Application of CDSPM and Qualitative Analysis for Dynamical Models

Abstract

In order to understand nonlinear partial differential equations (NLPDEs), physicists and mathematicians need to study exact solutions. Many analytical methods have been pro- posed to acquire solutions for NLPDEs such as Korteweg de Vries equation (KdV), Sin- Gordon equation, nonlinear Schr¨odinger equation (NLSE), and all these 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. The Complete Discrimination System for Polynomial Methods (CDSPM) is a collection of explicit expressions developed with the coefficients of a polynomial that contains real or symbolic variables. Liu was the first who suggested CDSPM technique. Using this method CDSPM we find both the exact, solitary wave (SW) solutions and sensitivity of nonlinear equation. Additionally, in a mathematical framework, sensitivity analysis examines to see how modifications to a system’s parameters affect its behavior. This thesis examines the complex wave patterns of the Gerjikov-Ivanov equation (GIE), commonly known as the derivative nonlinear Schr¨odinger equation (DNLSE) and analyt- ical solitons solutions for the cubic-quintic time-fractional nonlinear non-paraxial pulse transmission model. These modifiable model are significant because they are applied in fiber optics communication, nonlinear optics, and optical processing of signals. By using CDSPM approach we explore several solutions, such as rational, SW, and Jacobi elliptic function (JEF). The CDSPM technique is utilized to examine quasi-periodic behavior, bi- furcation behavior, critical solution conditions and sensitivity analysis. We also explores the sensitivity analysis and quasi periodic behaviour of our governing models at various initial values. In addition, a number of methods for detecting quasi periodic behaviour will be addressed, including 3D and 2D graphs, time series, and Poincar`e maps.