Optical solitons for the higher-order cubic-quintic nonlinear Schrödinger equation with a PT-symmetric potential

In nature, most of the systems are nonlinear and as a result of this fact, those systems are modeled by nonlinear systems of equations. Some of the most remarkable progress in nonlinear science is observed in wave propagation phenomena. Often, research on a given nonlinear system begins by investigating a one-dimensional partial differential equation (PDE) as an approximation to an experimental system in optics, fluid dynamics, plasma physics and biology. Many of these nonlinear equations have known nonlinear wave type solutions and some are commonly referred to as soliton solutions. Solitons are localized waves that collide elastically, suffering only a shift in phase. The history of solitons dates back to 1834, the year in which John Scott Russell observed a wave form propagated for several kilometers in a shallow canal of Scotland without being distorted. Solitons represent the solutions of nonlinear wave-type partial differential equations, including sine-Gordon, Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations. In this thesis, we explore the theoretical and numerical analysis of optical solitons of a higher order cubic quintic nonlinear Schrödinger equation (CQNLS) with a fourth-order dispersion term (4OD) in aPT -symmetric potential. In Chapter 1, the historical background of optical soliton research is briefly given. The application areas and the mechanism of the NLS 4OD equation are discussed, and the general properties ofPT -symmetric potentials are argued. In this chapter, the aim of the thesis, literature review and hypothesis of the thesis are given, respectively. In Chapter 2, spectral renormalization method (SR), the numerical method which is used to obtain localized soliton solutions is explained. The modification of this method is given in order to apply to CQNLS equation with a fourth order dispersion term and an external potential. Then, the Split-step Fourier method is given for nonlinear stability analysis. Chapter 3 is dedicated to (1+1)D 4OD cubic-quintic NLS equation without an external potential. Exact and numerical solution of the equation are analyzed and the produced results are shown by some graphs. Lastly, the nonlinear stability of the soliton solutions are investigated for various parameters of the considered equation and the results are compared. Chapter 4 includes studies of exact soliton solution of the (1+1)D 4OD cubic-quintic NLS equation with a PT -symmetric potential. This PT -symmetric potential is introduced and for different values of parameters, soliton solutions are found in this potential. For this various values of parameters of the equation, exact and numerical results are compared, the effect of the eigenvalue of the numerical solutions are figured out and the maximum amplitude of the solitons are discovered. For final, the nonlinear stability of the produced solitons are demonstrated in terms of various parameters. Result of this dissertation are summarized in Chapter 5. In this thesis, MATLABR2010a computer programme is used and all of the results are produced by the use of this programme.