On universality of critical behaviour in Hamiltonian PDEs

TL;DR

This paper investigates the universal nature of singularities in Hamiltonian PDEs, showing that solutions near critical points can be described by classical singularity theory and Painleve' equations, revealing deep structural insights.

Contribution

It provides a comparative analysis of singularities in Hamiltonian PDEs and their perturbations, linking local structures to classical and Painleve' singularity solutions.

Findings

Singularities in unperturbed systems are described by classical singularity theory.

Perturbed systems' singularities correspond to special Painleve' solutions.

The study highlights the universal features of critical behaviour in Hamiltonian PDEs.

Abstract

Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimension. For the systems of order one or two we describe the local structure of singularities of a generic solution to the unperturbed system near the point of "gradient catastrophe" in terms of standard objects of the classical singularity theory; we argue that their perturbed companions must be given by certain special solutions of Painleve' equations and their generalizations.