Multi shocks in Reaction-diffusion models

TL;DR

This paper classifies reaction-diffusion models on a one-dimensional lattice that can have double-shocks, investigates their evolution, and analyzes conditions under which shocks vanish or persist over time.

Contribution

It identifies four independent models with double-shocks and studies their shock dynamics, including conditions for shock disappearance and long-term behavior.

Findings

Only four models possess double-shocks on a 1D lattice.

Double-shocks can vanish, leading to shock-free states.

No stationary double-shocks exist in these models.

Abstract

It is shown, concerning equivalent classes, that on a one-dimensional lattice with nearest neighbor interaction, there are only four independent models possessing double-shocks. Evolution of the width of the double-shocks in different models is investigated. Double-shocks may vanish, and the final state is a state with no shock. There is a model for which at large times the average width of double-shocks will become smaller. Although there may exist stationary single-shocks in nearest neighbor reaction diffusion models, it is seen that in none of these models, there exist any stationary double-shocks. Models admitting multi-shocks are classified, and the large time behavior of multi-shock solutions is also investigated.