A nonexistenee theorem of small periodic traveling wave solutions to the generalized Boussinesq equation

Abstract

The generalized Boussinesq equation, u_tt - u_xx+ [f(u)]_xx + u_xxxx = 0, and its periodic traveling wave solutions are considered. Using the transform z = x - ω 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 equation is then established by using the Green 's function method. This integral equation generates compact operators in (C_2T, ∥ ⋅ ∥) , a Banach space of real-valued continuous periodic functions with a given period 2T. We prove that for small T > 0, there exists an r > 0 such that there is no nontrivial solution to the integral equation in the ball B(0, r) ⊆ C_2T. And hence, the generalized Boussinesq equation has no 2T-periodic traveling wave solutions having amplitude less than r