Optical and soliton solutions with modulation stability analysis of some nonliear partial differential equations

This thesis consists of five chapters. In the first chapter, the introduction provides general information on differential equations, the concept of waves, the definition and history of solitons, optical solitons and their types, as well as nonlinear Schrödinger equations. In the second chapter, fundamental definitions are presented, including ordinary and partial differential equations, the balancing term, and several types of linear and nonlinear partial differential equations. In the third chapter, the methods used in this thesis are examined in detail, including the Extended Sine-Gordon Method (ESGM), the Unified Method (UM), the Nucci Reduction Method (NRM) and Modulation Stability Analysis (MSA).In the fourth chapter, firstly, the analytical solutions of the (1+1)dimensional dissipative nonlinear Schrödinger equation were obtained using the Generalized Simple Equation Method (GSGM), and the stability analysis of the solutions was investigated. Secondly, the Unified Method was applied to a system composed of the nonlinear Schrödinger equation (NLSE), the Lakshmanan–Porsezian–Daniel (LPD) equation and the Sasa–Satsuma (SS) equation. Various types of optical soliton solutions of the investigated model were obtained and the modulation instability (MI) was analyzed. Finally, the (3+1) dimensional Vakhnenko–Parkes (VP) equation was solved using the Nucci reduction method, leading to the discovery of different soliton solutions. In the fifth chapter, general conclusions regarding the soliton and optical soliton solutions, as well as the stability analyses obtained throughout the thesis were presented. In this study the software tools Mathematica, Maple, and MATLAB were utilized for both computational purposes and graphical visualizations. Keywords: Nonlinear differential equations, Solitons, Optical soliton, Extended sine-Gordon method, Unified method, Nucci reduction method, Analytical methods, Modulation stability analysis.