Qualitative Research On Traveling Wave Solution Of Nonlinear Reaction-diffusion Equation

Recently, reaction-diffusion equations have been gradually becoming one of themost important research of the modern mathematics. Many studies concentrate on thedynamic property of the monotone and bistable time periodic reaction-diffusionequation. As a special solution of the reaction-diffusion equation, traveling wavesolutions reflect many properties of equations and have been applied on many scientificfields. This paper directly focuses on the existence, uniqueness and stability of theLotka-Volterra cooperating reaction-diffusion equation when it is under the bistablecondition.This paper state the condition when the bistable traveling wave solution ofLotka-Volterra cooperating reaction-diffusion equation is existed. We undergoes atransformation to the system and simplifies some proof. Firstly, focus on the property ofthe equilibrium points by the definition of above strong stability and below strongstability. Secondly, verify the existence of the time periodic traveling wave solutionwhich links the two stable points. Thirdly, proof the uniqueness of the existed timeperiodic traveling wave solution. Finally, examine the stability of time periodictraveling wave solution and propose a case.