Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type.

Abstract

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.