Symmetry Analysis and Conservation Laws for Some Systems of Nonlinear Partial Differential Equations

Abstract

The work complied in this thesis includes the investigation of nonlinear partial differential equations (PDEs) of integer and fractional order representing some physical phenomena for exact solutions, symmetries, and conservation laws. Linear dispersion analysis of fractional order PDEs is carried out to identify the normal/anomalous dispersion of waves. The techniques to retrieve solutions have thoroughly described and successfully implemented. The thesis consists of five chapters compiled for the investigation of seven nonlinear PDEs which are (2+1)-dimensional new coupled Zakharov-Kuznetsov (ZK) system, generalised $7^{th}$ order Korteweg and de Vries (KdV) equation, new coupled ZK system as well as Wu-Zhang system in (2+1)-dimensions having time derivatives of fractional order, time fractional $5^{th}$ order equation from Burgers hierarchy, space-time fractional potential Yu-Toda-Sasa-Fukuyama (YTSF) equation and space-time fractional Maccari model in (2+1)-dimensions. Thesis is organised into five chapters. The chapter 1 introduces some important nonlinear PDEs of integer and fractional orders. The physical phenomena inherited by different types of PDEs are tabulated. Brief literature reviews on Lie group of transformations, methods for finding exact solutions, and conservation laws are presented. Introduction of linear dispersion analysis is briefed out. The frame work of the thesis is also presented systematically in this chapter. The chapter 2 consists the preliminaries including some definitions, theorems related to Lie group theory, conservation laws, exact solutions, and dispersion analysis. Lie infinitesimal criterion to examine integer and time fractional PDEs is presented in an algorithmic way.