Self-similar solutions for reversing interfaces in the nonlinear diffusion equation with constant absorption

TL;DR

This paper constructs self-similar solutions for reversing interfaces in a nonlinear diffusion equation with constant absorption, using invariant manifolds and numerical methods to analyze interface behavior.

Contribution

It introduces a novel approach combining invariant manifolds and numerical schemes to find self-similar solutions for reversing interfaces in nonlinear diffusion.

Findings

Rich set of self-similar solutions obtained

Effective connection of asymptotic behaviors demonstrated

Applicable to both reversing and anti-reversing interfaces

Abstract

We consider the slow nonlinear diffusion equation subject to a constant absorption rate and construct local self-similar solutions for reversing (and anti-reversing) interfaces, where an initially advancing (receding) interface gives way to a receding (advancing) one. We use an approach based on invariant manifolds, which allows us to determine the required asymptotic behaviour for small and large values of the concentration. We then `connect' the requisite asymptotic behaviours using a robust and accurate numerical scheme. By doing so, we are able to furnish a rich set of self-similar solutions for both reversing and anti-reversing interfaces.