Complete integrability of shock clustering and Burgers turbulence

TL;DR

This paper demonstrates that the Lax equations governing shock clustering and Burgers turbulence are Hamiltonian, completely integrable, and solvable via inverse scattering, providing a rigorous framework for understanding the evolution of random initial data in scalar conservation laws.

Contribution

It shows that the Lax equations are Hamiltonian, integrable, and can be linearized, extending the theory of integrable systems to shock clustering and Burgers turbulence.

Findings

Lax equations are Hamiltonian and describe a least action principle.

The equations are completely integrable and linearizable via loop-group factorization.

Zero-curvature equations are solvable through inverse scattering techniques.

Abstract

We consider scalar conservation laws with convex flux and random initial data. The Hopf-Lax formula induces a deterministic evolution of the law of the initial data. In a recent article, we derived a kinetic theory and Lax equations to describe the evolution of the law under the assumption that the initial data is a spectrally negative Markov process. Here we show that: (i) the Lax equations are Hamiltonian and describe a principle of least action on the Markov group that is in analogy with geodesic flow on $SO(N)$; (ii) the Lax equations are completely integrable and linearized via a loop-group factorization of operators; (iii) the associated zero-curvature equations can be solved via inverse scattering. Our results are rigorous for $N$-dimensional approximations of the Lax equations, and yield formulas for the limit $N \to \infty$. The main observation is that the Lax equations are a $N \to \infty$ limit of a Markovian variant of the $N$-wave model. This allows us to introduce a variety of methods from the theory of integrable systems.