Existence of traveling wave solutions for a generalized Burgers–Fisher equation with weak convection

Abstract

The existence of traveling wave solutions for a generalized Burgers–Fisher equation with weak convection term is investigated in current paper. The corresponding traveling wave equation is converted to a regularly perturbed Hamiltonian system by rescaling of the wave speed. Then by using Melnikov’s method, we show that the generalized Burgers–Fisher equation contains kink and anti-kink wave solutions with small wave speeds, and the wave speed selection principle is presented as well. By analyzing the ratio of Abelian integrals, we also prove that the generalized Burgers–Fisher equation admits periodic wave solutions whose rescaled limit wave speed is monotonically increasing. Moreover, the upper and lower bounds of the limit wave speed are given, and the relationship between wave speed and wavelength is established. Numerical simulations are performed to illustrate our theoretical analysis.