Weiss-approach to pair of coupled non-linear reaction-diffusion equations

TL;DR

This paper applies the Weiss-algorithm to a coupled non-linear reaction-diffusion system, revealing its conditional Painlevé property, and constructs auto-Bäcklund transformations to generate special solutions.

Contribution

It introduces the use of the Weiss-algorithm for analyzing coupled reaction-diffusion equations and constructs new solution families based on the Painlevé analysis.

Findings

System has only conditional Painlevé property.

Constructed auto-Bäcklund transformations.

Derived one and two-parameter families of solutions.

Abstract

We consider a pair of coupled non-linear partial differential equations describing a biochemical model system. The Weiss-algorithm for the Painle\'{e} test, that has been succesfully used in mathematical physics for the KdV-equation, Burgers equation, the sine-Gordon equation etc., is applied, and we find that the system possesses only the "conditional" Painlev\'{e} property. We use the outcome of the analysis to construct an auto-B\"{a}cklund transformation, and we find a variety of one and two-parameter families of special solutions.