Isolated periodic wave trains and local critical wave lengths for a nonlinear reaction-diffusion equation

Abstract

In this paper, we consider the bifurcation of small amplitude isolated periodic wave trains (SAIPWT) and the monotonicity of wave length of the periodic wave trains (PWT) for a reaction-diffusion equation. By the travelling wave transformation, the reaction-diffusion equation is transferred into its travelling wave system. Using the computer algebra system, we compute the first six singular point values for the travelling wave system, and we prove that the reaction-diffusion equation has at most 6 SAIPWT. Moreover, we study the local critical period bifurcation at the origin for the travelling wave system, and deduce the monotonicity of wave length l(h) (i.e. the wave length l as a function of the positive half wave height h) of the continuum of PWT for the reaction-diffusion equation.