## 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.