On periodic traveling wave solutions of Boussinesq equation

Abstract

This paper is concerned with periodic traveling wave solutions of a generalized Boussinesq equation in the form u t t = α u x x x x + ( f 0 ( u ) ) x x {u_{tt}} = \alpha {u_{xxxx}} + {({f_0}(u))_{xx}} . The basic approach to this problem is to establish an equivalence relation between a corresponding periodic boundary value problem for the traveling wave solution and a Hammerstein type integral equation. This integral representation generates a compact operator in the space of continuous periodic functions of the given period. It is shown by restricting the integral operator on a suitable domain that the Boussinesq equation has the trivial solution as well as a nonconstant periodic traveling wave solution. Special attention is given to the traditional Boussinesq equation where f 0 ( u ) = a u 2 {f_0}(u) = a{u^2} . Both of the so called “good” and “bad” Boussinesq equation are treated and the existence of nonconstant traveling wave solution is discussed.