Advances in Mathematical Physics

Volume 2015, Article ID 789269, 6 pages

http://dx.doi.org/10.1155/2015/789269

## Stability of Negative Solitary Waves for a Generalized Camassa-Holm Equation with Quartic Nonlinearity

^{1}Department of Basic Courses, Guangzhou Maritime Institute, Guangdong 510725, China^{2}Department of Mathematics, Foshan University, Guangdong 528000, China

Received 6 October 2015; Revised 1 December 2015; Accepted 8 December 2015

Academic Editor: Christian Engstrom

Copyright © 2015 Shan Zheng and 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 stability of negative solitary waves to a generalized Camassa-Holm equation with quartic nonlinearity. We obtain the existence of negative solitary waves for any wave speed and some of their qualitative properties and then prove that they are orbitally stable by using a method proposed by Grillakis et al.

#### 1. Introduction

The Camassa-Holm equation (CH for short)was first derived by the method of recursion operators by Fokas and Fuchssteiner for studying completely integrable generalization of the KdV equation with bi-Hamiltonian structure in [1] and later proposed physically by Camassa and Holm as a model for unidirectional propagation of shallow water waves over a flat bottom in [2], and it was shown that the CH equation is completely integrable and possesses an infinite number of conservation laws. It is very different from the KdV equation that the CH equation has peakon solution and breaking waves (see [2–4]). As mentioned in [5], it is interesting to find that both phenomena of soliton interaction and wave breaking can be exhibited in one mathematical model of shallow water waves. The CH equation was well studied in view of mathematical point and a lot of results were established. For example, the Cauchy problem for CH and periodic CH equation were studied in [6–8], the global weak solutions and global conservative and dissipative solutions were obtained in [9–13], and the peakon and smooth solitary wave solutions were proved to be orbitally stable and interact like solitons [14–17] on the wave breaking; we refer to [3, 18–21].

Furthermore, considerable researches have studied the following generalized CH equation [22–27]:they have focused on the stronger nonlinear convection; that is, the nonlinear convection term in (1) has been changed to in (2), which makes the structure of their solutions change significantly. There are many new nonlinear phenomena arising from (2), such as compacton solitons with compact support, solitons with cusps, or peakons (cf. [28–36]). Four simple ansätze were proposed to obtain abundant solutions: compactons, solitary patterns solutions having infinite slopes or cusps, and solitary waves in [28]. By using bifurcation method, peakons and periodic cusp waves were studied in [32–34]; the explicit expressions of peakons for (2) are given in some special cases. In [35] some new exact peaked solitary waves were derived. By employing polynomial ansätze the periodic wave and peaked solitary waves of (2) were investigated in [36]. The Cauchy problem of (2) was studied in [37, 38] and the local existence was established. For and , the negative solitary wave to (2) was obtained and proved to be orbitally stable for any speed in [22]. Moreover, the stability problem of solitary wave to (2) was investigated as and in [39].

Our study is closely related to the results in [39]. For convenience, we write (2) when in the following form:In [39], when the parameter , (3) was shown to be Painlevé nonintegrable and to have positive solitary waves as wave speed . The solitary waves were proved to be unstable when the wave speed tends to the critical value and stable while the wave speed is a little bigger than the critical value.

However, when in (3), the stability problem of the solitary waves has not been solved yet. In this paper, we consider this stability problem. Unlike the results for positive solitary waves in [39], we show that there exist negative solitary waves for any wave speed . The point lying in our results is that we can actually determine that the scalar function (see below) is convex with respect to wave speed ; that is, all the negative solitary waves are orbitally stable.

#### 2. Preliminaries

##### 2.1. Hamiltonian System and Conservation Laws

Equation (3) can be rewritten in the following Hamiltonian form:where is a Hamiltonian operator, and (see below) denote Riesz representations of Fréchet derivatives of and , andis a functional of .

Another functional of is given bywhich can be treated as the kinetic energy of the waves. Both quantities and are critically important to the proof of solitary waves, which are shown to be conserved by the following lemma.

Lemma 1. *The functionals and defined above are conserved quantities under (3).*

* Proof. *Multiplying (3) by and integrating over we haveTo show that is invariant with respect to , we need to use the Hamiltonian structure of (3). It follows from (4) thatUsing the skew symmetry of Hamiltonian operator we obtain that

##### 2.2. Definition of Orbital Stability

As already observed by Benjamin and coworkers [40, 41], a solitary wave cannot be stable in the strictest sense of the word. To understand this, consider two solitary waves with different height, centered initially at the same point. Since the two waves have different amplitude and they have different velocity, as time passes the two waves will apart, no matter how small the initial difference was. However, in the situation just described, it is evident that two solitary waves with slightly differing height will stay similar in shape during the time evolution. An acceptable notion of stability is given by measuring the difference in shape. This sense of orbital stability was introduced by Benjamin [40]. We say a solitary wave is orbitally stable if a solution of (3) that is initially sufficiently close to a solitary wave will always stay close to a translation of the solitary wave during the time of evolution. A more mathematically precise definition is given as follows.

*Definition 2. *The solitary wave is stable if, for every , there exists a , such that if and is a solution of (3) for some with the initial value , then for all one has where is a translation of . The solitary is unstable if is not stable.

#### 3. Negative Solitary Waves and Well-Posedness

##### 3.1. Negative Solitary Waves and Their Properties

Letting , , and substituting it into (3), it follows thatthe prime denotes derivative with respect to . Integrating the above equation once we havewhere the integral constant takes zero due to the solitary waves vanish at infinity.

Lemma 3. *When the parameter in (3), for any wave speed , there exist negative solitary waves for (3).*

* Proof. *In fact, we only need to show the existence of homoclinic orbit corresponding to the negative solitary wave. Equation (12) can be rewritten as the following planar system:By using the transform , system (13) can be rewritten aswhich is equivalent to system (13) except the singular line ; it is more convenient to study (14) than (13). The first integral of (14) isSystem (13) has two equilibrium points: one at the origin and another given by . By the same bifurcation method used in [32–34], it is easy to determine that is a saddle point, is a center point, and there is a homoclinic orbit which corresponds to the negative solitary wave (see Figure 1). The homoclinic orbit proceeds from the saddle point , encircles the center , and returns to the origin. It passes through the point on -axis.