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. The various methodologies used for finding solutions in terms of solitary waves, exact travelling waves, and doubly periodic waves have thoroughly described. Also, method known as improved F-expansion is proposed for examining the space-time fractional PDEs and subsequently, applied to space-time fractional potential YTSF equation in chapter 5. The methods to derive conservation laws for nonlinear PDEs have been discussed in details and algorithms are constructed for the same. For fractional PDEs, the linear dispersion analysis is also suggested. The chapter 3 is devoted to study the integer order nonlinear PDEs with variable coefficients such as (2+1)-dimensional new coupled ZK system and generalised $7^{th}$ order KdV equation. The infinitesimal symmetries, symmetry groups, optimal system, invariants and reductions are systematically determined for new coupled ZK system. The variety of solutions in terms of Jacobi, trigonometric and hyperbolic functions are obtained, and analysed graphically to discuss the effect of arbitrary function on the wave profile. The generalised $7^{th}$ order KdV equation is also examined for Lie symmetries. Vector fields of the optimal system give solutions in an explicit form appeared as power series and involved Jacobi elliptic functions. The conservation laws are constructed for these equations by applying direct method and new conservation theorem with nonlinear self-adjointness. The chapter 4 presents the comprehensive investigation of nonlinear PDEs having time derivatives of fractional order. It includes new coupled ZK system in (2+1)-dimensions, Wu-Zhang system in (2+1)-dimensions and $5^{th}$ order equation from Burgers hierarchy. The Lie classical technique is adopted to examine Lie symmetries with the use of Riemann-Liouville fractional (RLF) order derivative and corresponding invariants for these equations. The dimensions of fractional PDEs are reduced from (2+1) to (1+1) using invariants. The solutions show bright, dark and singular solitary wave like character for new coupled ZK system. The methodology of exponential rational function method has been utilized to seek solutions of Wu-Zhang system. Solutions in form of power series have obtained for $5^{t h}$order equation from Burgers hierarchy. The solutions of these equations are discussed graphically. The conservation laws for the equations are obtained by new conservation theorem. These equations are also studied for deriving dispersion relations, group and phase velocities. The chapter 5 deals with important nonlinear PDEs from mathematical physics having space-time variations of fractional form such as potential YTSF equation and (2+1)-dimensional Maccari system. The improved F-expansion method suggested in chapter 2 for space-time fractional PDEs is applied to potential YTSF equation in this chapter and exact travelling waves are obtained as solutions. The Maccari system in (2+1)-dimensions is investigated using an extended Jacobi elliptic function expansion (EJEFE) method for deriving solutions having doubly periodic waves. The solutions of these equations are discussed graphically to show the influence of fractional parameters onto wave profile. The dispersion relations for space-time fractional PDEs are systematically derived and the anomalous/normal dispersion of waves is shown graphically. At last, the summary of the thesis and some concluding remarks are given of the work conducted in different chapters.