BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH FOURTH-ORDER DISPERSION AND DUAL POWER LAW NONLINEARITY

. For the nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity, by using the method of dynamical systems, we investigate the bifurcations and exact traveling wave solutions. Because obtained traveling wave system is an integrable singular traveling wave system having a singular straight line and the origin in the phase plane is a high-order equilibrium point. We need to use the theory of singular systems to analyze the dynamics and bifurcation behavior of solutions of system. For m > 1 and 0 < m = 1 n < 12 , corresponding to the level curves given by H ( ψ,y ) = 0 , the exact explicit bounded traveling wave solutions can be given. For m = 1 , corresponding all bounded phase orbits and depending on the changes of system’s parameters, all exact traveling wave solutions of the equation can be obtain.

1. Introduction. In 2018, [7] studied the exact solutions for the following nonlinear Schrödinger (NLS) equation with fourth-order dispersion and dual power law nonlinearity: iq t + aq xx − bq xxxx + c(|q| 2m + k 1 |q| 4m )q = 0, (1) by using the generalized projective Riccati equations method. This equation describes the propagation of optical pulse in a medium, and q(x, t) is the slowly varying envelope of the electromagnetic field, where a, b, c are real numbers. If b = 0, equation (1) reduces to the NLS. In addition if m = 1, equation (1) reduces to parabolic law nonlinearity, for which the exact solutions have been discussed in [Xu, 2011] using two direct algebraic methods. The coefficient of a represents the group velocity dispersion, while the coefficient of c represents the self-phase modulation with dual power law nonlinearity. The constant k 1 binds the two nonlinear terms and the exponent m governs the power law. Also, the coefficient b is the fourthorder dispersion term. For finding many new exact solutions, equation (1) has been studied in [10] using five different techniques, namely, the -expansion method, the improved Sub-ODE method, the extended auxiliary equation method, the new mapping method, and the Jacobi elliptic function method. In [7], the authors applied the generalized projective Riccati equations method to find some new soliton and periodic solutions.
Because the three papers [9], [10] and [7] did not study the dynamical behavior of system (8). In this paper, it is different from these references that we use the method of dynamical systems to investigate the bifurcations of the phase portraits of system (8) and find all possible exact solutions of system (8) depending on the changes of the parameter group (β, γ, k 1 ) and m. The results of this paper provide more complete understanding for the traveling wave solutions of equation (1). This paper is organized as follows. In section 2, we consider the bifurcations of phase portraits of system (8). In section 3, corresponding to the level curves defined by H(ψ, y) = 0, we give all possible exact solutions of system (8) and equation (1) for all m ≥ 1. In section 4, for the case m = 1, corresponding to all real level curves defined by H(ψ, y) = h, we calculate all possible exact explicit bounded solutions for system (8) and equation (1). In section 5, for 0 < m = 1 n < 1 2 , we discuss the exact solutions of system (8). As special case, corresponding to the homoclinic orbit defined by H(ψ, y) = 0, we obtain the solution of the form ϕ(ξ) = A cosh n (ωξ) = Asech n (ωξ) of the equation (5), which give rise to the geometric explanation of solutions found in [8].
2. The bifurcations of phase portraits of system (8) for m ≥ 1. We first consider the associated regular system of system (8) as follows: where m > 0, m ̸ = 1 2 and dζ = 2mψdξ, for ψ ̸ = 0. By the transformation (6), we only consider the positive solutions of ψ(ξ) in the right phase plane.

The exact traveling wave solutions of equation
(12) It is easy to see that for m > 1, if and only if h = 0, by using the first equation of system (8), we have It means that for m > 1, we only can obtain the exact solutions of system (8) defined by the real level curves H(ψ, y) = 0. All real level curves of given by H(ψ, y) = 0 are shown in Fig.5.

(22)
Therefore, by using the first equation of system (8) and (22), we can get all exact solutions for equation (1) with m = 1.
The closed branch family of the level curves defined by H(ψ, y) = h, h ∈ (h 1 , 0) is a periodic solution family of system (8), which has the same parametric representation as (35). Equation (1) has the exact solutions as (36).
For h ∈ (h 1 , 0), the level curves defined by H(ψ, y) = h has a family of closed branch enclosing the equilibrium point E 1 (ψ 1 , 0). Corresponding to this family of periodic orbits of system (8), it has the same parametric representation as (35). So that, equation (1) has the exact solutions as (36).
Corresponding to the homoclinic orbit of system (8)  , where 0 < ψ 1 < ψ M < ψ 2 < ψ a . It gives rise to the following exact solution of system (8): where ω 5 = √ 4|k1|γ 3 ψ a ψ M . Hence, we have the following exact solution of equation (1): Corresponding to the family of periodic orbits of system (8) defined by H(ψ, y) It gives rise to the following exact solution of system (8): . Therefore, we obtain the exact solution family of equation (1): The level curves defined by H(ψ, y) = h 2 contain two heteroclinic orbits connecting the equilibrium points O(0, 0) and E 2 (ψ 2 , 0), which have the same parametric representations as (16) with m = 1. Equation (1) has the exact solutions as (17) with m = 1.
Corresponding to the two closed branches of the level curves defined by H(ψ, y) = h, h ∈ (0, h 1 ) (see Fig.6 (b)), we have the same parametric representations of the (a) γ > 4k 1 β (b) γ = 4k 1 β Fig.8 The bifurcations of phase portraits of system (8) for m = 1 n , β > 0 By using the results in section 3, we have the following conclusions.