Two types of bounded traveling-wave solutions of a two-component Camassa–Holm equation

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Abstract

The Camassa–Holm equation (CH) is a well known integrable equation, with is used to describe the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of peakons from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in dynamics. In this paper, the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study CH2 equation. Two types of bounded traveling waves are found. One of them is called compacton, another is called generalized kink wave. The planar graphs of compactons and generalized kink awaves are simulated by using software Maple. Exact explicit parameter expressions of the compactons and implicit expressions of the generalized kink waves solutions are given.

Introduction

It is well known that the Camassa–Holm (CH) equation [1]mt+umx+2mux=0,wherem=u-α2uxxis a model for unidirectional nonlinear dispersive waves in shallow water. This equation has attracted a lot of attentions over the past decade due to its interesting mathematical properties, e.g., it is an integrable equation and admits peakons. Camassa and Holm [1] showed that Eq. (1.1) has peakon of the form u(x,t)=ce-|x-ct|/α. Boyd [2] gave three analytical representations of the Eq. (1.1) for the spatially periodic generalization of the peakon. Wazwaz [3] investigated a family of Camassa–Holm equations with distinct parameters, and obtained some traveling wave solutions. Lai [4] investigated a shallow water equation of Camassa–Holm type, owns nonlinear dissipative effect, established the existence and uniqueness of its local solution in Sobolev space Hs(R) with s>32. Cai et al. [5] studied a modified Camassa–Holm equation, and obtained several exact traveling solutions.

Dullin, Gottwald and Holm [6], [7] considered the following integrable shallow water equationmt+c0ux+umx+2mux=-γuxxx,wherem=u-α2uxx,which is called CH-γ equation, where α,c0 and γ are constants and α0. In [6], Dullin et al. identified how the dispersion coefficients for the linearized water waves appear as parameters in the isospectral problem for this inverse scattering transform integrable equation. They also determined how the linear dispersion parameters α,c0 and γ in Eq. (1.2) affect the isospectral content of its soliton solutions and the shape of its traveling waves. In [7], Dullin et al. showed the asymptotically equivalences of Eq. (1.2) and other shallow water wave equations. In [8], Guo and Liu obtained some peaked wave solutions of Eq. (1.2), and the limite property of the periodic cusp waves was also discussed. In [9], Guo and Liu obtained compacton and generalized kink wave solutions of Eq. (1.2). The compacton has been found in RH equation by Rosenau and Hyman [10]. The compactons mean the solitons with compact support, or the strict localization of solitary waves. The generalized kink wave is discovered by Liu [11]. Since it is defined on semifinal bounded domain and possesses some properties of the kink wave, so call it generalized kink wave. In [12], Xie and Cai obtained the compacton and the generalized kink wave solutions of EX-ROE. In [13], Xie and Wang obtained the compacton and the generalized kink wave solutions of CH-DP.

The Eq. (1.1) admits many integrable multicomponent generalizations [15], the most popular of which is two-component Camassa–Holm equation (CH2)mt+umx+2mux=-σρρx,wherem=u-α2uxx,ρt+(ρu)x=0,where σ=±1. This system appeared originally in [14] and its mathematical properties have been extensively studied many authors, e.g. [15], [16], [17], [18], [19], [20], [21], [22], [23]. The existence and exact parametric representations of solitary wave solutions, kink and anti-kink wave solutions, smooth and non-smooth periodic wave solutions of Eq. (1.3) were obtained by Li and Li [22].

Many methods have been used to investigate traveling wave solutions to nonlinear equations. The transformed rational function method [24] generates traveling wave solutions to nonlinear equations, and this method systematically presents traveling wave solutions. Moreover, the superposition principle has recently been used to construct subspaces of solutions to Hirota bilinear equation [25] and generalized bilinear equations [26]. Here, our aim in this paper is to use the bifurcation method of planar systems and simulation method of differential equations [5], [8], [9], [10], [11], [12], [13], [22] to investigate the compacton and the generalized kink wave solutions of the Eq. (1.3). The exact parameter expressions of compactons and the implicit expressions of generalized kink waves to the CH2 are obtained. The results of compacton and generalized kink wave solutions are new to the CH2 equation.

The rest of this paper is organized as follows. In Section 2, we firstly derive traveling wave equation and traveling wave system. Then we study the bifurcation of phase portraits of the traveling wave system. In Section 3, using the information of phase portraits, we make the numerical simulation for bounded integral curves of traveling wave equation. In Section 4, we obtain the parameter expressions of compactons and implicit expressions of the generalized kink waves from the bifurcation of phase portraits and numerical simulation. Finally, a short conclusion is given in Section 5.

Section snippets

Bifurcation phase portraits

Let u(x,t)=ϕ(x-ct)=ϕ(ξ),ρ(x,t)=ψ(x-ct)=ψ(ξ), with ξ=x-ct, where c is the wave speed. Then, second equation of (1.3) becomes-cψ+(ψϕ)=0,where “” is the derivative with respect to ξ. Integrating this equation once and setting the integration constant as A,A0, we haveψ(ξ)=Aϕ-c.The first equation of (1.3) becomesα2((ϕ-c)ϕ)=-12α2(ϕ)2-cϕ+32ϕ2+σψψ.Integrating this equation once and setting the integration constant as B, we have the following traveling wave equation.α2(ϕ-c)ϕ=-12α2(ϕ)2-cϕ+32ϕ2+

Numerical simulations of traveling wave equation

From the derivation in Section 2 we see that the bounded traveling waves of Eq. (1.3) correspond to the bounded integral curves of Eq. (2.4), and the bounded integral curves of Eq. (2.4) correspond to the orbits of system (2.5) in which ϕ=ϕ(ξ) is bounded. Therefore we can simulate the bounded integral curves of Eq. (2.4) by using the information of the phase portraits of system (2.5).

From Fig. 1, it is seen that ϕ=ϕ(ξ) is bounded in the following orbits of system (2.5):

(1) The homoclinic

The expressions of compactons and generalized kink waves

In this section we will search for the regions of parameter plane that compactons and the generalized kink waves appear, and derive their exact expressions. We assume that (ϕ0,0) is the initial point of an orbit of system (2.5), and from (2.8), it has expression H(ϕ,y)=h0, where h0=H(ϕ0,0).

LetG(ϕ)=(ϕ-c)(ϕ3-cϕ2+2Bϕ+h0)-Aσ.From H(ϕ,y)=h0, the following equation determines the orbit of passing through (ϕ0,0).α2(ϕ-c)2y2=G(ϕ).

By applying transformationdξ=α(ϕ-c)dτ,to the first expression of (2.5), we

Conclusion

In this paper, the bifurcation and global behavior on CH2 are studied, and the representations of compactons and generalized kink waves and the regions that compactons and generalized kink waves appear are obtained. In addition, their planar graphs are simulated for some parameters (see Fig. 2). It is worth mentioning that the solutions of compactons and generalized kink waves are new to the CH2, and all solutions presented in this paper have been verified as solutions of the PDEs by using

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  • This work was supported by the Natural Scienc Foundation of China (No.11171115)

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