Davydovych V. Reaction-diffusion systems: conditional symmetries, exact solutions and their properties

The thesis is devoted to constructing Q-conditional symmetry operators for two- and three-component diffusive Lotka-Volterra systems and for two general classes of nonlinear reaction-diffusion systems, finding exact solutions of diffusive Lotka-Volterra systems and of some other reaction-diffusion systems, and investigation of properties of the solutions obtained. The main results are as follows. Necessary and sufficient conditions for the existence of Q-conditional symmetries are found and new Q-conditional symmetry operators for the two-component diffusive Lotka-Volterra system are constructed. Under a common restriction, a complete description of Q-conditional symmetries of the first type for the three-component diffusive Lotka-Volterra system is obtained. Using the symmetries, new exact solutions for two- and three-component diffusive Lotka-Volterra systems are constructed and their properties are investigated. A possible biological interpretation of some exact solutions is presented. Under a common restriction, a complete description of Q-conditional symmetries of the first type for the nonlinear two-component reaction-diffusion systems with constant and non-constant coefficients diffusions is obtained. Exact solutions for solving a biologically (the interaction of two populations) and physically (the gravity-driven flow of thin films of viscous fluid) motivated reaction-diffusion systems are presented.