On critical behaviour in systems of Hamiltonian partial differential equations

TL;DR

This paper investigates the critical behavior of solutions to weakly dispersive Hamiltonian PDEs near gradient catastrophe points, proposing that they are described by Painlevé-I type equations, with numerical evidence supporting this conjecture.

Contribution

It introduces a novel description of critical phenomena in Hamiltonian PDEs using Painlevé equations and provides numerical validation for this approach.

Findings

Solutions near critical points are approximated by Painlevé-I solutions.

Numerical simulations support the conjecture about Painlevé equations governing critical behavior.

The study applies to systems like the nonlinear Schrödinger equation in the semiclassical limit.

Abstract

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\'e-I (P$_I$) equation or its fourth order analogue P$_I^2$. As concrete examples we discuss nonlinear Schr\"odinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.