Orbital Stability of Solitary Traveling Waves of Moderate Amplitude

Zhengyong Ouyang

doi:10.1155/2015/925715

Abstract

We consider the orbital stability of solitary traveling wave solutions of an equation describing the free surface waves of moderate amplitude in the shallow water regime. Firstly, we rewrite this equation in Hamiltonian form and construct two invariants of motion. Then using the abstract stability theorem of solitary waves proposed by Grillakis et al. (1987), we prove that the solitary traveling waves of the equation under consideration are orbital stable.

Highlights

We consider an equation for surface waves of moderate amplitude in the shallow water regime as follows: ut
Based on an equation first derived by Johnson [3], on the one hand, one can derive a Camassa-Holm equation at a certain depth below the fluid surface for small amplitude waves [4], on the other hand, for the free surface, a corresponding equation (1) can be derived for waves of moderate amplitude in the shallow water regime
Many results for waves of small amplitude have been obtained via the CH equation and its generalized forms

Summary

Introduction

We consider an equation for surface waves of moderate amplitude in the shallow water regime as follows: ut. The stability problems of the solutions for the CH equation and its generalized forms were investigated [16,17,18,19,20,21], orbital stability of smooth solitary waves, peaked solitary waves and multisolitons were proved. In the special case of parameters α = 4 and β = √12, the orbital stability of solitary traveling waves was proved by a method proposed by Grillakis et al [22, 23]. The notion of stability is orbital stability, which is the appropriate notion for model equations whose solitary waves are such that the height is proportional to the speed.

Preliminaries

Stability of Solitary Waves

Proof of Stability