Evidence of Soliton-Like Behavior of Solitary Waves in a Nonlinear Reaction-Diffusion System

Many systems of nonlinear reaction-diffusion equations have been found in which stable traveling solitary waves annihilate one another on collision. A two-component reaction-diffusion system with this property is constructed from the equations in a model of spreading cortical depression, in which the components are the concentrations of potassium and calcium ions in the extracellular compartment of nervous tissue. The solitary wave orbit is studied in relation to the source functions in the reaction diffusion system. The trajectories of the solutions are studied at various spatial points during the collision of the annihilating waves. During the collision interaction, parts of the plane of values of the components are visited that have not arisen during the passage of the solitary waves. The source functions are modified off the solitary wave trajectory, and it is found that two solitary waves emerge from the collision whose wave forms and speeds are identical to those of the colliding waves. The numerical computations thus suggest that the modified system of reaction-diffusion equations has solitary wave solutions with soliton-like properties. With slightly asymmetric initial data, when two solitary waves collided, only one emerged from the collision.