Bifurcations of self-similar solutions for reversing interfaces in the slow diffusion equation with strong absorption

TL;DR

This paper investigates the bifurcation phenomena of self-similar solutions in the slow diffusion equation with strong absorption, revealing how these solutions emerge from standing interfaces and characterizing their behavior near bifurcation points.

Contribution

It introduces a detailed analysis of bifurcations of self-similar solutions, utilizing asymptotic methods and linking them to hypergeometric functions, which advances understanding of reversing interfaces in nonlinear diffusion.

Findings

Bifurcations occur at points where hypergeometric functions truncate into polynomials.

Asymptotic dependencies of reversing interfaces are derived near bifurcation points.

Numerical results agree well with asymptotic predictions.

Abstract

Bifurcations of self-similar solutions for reversing interfaces are studied in the slow diffusion equation with strong absorption. The self-similar solutions bifurcate from the time-independent solutions for standing interfaces. We show that such bifurcations occur at the bifurcation points, at which the confluent hypergeometric functions satisfying Kummer's differential equation is truncated into a finite polynomial. A two-scale asymptotic method is employed to obtain the asymptotic dependencies of the self-similar reversing interfaces near the bifurcation points. The asymptotic results are shown to be in excellent agreement with numerical computations.