Bifurcation of traveling wave solutions for the BBM-like B(2, 2) equation

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Abstract

In this paper, we employ the bifurcation method of dynamical systems to investigate the BBM-like B(2, 2) equation. The phase portrait bifurcation of the traveling wave system corresponding to the equation is given. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth traveling wave solutions are given. Through some special phase orbits, Some solitary wave solutions expressed by implicit functions, periodic cusp wave solution, compacton solution and peakon solution are obtained.

Introduction

In recent years, nonlinear wave equations have played important roles in many scientific and engineering fields such as fluid mechanics, nonlinear optics and quantum mechanics. Thus, it has had considerable attention to find exact traveling wave solutions of those problems. Many methods have been presented to obtain new traveling wave solutions for many nonlinear equations such as the homogeneous balance method [1], the tanh function method [2], [3], sine–cosine method [4], [5], the Jacobi elliptic function method [6], [7], EXP function method [8], the bifurcation method of dynamical systems [9], [10], [11], [12] and so on. Among them, the bifurcation method has been proved to be a powerful mathematical tool to investigate the traveling wave solutions for nonlinear evolution equations [9], [10], [11], [12], [13].

BBM equation or regularized long-wave equation (RLW equation)ut+uux-uxxt=0was derived by Peregine [14], [15] and Benjamin et al. [16] as an alternative model to Korteweg–de Vries equation for small-amplitude, long wavelength surface water waves. The various generalized form of this equation have been considered by many authors [17], [18], [19]. Wazwaz [17] introduced a system of nonlinear variant RLW equationsut+aux-k(un)x+b(un)xxt=0and derived some compact and noncompact exact solutions by using the sine–cosine method and tanh method. Feng et al. [18] studied the following generalized variant RLW euqations:ut+aux-k(um)x+b(un)xxt=0.By using four different ansatzs, they obtained some new exact solutions such as compactons, solitary pattern solutions, solitons and periodic solutions. In order to understand the role of nonlinear dispersion in the formation of patterns in an undular bore, Shang [19] introduced a family of BBM-like equations with nonlinear dispersion, B(m, n) equationsut+(um)x-(un)xxt=0,m,n>1.He presented a new method called the extend sine–cosine method to seek the exact special traveling wave solutions. Especially, the compacton and solitary pattern solutions of the case m = n = 2 of Eq. (1.4) are given.

Motivated by the rich treasure of the case m = n = 2 of Eq. (1.4), in the present paper, we will employ the bifurcation method to study the traveling wave solutions for the case m = n = 2 of Eq. (1.4), i.e. the following BBM-like B(2, 2):ut+(u2)x-(u2)xxt=0.Two types of solitary wave solutions expressed as implicit functions, periodic cusp wave solution, peakon solution and compacton solution are obtained. To the best of our knowledge, the solitary wave solution, periodic cusp wave solution and peakon solution have not been reported in the literature.

The rest of this paper is organized as follows. In Section 2, we change the BBM-like B(2, 2) equation into the corresponding traveling wave system and discuss the bifurcation of the phase portraits of the traveling wave system. In Section 3, we give some analytic expressions of exact traveling wave solutions.

Section snippets

Bifurcation and phase portraits of traveling wave system

It is well known that a traveling wave solution of Eq. (1.5) with wave speed c is a solution of the form u(x,t)=φ(ξ) with ξ=x-ct. Substituting u(x,t)=φ(x-ct)=φ(ξ) into Eq. (1.5), we can obtain the following ODE-cφ+(φ2)+c(φ2)=0.Integrating Eq. (2.1) once with regard to ξ leads tog-cφ+φ2+2c(φ)2+2cφφ=0,where g is an integral constant.

Let y=φ, then Eq. (2.2) is equivalent to the following planar dynamical systemdφdξ=y,dydξ=cφ-φ2-g-2cy22cφ,with the first integralH(φ,y)=-cφ2y2-14φ4+c3φ3-g2φ2=h,

Some exact traveling wave solutions of Eq. (1.5)

In this section, through some special phase orbits, we will give some analytic expressions of traveling wave solutions of Eq. (1.5).

  • (1)

    For (c,g)A2, corresponding to two heteroclinic orbits (arc curves) defined by H(φ,y)=0 (see Fig. 2(c)), we havey=±12-cφ2-4c3φ+2gforφ+φ<0andφ=0for--g2cy-g2c,where φ+=2c3+4c2-18g3. Thus, by using the first equation of (2.3), we obtain the periodic cusp wave solution of Eq. (1.5)φ(ξ)=β+exp-|ξ-2nT|2-c+β-exp|ξ-2nT|2-c+2c3,ξ[(2n-1)T,(2n+1)T),where β±=-c3±2g2 and T=

Acknowledgements

The authors thank the anonymous reviewers for their helpful comments and suggestions which have helped us to improve it. This research was supported by National Natural Science Foundation of China(Grant No. 11102076), the Foundation of Qinlan Engineering of Jiangsu Province of China (Grant No. KYQ10004), the Dr. Start-up Fund of Jiangsu Teachers University of Technology (Grant No. KYY10064), and the Natural Science Foundation of Jiangsu Teachers University of Technology (Grant No. KYY10045).

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