On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi‐Triviality of bi‐Hamiltonian perturbations

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one‐dimensional systems of hyperbolic PDEs v _t + ?( v )]_x = 0. Under certain genericity assumptions it is proved that any bi‐Hamiltonian 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 tool is in constructing the so‐called quasi‐Miura transformation of jet coordinates, eliminating an arbitrary deformation of a semisimple bi‐Hamiltonian structure of hydrodynamic type (the quasi‐triviality theorem). We also describe, following 35], the invariants of such bi‐Hamiltonian structures with respect to the group of Miura‐type transformations depending polynomially on the derivatives. © 2005 Wiley Periodicals, Inc.