Conserved quantities, optimal systems and explicit solutions of certain partial differential equations

Abstract

In this thesis we study nine nonlinear partial differential equations (NLPDEs) from the point of view of classical Lie point symmetries and conserved quantities. These equations of choice have real world applications, primarily in the field of

fluid dynamics and plasma physics. One equation models wave propagation in a hyperelastic-rod. Several (3+1)-dimensional equations are studied in detail, these are the modified Korteweg-de Vries (mKdV) and Benjamin-Bona-Mahony (BBM) equations. We also study a (3+1)-dimensional generalised Kadomtsev-Petviashvili (KP) equation. Higher dimensional equations tend to be more apt models of nonlinear interrelations between physical quantities. Moreover, we study three systems, that is, the generalised coupled mKdV, Broer-Kaup-Kupershmidt (BKK) and coupled complex mKdV systems. We also explore two fifth-order nonlinear integrable models. Using Lie algebras, we obtain group invariant solutions, optimal systems of one dimensional subalgebras and their corresponding reductions. Techniques such as the extended Jacobi elliptic function, (G0=G)-expansion, power series solution, multiple exp-function methods are used in this work. Furthermore, we derive conservation laws for the underlying equations. Three and two dimensional renderings of selected solutions are provided. Techniques such as Noether's approach, multiplier method and Ibragimov's conservation theorem are used. In several instances we demonstrate explicitly the use of the first homotopy integral formula from variational calculus, which is sometimes attached to the multiplier method.