Nonlocally Related Partial Differential Equation Systems, the Nonclassical Method and Applications

Abstract

Symmetry methods are important in the analysis of differential equation (DE) systems. In this thesis, we focus on two significant topics in symmetry analysis: nonlocally related partial differential equation (PDE) systems and the application of the nonclassical method. In particular, we introduce a new systematic symmetry-based method for constructing nonlocally related PDE systems (inverse potential systems). It is shown that each point symmetry of a given PDE system systematically yields a nonlocally related PDE system. Examples include applications to nonlinear reaction-diffusion equations, nonlinear diffusion equations and nonlinear wave equations. Moreover, it turns out that from these example PDEs, one can obtain nonlocal symmetries (including some previously unknown nonlocal symmetries) from some corresponding constructed inverse potential systems. In addition, we present new results on the correspondence between two potential systems arising from two nontrivial and linearly independent conservation laws (CLs) and the relationships between local symmetries of a PDE system and those of its potential systems. We apply the nonclassical method to obtain new exact solutions of the nonlinear Kompaneets (NLK) equation ut = x −2 x4 ( αux + βu + γu 2 )) x , where α > 0, β ≥ 0 and γ > 0 are arbitrary constants. New time-dependent exact solutions for the NLK equation ut = x −2 ( x4 ( αux + γu 2 )) x , for arbitrary constants α > 0, γ > 0 are obtained. Each of these solutions is expressed in terms of elementary functions. We also consider the behaviours of these new solutions for initial conditions of physical interest. More specifically, three of these families of solutions exhibit quiescent behaviour and the other two families of solutions exhibit blow-up behaviour in finite time. Consequently, it turns out that the corresponding nontrivial stationary solutions are unstable.