Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 408586, 10 pages

http://dx.doi.org/10.1155/2015/408586

## Bifurcation Analysis and Solutions of a Higher-Order Nonlinear Schrödinger Equation

^{1}State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China^{2}School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 2 October 2014; Revised 8 December 2014; Accepted 8 December 2014

Academic Editor: Reza Jazar

Copyright © 2015 Yi Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2], which accounts for the group velocity dispersion and self-phase modulation. To increase the bit rate, it is often desirable to use shorter femtosecond pulses. However, when short pulses are considered, the equation can no longer represent the propagations of light pulses in fibers because higher-order dispersion terms and the non-Kerr nonlinearity effects cannot be neglected. This phenomenon can be expressed by a higher-order nonlinear Schrödinger equation [3]: where , , , , and are real constants. is a slowly varying envelope amplitude, represents the normalized retarded time (in the group velocity frame), and represents the normalized distance along the direction of propagation. comes from the group velocity dispersion (GVD). is proportional to the nonlinear index which originates from the Kerr effect. is the coefficients with the relevant work of the third-order dispersion. is related to self-steepening due to stimulated Raman scattering. The coefficient of the last term that is proportional to has its origin in the delayed Raman response . Generally speaking, can be estimated from the slope of the Raman gain and is defined as the first moment of the nonlinear response function [4]. In fact, should be an imaginary number, but many analytical studies have been done when is real, such as Painlevé property [5], inverse scattering transform [6], Hirota direct method, and conservation laws [7]. These researches verify its integrable nature and have obtained many exact wave solutions.

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 [8]. The nonlinearity due to fifth-order susceptibility can be obtained in many optical materials such as semiconductors and some transparent organic materials. Actually, it is also important to include some additional higher-order perturbation effects into the HNLS equation to analyze the solitary wave solution in a non-Kerr nonlinear medium.

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 [9–12]. By using this method of dynamical systems, we obtain the explicit expressions of the bounded traveling wave solutions for the equation and investigate the relation between the bounded orbit of the traveling wave system and the energy level . The new solutions correspond to the orbits on phase portraits and they include solitary, kink, and periodic wave solutions. Note that the existence of solitary wave solutions depends essentially on the model coefficients and therefore on the specific nonlinear features of the medium.

#### 2. Bifurcation and Phase Portraits

We consider the higher-order NLS equation with non-Kerr term [13]: where , , , , , , , and are real constants. , , and represent the coefficients of quintic non-Kerr nonlinearities. The quintic nonlinearities arise from the expansion of the refractive index of the light pulse. The polarizations induced through the susceptibilities give the cubic and quintic (non-Kerr) terms in a nonlinear Schrödinger equation. When , (2) reduces to (1).

Assume that (2) has the form of the exact solution where is the real-valued function and the parameters of , , , and are real constants to be determined later.

Substituting (3) into (2) and removing the exponential term, we change (2) into the form The real and imaginary parts of (4), respectively, are Integrating (6), we can get where is a constant. Equations (5) and (7) can be reduced to an equation if and By solving (8), we get or Then (5) and (7) reduce to the following planar dynamic system: where , , and .

Obviously, the above system (11) has the first integral

We suppose that is the coefficient matrix of the linearized system (11) at an equilibrium point and is the Jacobian determinant. By the bifurcation theory of planar dynamical system, we know that if , then the equilibrium point is a saddle point; if and , then it is a center point; if and , then it is a node; if and Poincaré index of the equilibrium point is , then it is a cusp point. By using the above facts to do qualitative analysis, we have the following.(1)If and or , , , and , then system (11) has only one equilibrium point . It is easy to find . So it is a saddle point (see Figure 1).(2)If , , , and , then system (11) has only one equilibrium point . It is easy to find that and . So it is a center point (see Figure 2).(3)If , , , and , then system (11) has three equilibrium points and . , , and Poincaré index of is equal to zero. So is a saddle point; are cusp points (see Figure 3).(4)If , , , and , then system (11) has three equilibrium points and . , , , and Poincaré index of is equal to zero. So is a center point; are cusp points (see Figure 4).(5)If and , then system (11) has three equilibrium points and . , , and . So is a center point; are saddle points (see Figure 5).(6)If and , then system (11) has three equilibrium points and . , , and . So is a saddle point; are center points (see Figure 6).(7)If , , , and , then system (11) has five equilibrium points , , and . , , , and . So and are saddle points; are center points. Furthermore, if , then (see Figure 7); if , then (see Figure 8); if , then (see Figure 9).(8)If , , , and , then system (11) has five equilibrium points , , and . , , , , and . So and are center points; are saddle points (see Figure 10).