The Traveling Wave Solutions and Their Bifurcations for the BBM-Like B(m, n) Equations

by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like B(3, 2) equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-like B(4, 2) equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.


Introduction
In recent years, the nonlinear phenomena exist in all fields including either the scientific work or engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, chemical kinematics, and chemical physics.Many nonlinear evolution equations are playing important roles in the analysis of the phenomena.
BBM equation or regularized long-wave equation (RLW equation) was derived by Peregine [12,13] and Benjamin et al. [14] as an alternative model to Korteweg-de Vries equation for smallamplitude, long wavelength surface water waves.
There are various generalized form related to (1).Shang [15] introduced a family of BBM-like equations with nonlinear dispersion which were called BBM-like (, ) equations as alternative model to the nonlinear dispersive (, ) equations [16][17][18].He presented a method called the extend sine-cosine method to seek exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of (2).When  =  = 2, (2) reduces to the BBM-like (2, 2) equation Jiang et al. [19] employed the bifurcation method of dynamical systems to investigate (3).Under different parametric conditions, they gave various sufficient conditions to guarantee the existence of smooth and nonsmooth traveling wave solutions.Furthermore, through some special phase orbits, they obtained some solitary wave solutions expressed by implicit functions, periodic cusp wave solution, compacton solution, and peakon solution.
and derived some compact and noncompact exact solutions by using the sine-cosine method and tanh method.Feng et al. [21] studied the following generalized variant RLW equations:   +   − (  )  + (  )  = 0. (5) By using four different ansatzs, they obtained some exact solutions such as compactons, solitary pattern solutions, solitons, and periodic solutions.Kuru [22][23][24] considered the following BBM-like equations with a fully nonlinear dispersive term: By means of the factorization technique, he obtained the traveling wave solutions of ( 6) in terms of the Weierstrass functions.
In the present paper, we use the bifurcation method and numerical simulation approach of dynamical systems to study the following BBM-like (, ) equations: For BBM-like (3, 2) equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blowup solution.We also reveal the relationships among these solutions theoretically.For BBM-like (4, 2) equation, we construct two elliptic periodic wave solutions and two hyperbolic blow-up solutions.This paper is organized as follows.In Section 2, we state our main results which are included in two propositions.In Sections 3 and 4, we give the derivations for the two propositions, respectively.A brief conclusion is given in Section 5.

Main Results and Remarks
In this section, we list our main results and give some remarks.Firstly, let us recall some symbols.The symbols sn  and cn  denote the Jacobian elliptic functions sine amplitude  and cosine amplitude .cosh , sinh , sech , and csch  are the hyperbolic functions.Secondly, for the sake of simplification, we only consider the case  > 0,  > 0, and  > 0 (the other cases can be considered similarly).To relate conveniently, for given constant wave speed , let Proposition 1.Consider BBM-like (3, 2) equation and its traveling wave equation For given constants  and , there are the following results.
The elliptic periodic blow-up solution  14 () becomes the hyperbolic smooth solitary wave solution  9 (), and for the varying process, see Figure 6.
Proposition 6.Consider BBM-like (4, 2) equation and its traveling wave equation For given constants  and , there are the following results.
Remark 7. In order to confirm the correctness of these solutions, we have verified them by using the software Mathematica; for instance, about  20 () the commands are as follows: (38)

The Derivations for Proposition 1
In this section, firstly, we derive the precise expressions of the traveling wave solutions for BBM-like (3, 2) equation.
Secondly we show the relationships among these solutions theoretically.Substituting (, ) = () with  =  −  into (9), it follows that Integrating (39) once, we have where  is an integral constant.
Letting  =   , we obtain the following planar system Under the transformation system (41) becomes (43) Clearly, system (41) and system (43) have the same first integral where ℎ is an integral constant.Consequently, these two systems have the same topological phase portraits except for the straight line  = 0. Thus, we can understand the phase portraits of system (41) from that of system (43).
On the other hand, solving equation we get three three singular points ( *  , 0) ( = 1, 2, 3), where According to the qualitative theory, we obtain the phase portraits of system (43) as Figures 9 and 10.
(7) When  >  and  =  0 , there are two special kinds of orbits Γ 7 surrounding the center point ( * 2 , 0) (see Figure 11(d)) and Γ 8 surrounding the center point ( * 3 , 0) (see Figure 12(a)), which are the boundaries of two families of closed orbits.Note that the periodic waves of (9) correspond to the periodic integral curves of (40), and the periodic integral curves correspond to the closed orbits of system (41).For given constants ,  and the corresponding initial value (0), we simulate the integral curves of (40) as shown in Figures 11 and 12.
From Figure 11, we see that when the initial value (0) tends to 0 − 0, the periodic integral curve tends to peakon.This implies that the orbit Γ 7 corresponds to peakon.On − plane, Γ 7 has the expression where Completing the integral in the above equation and noting that  = (), we obtain  11 () as (25).Similarly, from Figure 12, we see that when the initial value (0) tends to 0 + 0, the periodic integral curve tends to     =  9 () (see (23)) . (68) Hereto, we have completed the derivations for Proposition 1.

The Derivations for Proposition 6
In this section, we derive the precise expressions of the traveling wave solutions for BBM-like (4, 2) equation.Similar to the derivations in Section 3, substituting (, ) = () with  =  −  into (32) and integrating it, we have the following planar system: with the first integral where  and ℎ are the integral constants.