Bifurcations of Travelling Wave Solutions for the B ( m , n ) Equation

Using the bifurcation theory of dynamical systems to a class of nonlinear fourth order analogue of the B(m,n) equation, the existence of solitary wave solutions, periodic cusp wave solutions, compactons solutions, and uncountably infinite many smooth wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.


Introduction
Recently, Song and Shao [1] employed bifurcation method of dynamical systems to investigate bifurcation of solitary waves of the following generalized (2 + 1)-dimensional Boussinesq equation where , , α β γ and δ are arbitrary constants with 0 γδ ≠ .Chen and Zhang [2] obtained some double periodic and multiple soliton solutions of Equation (1.1) by using the generalized Jacobi elliptic function method.Further, Li [3] studied the generalized Boussinesq equation: ( ) by using bifurcation method.In this paper, we shall employ bifurcation method of dynamical systems [4]- [11] to investigate bifurcation of solitary waves of the following equation: Numbers of solitary waves are given for each parameter condition.Under some parameter conditions, exact solitary wave solutions will be obtained.It is very important to consider the dynamical bifurcation behavior for the travelling wave solutions of (1.3).In this paper, we shall study all travelling wave solutions in the parameter space of this system.Let ( ) ( ) ( ) where "'" is the derivative with respect to ξ .Integrating Equation (1.4) twice, using the constants of integration to be zero we find ( ) ( ) System (1.6) is a 5-parameter planar dynamical system depending on the parameter group ( ) , , , , m n p q r .For different m, n and a fixed r , we shall investigate the bifurcations of phase portraits of System (1.6) in the phase plane ( ) , y φ as the parameters , p q are changed.Here we are considering a physical model where only bounded travelling waves are meaningful.So we only pay attention to the bounded solutions of System (1.6).

Bifurcations of Phase Portraits of (1.6)
In this section, we study all possible periodic annuluses defined by the vector fields of (1.6) when the parameters , p q are varied.Let , Then, except on the straight lines 0 φ = , the system (1.6) has the same topological phase portraits as the following system ( ) Now, the straight lines and which imply respectively the relations in the ( ) ) and which imply respectively the relations in the ( ) ) , which imply respectively the relations in the ( ) , which imply respectively the relations in the ( ) , e e M y φ be the coefficient matrix of the linearized system of (2.1) at an equilibrium point ( ) , e e y φ .Then, we have By the theory of planar dynamical systems, we know that for an equilibrium point of a planar integrable system, if 0 J < then the equilibrium point is a saddle point; if 0 J > and Trace M y J y φ φ − > ,.then it is a node; if 0 J = and the index of the equilibrium point is 0 then it is a cusp, otherwise, it is a high order equilibrium point.
1) The case 0 q ≠ , We use Figure 2, Figure 3, Figure 4, and Figure 5 to show the bifurcations of the phase portraits of (2.1).
2) The case 0 q = .We consider the system ( ) with the first integral Figure 6 and Figure 7 show respectively the phase portraits of (2.3) for

Exact Explicit Parametric Representations of Traveling Wave Solutions of (1.6)
In this section, we give some exact explicit parametric representations of periodic cusp wave solutions.1).Suppose that ∈ , In this case, we have the phase portrait of (2.1) shown in Figure 2 (2-5).Corresponding to the orbit defined by ( ) , the arch curve has the algebraic equation ( ) Thus, by using the first Equation of (1.6) and (3.1), we obtain the parametric representation of this arch as follows: We will show in Section 4 that (3.10) gives rise to two periodic cusp wave solutions of peak type and valley type of (1.3).
2).Suppose that ∈ , In this case, we have the phase portrait of (2.1) shown in Figure 2 (2-4).corresponding to the orbit defined by ( ) , the arch curve has the algebraic equation Thus, by using the first equation of (1.6) and (3.3), we obtain the parametric representation of this arch as follows: ( We will show in Section 4 that (3.10) gives rise to a solitary wave solutions of peak type and valley type of (1.3).
3).Suppose that where  Thus, by using the first equation of (1.6) and (3.5), we obtain the parametric representation of this arch as follows: ; , ; where ( ) ; sn x k is the Jacobin elliptic functions with the modulo k , ( )( ) We will show in Section 4 that (3.6) gives rise to a smooth compacton solution of (1.3).4).Suppose that ∈ .In this case, we have the phase portrait of (2.1) shown in Figure 3 (  (5-1) (5-2) (5-3) (5-4) (5-5) (5-6) (5-7) (5-8) Figure 5.The phase portraits of (1.6) for m -n = 2l -1, n =2n 1 + 1, l, n 1 ∈Z + (5-1) r > 0, ( ) Thus, by using the first equation of (1.6) and (3.7), we obtain the parametric representation of this arch as follows: ( ) , the arch curve has the algebraic equation ( ) Thus, by using the first equation of (1.6) and (3.9), we obtain the parametric representation of this arch as follows: where ( ) ; sn x k is the Jacobin elliptic functions with the modulo k and 3, 4, 0, , n m r p q D D == < ∈  , In this case, we have the phase portrait of (2.1) shown in Figure 5 (5-6), corresponding to the orbit defined by ( ) , the arch curve has the algebraic equation ( ) Thus, by using the first equation of (1.6) and (3.11), we obtain the parametric representation of this arch as follows:  ( ) Thus, by using the first equation of (1.6) and (3.15), we obtain the parametric representation of this arch as follows: ( ) ( ) We will show in Section 4 that (3.16) gives rise to two periodic cusp wave solutions of peak type and valley type of (1.3).9).Suppose In this case, we have the phase portrait of (2.1) shown in Thus, by using the first equation of (1.6) and (3.17), we obtain the parametric representation of this arch as follows: ; , ; where ( ) ; sn x k is the Jacobin elliptic functions with the modulo k and ( )( ) )( ) We will show in Section 4 that (3.6) gives rise to a smooth compacton solution of (1.3).

