Stability of smooth periodic travelling waves in the Dullin-Gottwald-Holm equation

TL;DR

This paper investigates the existence, stability, and spectral properties of smooth periodic traveling wave solutions in the Dullin-Gottwald-Holm equation, providing new criteria and conditions for their stability.

Contribution

It introduces a concise method for establishing the existence of periodic solutions and develops a spectral stability criterion within a functional-analytic framework.

Findings

Existence of smooth periodic traveling solutions is confirmed.

Monotonicity of the period function is established using Chicone's criterion.

Conditions for orbital stability of solutions are identified.

Abstract

The existence of smooth periodic traveling solutions in the Dullin-Gottwald-Holm (DGH) equation and the monotonicity of the period function are clarified. By introducing two suitable parameters, we show the existence of periodic travelling solutions of DGH equation in a concise way. The monotonicity of period function with respect to different variables are proved by using Chicone's criterion and the method developed by Geyer and Villadelprat. The problem of the spectral stability of smooth periodic waves in the DGH equation is discussed. Within the functional-analytic framework, we obtain a criterion for the spectral stability of smooth periodic traveling waves in DGH equation. In addition, we show the smooth periodic travelling solutions are orbitally stable under certain conditions.