BIFURCATION AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR DISPERSIVE MK(M,N) EQUATION ∗

This paper investigated the generalized nonlinear dispersive mK(m,n) equation by the planar dynamical systems method, the bifurcations of the sys-tem with different parameter region of this equation are presented. Moreover, we find different kinds of exact explicit solutions like peak type solutions


Introduction
In the formation of patterns in liquid drops, nonlinear dispersive played an important role, Rosenau and Hyman [19] proposed the K(m,n) equation: (1.1) Wazwaz [21] introduced and studied the exact solutions of several generalized form of the mK(n,n) equation, the so-called mK(m,n) equation in one-, two-, and threedimensional spatial spaces.Recently, Yan [22] extended these equations to more general forms by making index of u in each term different and obtained compacton solutons, solitary wave solutions and periodic wave solutions.He et al. [10] considered the mK(n,n) equation by the method of planar dynamical systems and derived exact explicit solutions.Lai, He and Qing [12] studied the equation in a higher spaces and obtained explicit traveling wave solution in terms of sin, cos, sec and csc profiles.
Introducing the traveling wave transformation ξ = x − ct, let u(x, t) = u(ξ) and substituting it into Eq.(1.2), it follows that in which c ̸ = 0 is a constant and the prime represents d dξ .Integrating Eq.(1.3) once, we obtain in which g is an integral constant.
Introducing y = u ′ , then Eq.(1.4) is rewritten by the following planar system which has the first integral of Since all traveling wave solutions of (1.2) are determined by the phase orbits defined by the vector fields of system (1.5), we will analyze the bifurcations conditions and phase portraits of (1.5).
The rest of the paper is organized as follows.In Section 2, we seek for the equilibrium points of (1.5).In Section 3, we discuss the bifurcation of phase portraits of system (1.5) and obtain its traveling wave solutions.The paper is ended with the conclusion.

The analysis for generalized nonlinear dispersive mK(m,n) equation
Notice that the line l : u = 0 is a singular line, which made smooth system (1.5) have non-smooth traveling wave solutions.Therefore, we let dξ = nu n−1 dτ , then the system has the same phase portraits with system (1.5) except on the singular line u = 0.
For studying the singular points of system (2.1), let We can find out zero points of f (u) easily.Suppose that m > n(m < n has the similar results with this situation).
As a result, we get conclusions as follows.Letting (u i , y i ) be any equilibrium point of system (2.1), the coefficient matrix of linearized system (2.1) can be presented we can obtain that Substitute an equilibrium point of system (2.1) into (2.2) by the theory of planar dynamical system [13], if J < 0, the equilibrium point is a saddle point; if J > 0, the equilibrium point is a center point;If J = 0 and the index of equilibrium point is 0, then it is a cusp.Thus, we can know that when g = 0, (i 0) and A 3 (u 3 , 0) are two center points; if m − n = 2k + 2 and a < 0, c < 0, equilibrium point A 2 (u 2 , 0) and A 3 (u 3 , 0) are two saddle points.

Bifurcation sets and exact solutions of system (2.1)
In this section, we study the bifurcation set and exact solutions of the planar Hamiltonian system(2.1).
Theorem 3.1.When g = 0, we have, (i) When m − n = 2k + 1 and a > 0, c > 0, system (2.1) has a periodic family orbit and a homoclinic orbit, then Eq.( 1.2) has a periodic wave solution and a peak type solitary wave solution.When a < 0, c > 0, system (2.1) has a family periodic orbit and a homoclinic orbit, then Eq.( 1.2) has a periodic wave solution and a valley type solitary wave solution.The solitary wave solution has the expression (ii) when m − n = 2k + 1 and a > 0, c < 0, system (2.1) has a homoclinic orbit, then Eq.( 1.2) has a valley type solitary wave solution.When a < 0, c < 0, system (2.1) has a homoclinic orbit, then Eq.( 1.2) a peak type solitary wave solution.The solution is given by (iii) when m − n = 2k + 2 and a > 0, c > 0, system (2.1) has two family periodic orbits and two homoclinic orbit, then Eq.( 1.2) has two periodic wave solutions and a solitary wave solution.The expression of the solitary wave solution is (ii) Similarly, when m − n = 2k + 1 and a > 0, c < 0 or a < 0, c < 0 (see Figure 2(c) and Figure 2(d)), corresponding to H(u, y) = 0, we can obtain In terms of (3.5) and du dξ = y, the solitary wave solution (3.2) is obtained.(iii) In this situation, the phase portraits are performed in Figure 3(a), corresponding to homoclinic orbit which is defined by H(u, y) = 0, we can obtain Theorem 3.2.When g ̸ = 0, we have (i) when (c, g) ∈ A 3 , system (2.1) has three family periodic orbits.Thus, Eq.(1.2) has three periodic wave solutions.A periodic wave solution has the form (ii) when (c, g) ∈ B 1 , system (2.1) has two family periodic orbits.Then, Eq. (1.2) has two periodic wave solutions.A periodic wave solution has the form in which u 1 and u 2 are two real roots of equation c n u n −au m −g = 0 and u 1 > u 2 > 0. Proof.(i) When (c, g) ∈ A 3 (see Figure 4(c)), corresponding to H(u, y) = 0, we have (ii) When (c, g) ∈ B 1 (see Figure5(a)), corresponding to H(u, y) = − 14  3 , we have

Conclusion
In this paper, we use the bifurcation method to study exact traveling wave solutions of generalized nonlinear dispersive mK(m,n) equation.We derive exact solitary wave solutions, periodic wave solutions and compactons solutions.The bifurcation and phase portraits under different parameters are also given, furthermore, we can get the types of solutions easily.