SYMMETRY ANALYSIS AND CONSERVATION LAWS OF PHYSICAL MODELS ON CURVED SURFACES

Abstract

Physical models with non-flat background are important in biological mathematics. Most of the biological membranes are not flat in general. For example, membranes which convert energy in mitochondria and chloroplasts are tubes, buds and may be sheets. In most of the biological processes, the geometry of membranes is very important. The organization and shape of the membranes play a vital role in biological processes such as shape change, fusion- division, ion adsorption etc. A cell membrane is a system for exchange of energy and matter from the neighbourhood. Absorption and transformation of conserved quantities such as energy and matter from the environment are one of the characteristics of membranes. The shape of proteins, non zero curvature of membranes and involvement of conserved quantities lead one to discuss physical models on curved surfaces. Conservation laws play a vital role in science and also helpful to construct potential systems which can be used to calculate exact solutions of differential equations. Physical models on curved surfaces govern partial differential equation which need not to be derivable from variational principle. The partial Noether approach is the systematic way to construct the conservation laws for non-variational problems. The group classification and conservation laws for some partial differential equation on curved surfaces are presented in this dissertation. In particular some linear and nonlinear models of heat and wave equation on plane, cone, sphere are classified. The conservation laws for the (1 + 2)-dimensional heat equation on different surfaces are constructed via partial Noether approach and then the results are generalized for the (1+n)-dimensional case. The symmetry conservation laws relation is used to simplify the derived conserved vectors and exact solu- tions are constructed. We also extend these results to a special type of (1 + n)-dimensional linear evolution equation. Potential systems of some models from different sciences are also given. The similar analysis is performed for the (1 + 2)-dimensional wave equation on the sphere, cone and on flat surface. Furthermore, the nonlinear heat equation on curved surfaces is considered. A class of func- tions is found on the plane, sphere and torus, which is not only independent of the number of independent variables but also independent of the background metric. We consider whether the background metric or the nonlinearity have the dominant role in the infinitesimal gen- erators of heat equation on curved manifolds. Then a complete Lie analysis of the time dependent Ginzburg-Landau equation (TDGL model) is presented on the sphere and torus. In addition, for the (1 + n)-dimensional nonlinear wave equation (Klein Gordon Equation) it is proved that there is a class of functions which is independent from number of independent variables. Then for the (1 + 2)-dimensional wave equation it is proved that there is a class of functions which is invariant either the underlying space is a plane, sphere or torus.