Explicit Wave Solutions and Qualitative Analysis of the ( 1 + 2 )-Dimensional Nonlinear Schrödinger Equation with Dual-Power Law Nonlinearity

The nonlinear Schrödinger equation (NLSE) is studied in various areas of appliedmathematics, theoretical physics, and engineering. In particular, it appears in the study of nonlinear optics, plasma physics, fluid dynamics, biochemistry, and many other areas. This equation is completely integrable by the inverse scattering transform [1]. There have been various forms of this equation that arise in the study of different areas. They have different forms of nonlinearity. Some previous works concentrated on theKerr lawnonlinearitymedia [2–5]. However, as one increases the intensity of the incident light power to produce shorter (femtosecond) pulses, non-Kerr nonlinearity effects become prominent, and the dynamics of pulses should be described by the nonlinear Schrödinger family of equations with higher order nonlinear terms [6]. Hence, it is very important that all higher order effects are considered in the propagation of femtosecond pulses [7]. The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is given in [8–13]:


Introduction
The nonlinear Schrödinger equation (NLSE) is studied in various areas of applied mathematics, theoretical physics, and engineering.In particular, it appears in the study of nonlinear optics, plasma physics, fluid dynamics, biochemistry, and many other areas.This equation is completely integrable by the inverse scattering transform [1].There have been various forms of this equation that arise in the study of different areas.They have different forms of nonlinearity.Some previous works concentrated on the Kerr law nonlinearity media [2][3][4][5].However, as one increases the intensity of the incident light power to produce shorter (femtosecond) pulses, non-Kerr nonlinearity effects become prominent, and the dynamics of pulses should be described by the nonlinear Schrödinger family of equations with higher order nonlinear terms [6].Hence, it is very important that all higher order effects are considered in the propagation of femtosecond pulses [7].
The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is given in [8][9][10][11][12][13]: where (, , ) is a complex function that indicates the complex amplitude of the wave form,  2 = −1,  is an arbitrary nonzero constant, and  is a constant which indicates the saturation of the nonlinear refractive index.Biswas [8] obtained the 1-soliton solution and calculated a few conserved quantities.With the aid of symbolic computation, Zhang and Si [9] obtained soliton solutions, combined soliton solutions, triangular periodic solutions, and rational function solutions.Some soliton solutions are obtained in [10] using the / expansion method.For  = 1, the dark 1-soliton solution is obtained by the ansatz method, the invariance, conservation laws, and double reductions in [11].Bulut et al. [12] found some soliton solutions, rational and elliptic function solutions using the extended trial equation method.The bifurcation is considered in [13] and two implicit solutions are obtained (see (16) and (18) of [13]).In this paper, using the factorization technique [14], bifurcation theory of dynamical system, and phase portraits analysis [15][16][17], we present some explicit expressions of the smooth solitary, kink, and antikink waves of (1).
2 Advances in Mathematical Physics
The remainder of this paper is organized as follows.In Section 3, we consider bifurcation sets and phase portraits of (14).Some explicit smooth solitary, kink, and antikink wave solutions of (1) are presented in Section 4. A short conclusion will be provided in Section 5.
The proof of Proposition 3 is completed.

Conclusion
In this paper, we present some explicit smooth solitary wave solutions for (1) expressed in (20)-( 26), (49), and (50), and also some explicit kink and antikink wave solutions shown in (69).We will continue to discuss the properties of (1), and more results will be obtained.