Solving the shock wave solutions of reaction-diffusion equations based on Conley index theory

Based on Conley index theory, the shock wave solutions of a class of nonlinear reaction-diffusion equations were studied. Considering the diffusion coefficient as a system parameter, the existence of heteroclinic orbits of ordinary differential equations satisfied by traveling wave solutions is analyzed by using Conley index and Morse decompositions. The existence of saddle-focus and saddle-crunode style shock wave solutions of the reaction-diffusion equations is proved on the basis of an idea that the solitary waves and shock waves of partial differential equations correspond to the homoclinic orbits and heteroclinic orbits of ordinary differential equations. In particular, the existence and uniqueness of saddle-saddle style shock wave solutions are proved by using connection matrixes and transition matrixes, which are computed with Conley packages and Maple software by programming.