Constructing Soliton and Kink Solutions of PDE Models in Transport and Biology

TL;DR

This paper reviews methods for finding soliton, kink, and periodic traveling wave solutions in transport and biological PDE models, introducing a modified algebraic balance method and exponential series approximation.

Contribution

It introduces a modified algebraic balance method and an exponential series approximation for invariant traveling wave solutions in PDE models.

Findings

Effective approximation demonstrated on hyperbolic Burgers equation

Analytical solutions obtained for various wave patterns

Method extends to cases where analytical solutions are difficult

Abstract

We present a review of our recent works directed towards discovery of a periodic, kink-like and soliton-like travelling wave solutions within the models of transport phenomena and the mathematical biology. Analytical description of these wave patterns is carried out by means of our modification of the direct algebraic balance method. In the case when the analytical description fails, we propose to approximate invariant travelling wave solutions by means of an infinite series of exponential functions. The effectiveness of the method of approximation is demonstrated on a hyperbolic modification of Burgers equation.