The Existence of Smooth and Non-Smooth Travelling Wave Solutions of (1.6)
In this section, we use the results of Section 2 to discuss the existence of smooth and non-smooth solitary wave and periodic wave solutions.We first consider the existence of smooth solitary wave solution and periodic wave solutions.Theorem 4. 1 1).Suppose that ( ) 3).Suppose that ( ) 3) has a smooth solitary wave solution of peak type, corres- ponding to a branch of the curves 3) has two families of periodic wave solutions.When h varies from 1 h to 0, these periodic travelling waves will gradually lose their smoothness, and evolve from smooth periodic travelling waves to periodic cusp travelling waves, finally approach a periodic cusp wave of valley type and a periodic cusp wave of peak type defined by ( ) , 0 H y φ = of (1.7) (see Figure 2 (2-1)).

, , , H y h h h h
2).Suppose that 3) has two families of periodic wave solutions.When h varies from 0 to 1 h , these periodic travelling waves will gradually lose their smoothness, and evolve from smooth periodic travelling waves to periodic cusp travelling waves, finally approach a periodic cusp wave of valley type and a periodic cusp wave of peak type defined by 3) has two families of periodic wave solutions.When h varies from 0 to 3 h , these periodic travelling waves will gradually lose their smoothness, and evolve from smooth periodic travelling waves to periodic cusp travelling waves, finally approach a periodic cusp wave of valley type and a periodic cusp wave of peak type defined by 3) has two families of periodic wave solutions.When h varies from 0 to 1 h , these periodic travelling waves will gradually lose their smoothness, and evolve from smooth periodic travelling waves to periodic cusp travelling waves, finally approach a periodic cusp wave of valley type and a periodic cusp wave of peak type defined by ( ) , 0 H y φ = of (1.7) (see Figure 3 (3-3)).
Theorem 4.4 1).Suppose that 3) has a family of uncountably infinite many periodic traveling wave solutions; where h varies from 1 h to 0, these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 1 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-7)).c).If ( ) 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 2 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 2 h to 0, these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4)(5)(6)(7)(8)(9)(10)(11)).e).If ( ) 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 2 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-5)).f).If ( ) 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 2 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 4 (4-9)).
2).Suppose that ,then when ( ) 3) has two family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 1 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 2 (2-6)).Parallelling to Figure 2 (2-6), we can see the periodic travelling wave solutions implied in Figure 3 (3-6) and Figure 6 (6-2) have the same characters.
3).Suppose that 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 2 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-6)).h).If ( ) 0, 2, , r n p q D < ≥ ∈ ; then when ( ) 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 1 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-7)).i).If 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 1 h to 0, these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-3)).j).If 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 2 h , these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-5)).
4).Suppose that 3) has a family of uncountably infinite many periodic traveling wave solutions; when h varies from 1 h to 0, these periodic traveling wave solutions will gradually lose their smoothness, and evolve from smooth periodic traveling waves to periodic cusp traveling waves (see Figure 5 (5-5)).
Equation (1.3) has one family of uncountably infinite many periodic traveling wave solutions; when h varies from 0 to 1 h , these periodic travelling wave solutions will gradually lose their smoothness, and evolve from smooth periodic travelling waves to periodic cusp travelling waves (see Figure 7 ).
(1.5) is equivalent to the two-dimensional systems as follows
In this case, we have the phase portrait of (2.1) shown in Fig- ure 4 (4-5).corresponding to the orbit defined by ( ) curve has the algebraic equation 3-1), corresponding to the orbit defined by ( ) , 0 H y φ = to the equilibrium point ( ) 0,0 A , the arch curve has the algebraic equation

−
We will show in Section 4 that (3.10) gives rise to a smooth compacton solution of (1.3).6).Suppose that ( )3 4 in Section 4 that(3.20)gives rise to a smooth compacton solution of (1that.In this case, we have the phase portrait of (2.1) shown in Figure3(3-2) and (3-7), corresponding to the orbit defined by using the first equation of (1.6) and (3.13), we obtain the parametric representation of this arch as follows:

4 ( 4
1.7), Equation (1.3) has a smooth solitary wave solution of peak type, corresponding to a branch of the curves 1.7), Equation (1.3) has a smooth family of periodic wave solutions (see Figure equation (1.3) has a smooth solitary wave solution of peak type, corresponding to a branch of the curves Equation (1.3) has a smooth family of periodic wave solutions (see Figure 4 (4-3)).

=∈ 3 )
defined by (1.7), equation (1.3) has a smooth solitary wave solution of valley type, corresponding to a branch of the curves defined by (1.7), equation (1.3) has a smooth family of periodic wave solutions (see Figure4 (4-3)).has two periodic cusp wave solutions; corresponding to two branches of the curves

=
defined by (1.7), Equation (1.3) has two periodic cusp wave solutions; corresponding to two branches of the curves

=
defined by (1.7), Equation (1.3) has two periodic cusp wave solutions; corresponding to two branches of the curves

=
defined by (1.7), Equation(1.3)   has two periodic cusp wave solutions; corresponding to two branches of the curves