BIFURCATIONS OF TRAVELING WAVE SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH FOURTH-ORDER DISPERSION AND CUBIC-QUINTIC NONLINEARITY∗

For the nonlinear schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity, by using the method of dynamical systems, the dynamics and bifurcations of the corresponding traveling wave system are studied. Under different parametric conditions, twenty exact parametric representations of the traveling wave solutions are obtained.


Introduction
In this paper, we consider the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity as follows: (1.1) This equation govern wave dynamics of optical fiber system (see [8]). The authors of [8] derived analytic soliton solutions ( bright and dark soliton and other soliton solutions ) of equation (1.1), by using an algebraic method with an auxiliary equation.
As far as we are concerned, the dynamical behavior of traveling wave solutions of equation (1.1) has not been studied in the literature. In this paper, we investigate the bifurcations of phase portraits of traveling system and the dynamical behavior of traveling wave solutions for equation (1.1) by applying the method of dynamical systems (see [2][3][4][5][6][7][9][10][11][12][13] ).
(ii) Under different parameter conditions, corresponding to the level curves defined by H(φ, y) = h s with different types, system (1.5) has periodic wave solutions, kink wave solutions, periodic peakon solutions, and peakon solutions with exact parametric representations given by (3.4)-(3.23).
The proof of the theorem 1.1 will be seen in next two sections. This paper is organized as follows. In section 2, we consider the bifurcations of phase portraits of system (1.5). In section 3 we give all possible exact solutions of φ(ξ), under different parameter conditions.
It is easy to see from the right hand of h s that if ∆ 1 = q 2 − 6pr > 0, then, when , we have h s = 0. We next consider more interesting cases. Assume that a > 0, pr > 0, ∆ 1 > 0 and for a fixed parameter group (p, q, r) such that system (2.1) has five equilibrium points on the φ−axis, then, by varying the parameter a > 0, i.e., by changing the relative positions between the singular straight lines φ = ∓ √ a and the equilibrium points E j∓ (∓φ j , 0), (j = 1, 2), O(0, 0), on the basis of qualitative analysis, under different parameter conditions, we have the bifurcations of phase portraits of system (2.1) shown in Fig. 1(a)-(g) and Fig. 2(a)-(g). (a) 0 < a < φ 2 Figure 2. The bifurcations of phase portraits of system (1.5) when p < 0, q > 0, r < 0 and ∆1 > 0.

Some exact traveling wave solutions of equation (1.1)
In this section, we calculate some possible exact parametric representations of the orbits defined by H(φ, y) = h of system (1.5) and give some traveling wave solutions for equation (1.1). We see from (1.6) that (3.1) By using the first equation of (1.5), we have .
(3.3) Obviously, for a general h, we can not obtain the exact parametric representations for the level curves defined by (1.6) since ψG(ψ) is a fifth polynomial and the right hand of (3.3) is a hyperelliptic integral. Only in some special cases, we can get the exact parametric representations.
When f ( √ a) > 0, the level curves defined by H(φ, y) = h s in Fig. 1 can be shown in Fig. 3 (a)-(d). (i) Corresponding to the level curves defined by H(φ, y) = h s in Fig. 3 (a), there exists an oval orbit passing through two singular straight lines φ = ∓φ s and intersecting the φ−axis at points (∓φ M , 0). In addition, there are two open orbits passing through the φ−axis at points (∓φ L , 0). In this case, we have Thus, for the oval orbit, (3.3) reduces to It follows the parametric representation of a periodic solution of equation (1.1): where , and K(k) is the complete elliptic integral of the first kind, dn(·, k), sn(·, k), cn(·, k) are Jacobian elliptic functions (see [1]).
Notice that the area of the oval in Fig. 3 (a) is partitioned into three parts by the two singular straight lines φ = ∓ √ a. The right arch is the limit curve of the family of periodic orbits of system (1.5) enclosing the equilibrium point (φ 1 , 0), which gives rise to a lower periodic peakon solution (see Fig. 4 L −a , k . The left arch is the limit curve of the family of periodic orbits of system (1.5) enclosing the equilibrium point (−φ 1 , 0), which gives rise to a upper periodic peakon solution (see Fig. 4

(a)) of equation (1.1) with the parametric representation
The middle two curves are the limit curves of the family of periodic orbits of system (1.5) enclosing the equilibrium point O(0, 0), they give rise to a sawtooth cusp wave solution (see Fig. 4 (c)) of equation (1.1).
(ii) Corresponding to the level curves defined by H(φ, y) = h s in Fig. 3 (b), there exists an oval orbit tangent to two singular straight lines φ = ∓ √ a at the points (∓φ 1 , 0). In addition, there are two open orbits passing through the φ−axis at the points (∓φ L , 0). In this case, we have G(φ) = (φ L − φ 2 )(φ 1 − φ 2 ) 3 . Thus, for the oval orbit, (10) reduces to , where ψ L = φ 2 L and ψ 1 = φ 2 1 . It follows the parametric representation of a periodic solution of equation (1.1): where Corresponding to the level curves defined by H(φ, y) = h s in Fig. 3 (c), there exist a closed orbit intersecting the φ−axsis at two points (∓φ l , 0) and two arches enclosing the equilibrium points (∓φ 2 , 0) and intersecting the φ−axis at the points (∓φ m , 0), respectively. In this case, we have where . It gives the following lower periodic peakon solution: (3.9) where ξ 02 = tn −1 For the left arch orbit, we have It gives rise to an upper periodic peakon solution.
(i) Corresponding to the level curves defined by H(φ, y) = h s in Fig. 5 (a), there exist two oval orbits contacting to the singular straight lines φ = ∓ √ a and intersecting the φ−axis at the points (∓φ M , 0), respectively. In this case, we have . It implies the two periodic solutions of equation (1.1): where (ii) Corresponding to the level curves defined by H(φ, y) = h s in Fig. 5 (b), there exist two oval orbits passing through the singular straight lines φ = ∓ √ a and enclosing the equilibrium points (∓φ 1 , 0) and (∓φ 2 , 0), respectively. Meanwhile, the two oval orbits intersect the φ−axis at four points (∓φ m , 0) and (∓φ M , 0). Now,  Figure 5. The level curves defined by H(φ, y) = hs of system (1.5). where Consider the arch in the right side of the singular straight line φ = √ a which is the limit curve of a family of periodic orbits enclosing the equilibrium point (φ 2 , 0). We see from (3.14) that the arch curve has the following parametric representation:  Fig. 3 (b)).
Similarly, corresponding to the arch in the left side of the singular straight line φ = − √ a, the the parametric representation gives rise to a periodic peakon solution of equation (1.1) (similar to Fig. 3 (a)).