Existence of travelling wave front solutions of a two-dimensional anisotropic model

This paper considers a two-dimensional anisotropic modelψt=rψ-1k04Δ+k022ψ-c̃k04∂y4ψ+2ηk04∂x2∂y2ψ-ψ3,introduced by Pesch and Kramer (1986) [28]. Assume that ψ travels with a speed c in the propagation direction x and is periodic in the transverse direction y. This model is formulated as a spatial dynamic system in which the variable x is a time-like variable. A center-manifold reduction technique and a normal form analysis are applied to show that this dynamic system can be reduced to a system of ordinary differential equations. A bifurcation analysis yields the persistence of the heteroclinic orbit for the reduced system when higher order terms are added and the speed c is small enough, which establishes the existence of travelling wave front solutions. In order to overcome the difficulty caused by the irreversibility, some appropriate constants are adjusted.