BIFURCATIONS OF TRAVELING WAVE SOLUTIONS FOR A GENERALIZED CAMASSA-HOLM EQUATION

In this paper, the traveling wave solutions for a generalized CamassaHolm equation ut−uxxt = 12 (p+1)(p+2)u ux− 1 2p(p−1)u ux−2puuxuxx− uuxxx are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized CamassaHolm equation are given for p either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.


Introduction
During the past few decades, more and more mathematicians paid attentions to certain nonlinear evolution equations which possess solitary wave solutions, such as peaked wave solutions [13, 17-19, 22, 23].The well-known Camassa-Holm (CH) equation [3], which describes nonlinear dispersive waves is in the following form (1.1) The CH equation arises from the theory of shallow water waves [3,4] and provides a model of wave breaking for a large class of solutions [5,6].Particularly, the CH equation (1.1) has peakon solution with the form u(x, t) = ce −|x−ct| .The name peakon, which means traveling wave with slope discontinuities, is used to distinguish them from general traveling wave solutions since they have a corner at the peak of height c, where c is the wave speed.However, it is necessary to point out that a peakon solution is not a classical solution, but a weak solution in the sense of satisfying a integral formulation, which is introduced in [14].
Recently, some generalizations of the CH equation admitting breaking wave solutions have been studied, such as the b family equation [9,10] has been studied for possessing peakon solutions.Furthermore, Wazwaz [20] studied nonlinear dispersive variants CH(n, n) of the generalized Camassa-Holm equation in (1 + 1), (2 + 1) and (3 + 1) dimensions, respectively.Motivated by the literature, the present paper focuses on the following generalized Camassa-Holm equation [1]: where p ∈ Z and p = 0 is the parameter with the nonlinear terms, and derived by Anco, Recio, Gandarias and Bruzon [1], which is considered by one of the related Hamiltonian structures of the Camassa-Holm equation.(1.3) is exactly the Camassa-Holm equation (1.1) when p = 1.In this paper, all the bifurcations of the generalized Camassa-Holm equation (1.3) are given for p either positive or negative integer.
The present paper focuses on the bifurcation analysis (see [7,8,11,12,14,15,21,24,25]) of equation (1.3).Let u(x, t) = u(x − ct) = u(ξ), where c is the wave speed, substituting it into (1.3) and integrating once, the partial differential equation (1.3) can be transformed to the following ordinary differential equation where is the derivative with respect to ξ, and g is an integral constant.Eq. (1.4) is equivalent to the following ODE system as follows with the first integral Eq. (1.5) is a 3-parameter planar dynamical system depending on the parameter group (c, g, p).For different c, g and a fixed p, the bifurcations of (1.5) in the phase plane (u, y) would be shown as the parameters (c, g) are changed.
Notice that when c + u p = 0, the right hand side of the second equation in (1.5) is not continuous.In other words, on the straight line c + u p = 0, u (ξ) has no definition, it implies that the smooth system (1.3) sometimes has non-smooth traveling wave solutions.The existence of singular straight line for a traveling wave equation would lead to traveling waves losing their smoothness.

Bifurcations of Phase Portraits of (1.5)
In this section, we study the phase portraits for equation (1.5).First, imposing the transformation dτ = (c+u p )dξ to Eq. (1.5), it yields the following nonlinear system Obviously, Eq. (2.1) has the same topological phase portraits as Eq.(1.4) except for the straight lines c + u p = 0. Eqs.(2.1) and (1.5) are integrable because of the same first integral (3.2).For a fixed h, Eq. (1.6) determines a set of invariant curves with different branches.As h is varied, Eq. (1.6) defines different families of orbits with different dynamical behaviors.On the following, g = 0 and g = 0 will be considered separately.
For our convenience, denote that Using the qualitative analysis above, we can obtain the bifurcation curves and phase portraits under various parameter conditions.Figs.2-Figs.4 show the phase portraits of (2.1) in the cases g = 0 and g = 0, respectively.
, respectively, we can obtain similar bifurcations of phase portraits of (3.1).To save space, we omit them.

Some Exact Parametric Representations of Traveling Wave Solutions of (1.3)
In this section, we provide some parametric representations for heteroclinic orbits, periodic orbits and homoclinic orbits defined by (1.5) in different parameter conditions besides p = 1.Obviously, (2.1) has the same orbits as (1.5) except for c + u p = 0.The transformation of variables dτ = (c + u p )dξ leads to some differences between the parametric representations of orbits for (1.5) and (2.1) when c + u p = 0. Notice that H(u, y) = h is defined by (1.6), and in order to find the parametric representation of (1.5), (1.6) can be cast into (1) Denote Γ h is the open curve family determined by H(u, y) = h, h ∈ (h 0 , h 3 ).A γ and B γ are called turning points on the curve Γ h , and C(φ m , 0) stands for the intersection of Γ h and the y-axis (see Fig. (2-1)).The detailed discussion on the existence of peakon can be seen in [14].
For Fig.
Figure 6.Profiles of periodic wave solutions and smooth solitary wave solutions.
(3) Suppose that p = 2m, (c, g) ∈ (A3).In this case, we have the phase portrait of (2.1) shown in Fig. .For concrete value, taking p = 2, Eq.(1.6) can be changed to where Thus, by the first equation of (1.5) and [2], we obtain the following parametric representation: where (3.5) gives rise to a smooth solitary solution of system (1.5) (see Fig. ).
(I) Corresponding to the curve triangle defined by H(u, y) = h s , by (3.3) there are an arch heteroclinic orbits connecting to two equilibrium points E(u s+ , Y + ) and E(u s+ , Y − ).Eq. (1.6) can be rewritten as where t 8 < 0 Thus, we obtain the following parametric representation of the periodic peakon wave solution is obtained: which gives rise to periodic peakon solution of (1.3) (see Fig. (7-2)), which is introduced in detail by Li and Zhu in Theorem A of [16].
Particularly, when u > −c, the direction of the vector field defined by (1.5) is opposite to the direction of the vector field defined by (2.1), they gives rise to an unbounded solution of system (1.5).
(II) Corresponding to the orbits defined by H(u, y) = h ∈ (h 1 , h s ), by Eq. (2.1) there are a family of periodic orbits enclosing the equilibrium point E 1 (u 1 , 0).Eq. (1.6) can be rewritten as with − 1 c < t 9 < t 10 < 0. Thus, by the first equation of (1.5) and from the formulas 258.00, 258.11, 340.04 in [2], we have the following parametric representation of the periodic wave solutions (see Fig. where k 2 5 = t10−t9 −t9(ct10+1) , ϕ 5 = arcsin u(ct10+1) (u−t10) .If h → h s , the periodic is going to the curve triangle.This fact indicates that the wave profile of u(ξ) determined by the periodic orbit is a periodic peakon wave, which will be solved on the following, and the curve triangle gives rise to a periodic peakon wave solution such like Eq. (3.11).

Conclusion
In summary, the bifurcations of the generalized Camassa-Holm equation are investigated by using the bifurcation theory and phase portraits of (1.6).It shows that the system (1.3) has peakon, peakon periodic solutions, solitary wave solution, kink and anti-kink wave solutions, and periodic wave solutions under some parameter conditions.Finally, we describe the dynamical behaviors of (1.3) and explain the reason for the existence of non-smooth traveling wave.From the above analysis, it can be known that this method can be suitable for other nonlinear wave equations.

p+1p
, which implies the following relations in the (c, g)-parameter plane

Figure 7 .
Figure 7. Profiles of kink solution, periodic peakon solutions and periodic wave solutions..