Structural Stability for a Class of Nonlinear Wave Equations

Abstract

In this paper we discuss the structural stability of an initial value problem defined for the equation ut-utxx+auux=buxuxx+uuxxx (i.1) where a, b are constants, x Ğ„ ℝ , t Ğ„ ℝ. For the choices of a and b , (i.1) describe the nonlinear shallow water waves. Upper and lower bounds are derived for energy decay rate in every finite interval [0,T] which reveals that only the lower bound of the energy decays exponentially. Key Words: Degasperis-Procesi equation, Camassa-Holm equation, traveling wave