Symmetries and Exact Solutions of Some Nonlinear Partial Differential Equations

Abstract

The thesis entitled SYMMETRIES AND EXACT SOLUTIONS OF SOME NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS is devoted to find symmetries and exact solutions of some nonlinear partial differential equations (PDEs) which represent some physically relevant systems. The thesis comprises eight chapters. Chapter 1 is introductory and consist of prerequisites of the present work. It presents primarily the methodologies utilized in the thesis and a brief account of the related studies made by various authors in the field. In Chapter 2, we have investigated the symmetries and invariant solutions of b-family equation and modified b-family equation. Firstly, the Lie group method is utilized for the purpose of obtaining the group infinitesimals of b-family equation. The basic fields of the optimal system lead to reductions that are inequivalent with respect to the symmetry transformations. Secondly, we used direct method introduced by Clarkson and Krusksal to find symmetries of bfamily equation. We obtain the exact solutions of b-family equation corresponding to reduced ordinary differential equations (ODEs). In this chapter, We have also investigated the symmetries of modified b-family equation, which describe the balance between the convection and the stretching for small viscosity in the dynamics of 1D nonlinear waves in fluids. We have shown that only non constant similarity reduction obtainable either by Lie classical method or Direct method due to Clarkson and Kruksal, is travelling wave solution of the equation. In Chapter 3, coupled Higgs field equation and Hamiltonian amplitude equation are studied using the Lie classical method. Symmetry reductions and exact solutions are reported for Higgs equations and Hamiltonian amplitude equation. We also establish the travelling wave solutions involving parameters of the coupled Higgs equations and Hamiltonian amplitude equation by using (G’/G )-expansion method. The travelling waves solutions expressed by hyperbolic, trigonometric and the rational functions are obtained. In chapter 4, we considered the variable coefficient form of (2+1)-dimensional Zakharov-Kuznetsov modified equal width equation (vcZKMEW). By using Lie group analysis, symmetries for the equations are obtained. Using symmetries, the vcZKMEW equation is reduced to two dimensional partial differential equation. Exact solutions of reduced two dimensional PDE are obtained and corresponding exact solutions of vcZKMEW equation are shown. In Chapter 5, the variable coefficients version of the Benjamin-Bona-Mahony (BBM) equation has been investigated for symmetries and some interesting exact solutions have been derived. The Painlev´e analysis of an ODE has also been performed which shows that it is not integrable. In Chapter 6, the Painlev´e analysis of (2+1)- dimensional variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlev´e property. Similarity reductions of the VCBK equation to one dimensional partial differential equations including Burgers equation are investigated. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ODEs deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained. In Chapter 7, the variable-coefficients Gardner (vc-Gardner) equation is considered. By using the Painlev´e analysis and Lie group analysis, the Painlev´e properties and symmetries for the equations are obtained. The exact solutions generated from the symmetries and Painlev´e analysis are presented.