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The purpose of this paper is to investigate a higher-order nonlinear Schrödinger equation with non-Kerr term by using the bifurcation theory method of dynamical systems and to provide its bounded traveling wave solutions. Applying the theory, we discuss the bifurcation of phase portraits and investigate the relation between the bounded orbit of the traveling wave system and the energy level. Through the research, new traveling wave solutions are given, which include solitary wave solutions, kink wave solutions, and periodic wave solutions.

In the past decades, communication systems have scored a great growth of the transmission capacity. Due to the undamped and unchanged characteristics in a far distance, optical solitons are the focus of many research groups during the past decades and stand a good chance to be the main information carriers in telecommunications in the future. Ignoring optical losses, the wave dynamics of nonlinear pulse propagation in a monomode fiber is described by the nonlinear Schrödinger equation [

Laser spectroscopic techniques have been widely used in all fields of science. It can help us observe the physical processes in materials and molecules which occur on a femtosecond time scale by using ultrashort lasers. The pulses can also be applied in telecommunication and ultrafast signal routing systems. Research indicates that non-Kerr nonlinear effects begin to have some effects when the pulse width becomes narrower and the intensity of the incident light field becomes stronger. The influence is described by the NLS family of equations with nonlinear terms [

In this paper, with the aid of Mathematica, we study the new traveling wave solutions for a higher-order NLS equation that contains the non-Kerr nonlinear terms, which describes propagation of very short pulses in highly nonlinear optical fibers by using different elliptic functions. The bifurcation theory method is widely used to solve differential equations [

We consider the higher-order NLS equation with non-Kerr term [

Assume that (

Substituting (

Obviously, the above system (

We suppose that

If

If

If

If

If

If

If

If

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

The phase portraits of (

For a fixed

To facilitate further analysis, we investigate the relation between the bounded orbit of (

Put

The phase portraits of (

The phase portraits of (

The phase portraits of (

Observe that the energy curve

Assume that

Assume that

Assume that

(ii) If

(iii) If

When the energy level

(iv) If

When the energy level

(ii) If

Substituting (

(ii) If

We will draw the figures of some solutions under the special conditions. Figure

By using the bifurcation theory method of dynamical systems, we successfully obtain 10 phase portraits for the corresponding dynamic system of (

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

The authors express their sincere thanks to the editor and reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant no. 11272023, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) under Grant no. IPOC2013B008, and by the Fundamental Research Funds for the Central Universities of China under Grant no. 2011BUPTYB02.