Advances in Mathematical Physics

Volume 2015 (2015), Article ID 925715, 7 pages

http://dx.doi.org/10.1155/2015/925715

## Orbital Stability of Solitary Traveling Waves of Moderate Amplitude

Department of Mathematics, Foshan University, Foshan, Guangdong 528000, China

Received 2 December 2014; Accepted 24 January 2015

Academic Editor: Stephen C. Anco

Copyright © 2015 Zhengyong Ouyang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

In this paper, we consider an equation for surface waves of moderate amplitude in the shallow water regime as follows: where the parameters and denote amplitude and shallowness. The nonlinear evolution equation (1) arises as an approximation of the Euler equations [1, 2]. 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. The Camassa-Holm (CH) equation is completely integrable and bi-Hamiltonian, which possesses soliton, peakon, and compacton solutions [5–12], especially breaking waves; that is, the solution remains bounded but its slope becomes unbounded [13]. Besides, the solutions of CH equation also include global weak solutions [11, 14, 15]. The stability problems of the solutions for the CH equation and its generalized forms were investigated [16–21], orbital stability of smooth solitary waves, peaked solitary waves and multisolitons were proved. As described the above, many results for waves of small amplitude have been obtained via the CH equation and its generalized forms. However, it is interesting and necessary to study waves of moderate amplitude. Therefore, the moderate amplitude wave equation (1) was investigated and some results were obtained. For instance, the problems of local well-posedness were considered and the existence and uniqueness of solutions were proved [2]. Only in the special case of parameters and , the orbital stability of solitary traveling waves was proved by a method proposed by Grillakis et al. [22, 23]. Nevertheless, the stability problems of solutions for (1) are not solved yet when parameters and take any values.

In this letter, by a solitary wave we mean a localized traveling wave. 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. Indeed, in this case the only type of stability is that of the shape of the wave, a fact that is captured by the notion of orbital stability, as pioneered by considerations made in celestial mechanics. Set a solitary wave , , where is wave speed. Substituting into (1), we have By integration with respect to , we obtain from (2) that where is an integration constant. Due to the solutions considered which satisfy the property that they are localized and that and its derivatives decay at infinity, in (3) the constant of integration .

#### 2. Preliminaries

In (1), the so called amplitude and shallowness parameters and relate the average length , amplitude , and water depth satisfying , (see Figure 1).