Asymptotic scaling symmetries for nonlinear 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.