The role of self-similarity in singularities of PDE's

TL;DR

This paper reviews how self-similarity and dynamical systems analysis help understand singularities in PDEs, highlighting stable fixed points, limit cycles, and travelling waves as key behaviors.

Contribution

It introduces a unified approach using similarity transformations and dynamical systems to classify and analyze singularities in evolution equations.

Findings

Self-similar behavior often corresponds to fixed points or simple attractors.

Analysis near fixed points reduces the complexity of singularity characterization.

Examples include stable fixed points, limit cycles, and travelling waves.

Abstract

We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a \textit{dynamical system}. We point out that analysing the dynamics close to the fixed point is a useful way of characterising the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. For each "class" of singularity, we give detailed examples.