Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

In this paper, the dynamical properties and the classification of single traveling wave solutions of the coupled nonlinear Schrödinger equations with variable coefficients are investigated by utilizing the bifurcation theory and the complete discrimination system method. Firstly, coupled nonlinear Schrödinger equations with variable coefficients are transformed into coupled nonlinear ordinary differential equations by the traveling wave transformations. Then, phase portraits of coupled nonlinear Schrödinger equations with variable coefficients are plotted by selecting the suitable parameters. Furthermore, the traveling wave solutions of coupled nonlinear Schrödinger equations with variable coefficients which correspond to phase orbits are easily obtained by applying the method of planar dynamical systems, which can help us to further understand the propagation of the coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Finally, the periodic wave solutions, implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions of the coupled nonlinear Schrödinger equations with variable coefficients are constructed.


Introduction
The coupled nonlinear Schrödinger (CNLS) equations [1][2][3] are a very important mathematical physical model in the fields of quantum mechanics, nonlinear optics, optical fiber communication, Bose-Einstein condensate, fluid mechanics, and so on. Because of its importance, many new methods and techniques have been proposed to study the CNLS equation, which include the Bäcklund transformation, the variational iteration method, the extended unified method, and the Hirota method [4][5][6][7][8][9]. However, the study of CNLS equations with variable coefficients [10][11][12][13][14] has more important theoretical significance and practical value than CNLS equations with constant coefficients because it usually describes the inhomogeneous effects of nonlinear optical pulse propa-gations in the real optical fiber communication system. The CNLS equations with variable coefficients are expressed as follows: where ψ = ψðt, xÞ and ϕ = ϕðt, xÞ denote the slowly varying amplitudes of two orthogonally polarized optical pluses, which are complex-valued functions with the retarded time t and the normalized propagation distance x. ðt, xÞ ∈ ½0, T × ½a, b, T ≥ 0, −∞≤a ≤ b≤+∞. The coefficients βðxÞ and g ðxÞ are integrable real functions of x, which represent the group velocity dispersion and the Kerr nonlinearity, respectively. Obviously, the integrability method, the modified sine-Gordon equation method, the unified method, the improved F-expansion method, and the Hirota bilinear method have used to establish the traveling wave solutions of the CNLS equation with variable coefficients in Ref. [10][11][12][13][14], respectively. In this paper, we investigate the dynamical properties and the classification of single traveling wave solutions of the CNLS equation with variable coefficients based on the bifurcation theory and the complete discrimination system method. The synopsis of the article is organized as follows. In Section 2, the bifurcation theory is employed to investigate the CNLS equations with variable coefficients; phase portraits and some Jacobian elliptic function solutions are obtained. In Section 3, some single traveling wave solutions of the CNLS equations with variable coefficients are obtained by using the complete discrimination system method. In Section 4, we present some research results.

Bifurcations of Phase Portraits of System (9)
In order to analyze the dynamic behavior of Equation (1), we first introduce the following traveling wave transformation: where Ψ and Φ are a real valued function, μ and α are arbitrary constants, and λðxÞ and θðxÞ are arbitrary function of x . Substituting (2) into (1), we get the coupled nonlinear differential equations Next, we decompose real parts and imaginary parts of (3) and put zero for imaginary parts of Equations (3), where the imaginary part of Equations (3) is λ ′ ðxÞ = 2αβðxÞ. Then, we set λðxÞ = 2α Ð x a βðςÞdς, a ≤ x ≤ b. Therefore, ξ = μðt − 2α Ð x a βðςÞdςÞ. Using the coefficients of Ψ ′ ′ (or Φ ′ ′) as the normalization coefficient, we have where c 1 and c 2 are constants and θ 0 is the integral constant.
On the other hand, the real parts of Equations (3) are simplified as Let dΨ/dξ = p, dΦ/dξ = q; system (5) is transformed into a four-dimensional dynamical system as follows: In order to remove the coupled relationship of (5), we consider the dynamical behavior of system (6) in the subma- (6), we obtain two completely uncoupled planar dynamical systems: Systems (7) and (8) are the same if and only if they satisfy l = ±1. Then, system (7) can be rewritten as the following planar dynamical system which is equivalent to the following Hamiltonian system: Suppose f ðΨÞ = −2c 2 Ψ 3 − c 1 Ψ, and further assume that M i ðΨ i , 0Þði = 0, 1, 2Þ is an equilibrium points of system (9), 2 Advances in Mathematical Physics then the eigenvalues of system (9) at the equilibrium points are expressed as λ 1,2 = ± ffiffiffiffiffiffiffiffiffiffiffiffi f ′ðΨÞ q . Thus, we can easily get three zeros of f ðΨÞ when c 1 c 2 < 0, which are Ψ 0 = 0, . We can easily obtain one zero of f ðΨÞ when c 1 c 2 > 0, which is Ψ 3 = 0. By the bifurcation theory of planar dynamical systems, the equilibrium The phase portraits of (9) depending on different parameters c 1 and c 2 are shown in Figure 1.
Substituting (20) into dΨ/dξ = p and integrating them, we obtain two families of periodic solutions Namely,

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Namely, then the solutions of Equation (1) can be shown as Case 3. D 2 > 0, D 3 = 0, D 4 = 0, and E 2 > 0. GðφÞ has two real roots of multiplicities of two, namely, where u > v. Substituting (35) into (27), we can obtain When φ > u or φ < v, we can obtain the solution of Equation (23) as follows: then we have When v < Φ < u, we can gain the solution of Equation (23) as follows: Similarly, we can have Case 4. D 2 > 0, D 3 > 0, and D 4 = 0. GðφÞ has two real roots and real roots with multiplicities of two, namely, where ρ i (i = 1, 2, 3) are real numbers and ρ 2 > ρ 3 .