An analysis of symmetries and conservation laws of some classes of PDEs that arise in mathematical physics and biology.

Abstract

In this thesis, the symmetry properties and the conservation laws for a number of well-known PDEs which occur in certain areas of mathematical physics are studied. We focus on wave equations that arise in plasma physics, solid physics and fluid mechanics. Firstly, we carry out analyses for a class of non-linear partial differential equations, which describes the longitudinal motion of an elasto-plastic bar and anti-plane shearing deformation. In order to systematically explore the mathematical structure and underlying physics of the elasto-plastic flow in a medium, we generate all the geometric vector fields of the model equations. Using the classical Lie group method, it is shown that this equation does not admit space dilation type symmetries for a speci fic parameter value. On the basis of the optimal system, the symmetry reductions and exact solutions to this equation are derived. The conservation laws of the equation are constructed with the help of Noether's theorem We also consider a generalized Boussinesq (GB) equation with damping term which occurs in the study of shallow water waves and a system of variant Boussinesq equations. The conservation laws of these systems are derived via the partial Noether method and thus demonstrate that these conservation laws satisfy the divergence property. We illustrate the use of these conservation laws by obtaining several solutions for the equations through the application of the double reduction method, which encompasses the association of symmetries and conservation laws. A similar analysis is performed for the generalised Gardner equation with dual power law nonlinearities of any order. In this case, we derive the conservation laws of the system via the Noether approach after increasing the order and by the use of the multiplier method. It is observed that only the Noether's approach gives a uni ed treatment to the derivation of conserved vectors for the Gardner equation and can lead to local or an in finite number of nonlocal conservation laws. By investigating the solutions using symmetry analysis and double reduction methods, we show that the double reduction method yields more exact solutions; some of these solutions cannot be recovered by symmetry analysis alone. We also illustrate the importance of group theory in the analysis of equations which arise during investigations of reaction-diffusion prey-predator mechanisms. We show that the Lie analysis can help obtain different types of invariant solutions. We show that the solutions generate an interesting illustration of the possible behavioural patterns.