On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour

TL;DR

This paper investigates how Hamiltonian perturbations affect hyperbolic conservation laws near critical points, revealing a universal behavior described by a special solution to an integrable fourth order ODE.

Contribution

It demonstrates the universality of critical behavior in Hamiltonian-perturbed hyperbolic systems, independent of specific perturbations or solutions.

Findings

Critical behavior near gradient catastrophe is universal.

Solution behavior is governed by a special integrable ODE.

Universality holds for generic perturbations and solutions.

Abstract

Hamiltonian perturbations of the simplest hyperbolic equation $u_t + a(u) u_x=0$ are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE.