Exact travelling wave solutions of non-linear reaction-convection-diffusion equations -- an Abel equation based approach

TL;DR

This paper derives exact travelling wave solutions for nonlinear reaction-convection-diffusion equations using Abel equation methods, generalizing classical models like Fisher-Kolmogorov and analyzing solutions with numerical and semi-analytical techniques.

Contribution

It introduces an Abel equation based approach to find exact solutions for a broad class of nonlinear reaction-convection-diffusion equations, including generalizations of standard models.

Findings

Exact travelling wave solutions for generalized reaction-convection-diffusion equations.

Reduction of complex PDEs to Abel type equations using integrability conditions.

Numerical and semi-analytical analysis of solutions.

Abstract

We consider quasi-stationary (travelling wave type) solutions to a general nonlinear reaction-convection-diffusion equation with arbitrary, autonomous coefficients. The second order nonlinear equation describing one dimensional travelling waves can be reduced to a first kind first order Abel type differential equation By using two integrability conditions for the Abel equation (the Chiellini lemma and the Lemke transformation), several classes of exact travelling wave solutions of the general reaction--convection-diffusion equation are obtained, corresponding to different functional relations imposed between the diffusion, convection and reaction functions. In particular, we obtain travelling wave solutions for two non-linear second order partial differential equations, representing generalizations of the standard diffusion equation and of the classical Fisher--Kolmogorov equation, to which they reduce for some limiting values of the model parameters. The models correspond to some specific, power law type choices of the reaction and convection functions, respectively. The travelling wave solutions of these two classes of differential equations are investigated in detail by using both numerical and semi-analytical methods.