mechanics, plasma physics, solid-state physics, and mathematical biology [ – ]. ). one objective of this paper is to construct one- and two-periodic wave solutions for the extended ( + )-dimensional shallow water wave equation Another objective of the paper is to investigate the asymptotic behavior of the quasi-periodic wave solutions. In Sections and , we apply the Riemann theta function to construct one- and two-periodic wave solutions for Eq ; μN = , , θ(ξ , , |τ ) and θ(ξ , , |τ ) are quasi-periodic solutions of the bilinear equation. Proposition plays an important role in constructing quasi-periodic wave solutions for coupled bilinear equations.

One-periodic waves and asymptotic properties

Conclusions