Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation

Kenneth L Jones,Yunkai Chen

doi:10.1155/s0161171299226439

Abstract

The generalized forced Boussinesq equation,utt−uxx+[f(u)]xx+uxxxx=h0, and its periodic traveling wave solutions are considered. Using the transformz=x−ωt, the equation is converted to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relation between the ordinary differential equation and a Hammerstein type integral equation is then established by using the Green's function method. This integral equation generates compact operators in a Banach space of real-valued continuous periodic functions with a given period2T. The Schauder's fixed point theorem is then used to prove the existence of solutions to the integral equation. Therefore, the existence of nonconstant periodic traveling wave solutions to the generalized forced Boussinesq equation is established.

Highlights

In the 1870’s, Boussinesq derived some model equations for the unidirectional propagation of small amplitude long waves on the surface of water [2]
Boussinesq’s theory was the first to give a satisfactory and scientific explanation of the phenomenon of solitary waves discovered by Scott Russell [8]
The original equation obtained by Boussinesq is not the only mathematical model for small amplitude, planar, and undirectional long waves on the surface of shallow water

Summary

Introduction

In the 1870’s, Boussinesq derived some model equations for the unidirectional propagation of small amplitude long waves on the surface of water [2]. We shall prove an existence theorem of periodic traveling wave solutions to this equation following the idea of Liu and Pao [5], and Chen and He [4]. The main result, the existence of periodic traveling wave solutions to the generalized forced Boussinesq equation is established. It is obvious that any solution U (z) of the boundary value problem consisting of equations (2.2), (2.3), and (2.4) can be extended to a 2T -periodic traveling wave solution to the original Boussinesq equation (2.1). This, in turn, leads to the existence of 2T -periodic traveling wave solution U (z) to the generalized forced Boussinesq equation. [10], we can obtain the following results from equations (3.1), (3.2), and (3.3)

Ꮽiv dᏭiv dz

It is worth nothing that as long as