On Hamiltonian perturbations of hyperbolic systems of conservation laws

TL;DR

This paper investigates the structure of Hamiltonian perturbations in hyperbolic PDE systems, demonstrating that bihamiltonian perturbations can be simplified via coordinate transformations, and explores invariants under Miura-type transformations.

Contribution

It proves that bihamiltonian perturbations can be eliminated order-by-order through rational coordinate changes and describes invariants under polynomial Miura transformations.

Findings

Bihamiltonian perturbations are eliminable by coordinate transformations.

Construction of quasi-Miura transformations for perturbation elimination.

Identification of invariants under Miura-type transformations.

Abstract

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.