Existence of periodic traveling wave solutions to the generalized forced Boussinesq equation (Q1587826)

Summary: The generalized forced Boussinesq equation, \[ u_{tt}- u_{xx}+ [f(u)]_{xx}+ u_{xxxx}= h_0, \] and its periodic traveling wave solutions are considered. Using the transform \(z= x-\omega t\), 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 inequation 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 period \(2T\). 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.