Study of Stochastic Nonlinear Schr¨odinger Equation for Optical Soliton Solutions

Abstract

It finds extensive application in the analysis of nonlinear systems for the detection of ap- proximate solutions and stability analysis. In order to understand nonlinear partial differ- ential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many NLPDEs such as Korteweg de Vries equation (KdV), Sin-Gordon equation, non- linear Schr¨odinger equation (NLSE), and other equations possess solitons solutions. A soliton, which arises as a result of dispersion and nonlinearity, is a wave that retain their original features while propagating from one medium to another. This thesis discusses the stochastic variants of the well-known non linear Kairat-II and Kairat-X (NLKIIKX) equa- tions in (3+1) space dimensions, those being soliton theory’s major models. Adding the Wiener process to simulate random fluctuations, we study the dynamics of these equations within a probabilistic system. With enhanced direct algebraic technique (EDAT) and novel projective Ricatti equation approach (NPREA), we derive exact soliton solutions for these stochastic equations. The behavior of a stochastic differential equation (SDE) is fundamen- tally different from a deterministic equation due to the presence of randomness and gain an understanding about random fluctuation-influenced complex system behavior. At last, by using these two methods, we shall develop three-dimensional and two-dimensional plots in order to show soliton solutions. It is used to simulate randomness-influenced systems, e.g., stock markets or noisy physical systems. We will also study the energy balance approch (EBA) which examines a system by comparing the rate of input and output energy to find the steady-state or dynamic state.