Integrable viscous conservation laws

TL;DR

This paper extends the Dubrovin-Zhang approach to classify integrable viscous conservation laws, introduces the viscous central invariant, and explores their hierarchies and universal behaviors, including a viscous Painlevé equation.

Contribution

It proposes a novel classification framework for viscous conservation laws using a single functional parameter, the viscous central invariant, and constructs associated hierarchies and solutions.

Findings

Normal forms are parameterized by one viscous central invariant.

Identified a viscous analog of the Camassa-Holm equation.

Derived the viscous Painlevé I2 equation for critical behavior.

Abstract

We propose an extension of the Dubrovin-Zhang perturbative approach to the study of normal forms for non-Hamiltonian integrable scalar conservation laws. The explicit computation of the first few corrections leads to the conjecture that such normal forms are parameterized by one single functional parameter, named viscous central invariant. A constant valued viscous central invariant corresponds to the well-known Burgers hierarchy. The case of a linear viscous central invariant provides a viscous analog of the Camassa-Holm equation, that formerly appeared as a reduction of a two-component Hamiltonian integrable systems. We write explicitly the negative and positive hierarchy associated with this equation and prove the integrability showing that they can be mapped respectively into the heat hierarchy and its negative counterpart, named the Klein-Gordon hierarchy. A local well-posedness theorem for periodic initial data is also proven. We show how transport equations can be used to effectively construct asymptotic solutions via an extension of the quasi-Miura map that preserves the initial datum. The method is alternative to the method of the string equation for Hamiltonian conservation laws and naturally extends to the viscous case. Using these tools we derive the viscous analog of the Painlev\'e I2 equation that describes the universal behaviour of the solution at the critical point of gradient catastrophe.