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Spatially discrete reaction-diffusion equations with discontinuous hysteresis
Pavel Gurevich and Sergey Tikhomirov
We deal with a class of lattice dynamical systems, namely hysteretic reaction-diffusion equations that are continuous in time and discrete in space. Our primary goal is to analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling. The rattling is characterized by a specific behavior of the solution profile: while evolving in time, it oscillates up- and downwards and "switches" hysteresis at nodes satisfying a certain rule. In the one-dimensional case, this switching rule makes the profile shape two hills propagating outwards the origin with decreasing velocity. In a prototype case, we prove the existence of rattling and identify the propagation velocity.