Front motion for phase transitions in systems with memory

Details Front motion for phase transitions in systems with memory Front motion for phase transitions in systems with memory

2000-02-22

arxiv.org/abs/nlin/0002039v1

Nonlinear Sciences

Pattern Formation and Solitons

Scientific paper

10.1016/S0167-2789(00)00132-9

We consider the Allen-Cahn equations with memory (a partial integro-differential convolution equation). The prototype kernels are exponentially decreasing functions of time and they reduce the integrodifferential equation to a hyperbolic one, the damped Klein-Gordon equation. By means of a formal asymptotic analysis we show that to the leading order and under suitable assumptions on the kernels, the integro-differential equation behave like a hyperbolic partial differential equation obtained by considering prototype kernels: the evolution of fronts is governed by the extended, damped Born-Infeld equation. We also apply our method to a system of partial integro-differential equations which generalize the classical phase field equations with a non-conserved order parameter and describe the process of phase transitions where memory effects are present.

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