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Existence of travelling wave front solutions of a two-dimensional anisotropic model. (English) Zbl 1358.34047

Summary: This paper considers a two-dimensional anisotropic model \[ \psi_t=r_\psi -\frac{1}{k_0^4}\left(\Delta+k_0^2\right)^2\psi-\frac{\tilde c}{k_0^4}\partial_y^4\psi+\frac{2\eta}{k_0^4}\partial_ x^2\partial_ y^2\psi-\psi^3, \] introduced by Pesch and Kramer (1986). Assume that \(\psi\) 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.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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