^{1}

^{2}

^{2}

^{1}

^{2}

The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is studied using the factorization technique, bifurcation theory of dynamical system, and phase portraits analysis. From a dynamic point of view, the existence of smooth solitary wave, and kink and antikink waves is proved and all possible explicit parametric representations of these waves are presented.

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 [

The

In this paper, using the factorization technique [

Throughout this paper, we always suppose that

Using transformation

From (

Taking transformation

Using Proposition 2 of [

Solving (

Some special solutions of (

Letting

For a fixed

The remainder of this paper is organized as follows. In Section

We will use the following notations:

Let

For an equilibrium point

Using the properties of equilibrium points and the bifurcation method, we can obtain four bifurcation curves of (

Bifurcation sets and phase portraits of (

Bifurcation sets and phase portraits of (

In this section we present all possible explicit smooth solitary, kink, and antikink wave solutions of (

When

When

When

When

From Figures

From Figures

From Figures

From Figures

The proof of the Proposition

Smooth solitary waves of (

When

When

When

When

When

When

From Figures

From Figures

From Figures

From Figures

From Figure

From Figure

The proof of the Proposition

When

From Figures

The proof of Proposition

Kink and antikink waves of (

The implicit solutions in [

In this paper, we present some explicit smooth solitary wave solutions for (

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grant no. 11461022 and the Major Natural Science Foundation of Yunnan Province, China, under Grant no. 2014FA037.