Well-Posedness and Orbital Stability of Periodic Traveling Waves for Schamel's Equation

Abstract

In this paper we will prove results of global well-posedness and orbital stability of periodic traveling-wave solutions related to the Schamel equation. The global solvability will be established by using compactness tools. In order to overcome the lack of smoothness of the nonlinearity, we approximate the original problem by regularizing the nonlinear term through a convenient polynomial near the origin. By following an adaptation of the well-known method introduced by Grillakis, Shatah, and Strauss, the orbital stability will be determined by proving, under certain conditions, that the periodic waves minimize an augmented Hamiltonian.