Asymptotic scaling symmetries for nonlinear PDEs

TL;DR

This paper introduces the concept of asymptotic symmetries in nonlinear PDEs, providing a geometric framework and applying it to anomalous diffusion equations to explain observed scale invariance.

Contribution

It defines asymptotic symmetries for PDEs, especially scaling and translation, and applies this to explain numerical observations in anomalous diffusion models.

Findings

Analytical explanation of asymptotic scale invariance in Richardson-like equations

Application of asymptotic symmetry framework to optical lattice diffusion equations

Geometrical construction allows extension to more general PDEs

Abstract

In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large $x$ and/or $t$) invariant under a group $G$ which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations -- and solution-preserving maps -- we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physical relevance, with scaling and translation symmetries of scalar equations. We apply the general discussion to a class of ``Richardson-like'' anomalous diffusion and reaction-diffusion equations, whose solution are known by numerical experiments to be asymptotically scale invariant; we obtain an analytical explanation of the numerically observed asymptotic scaling properties. We also apply our method to a different class of anomalous diffusion equations, relevant in optical lattices. The methods developed here can be applied to more general equations, as clear by their geometrical construction.