Symmetry Classes of Quasilinear Systems in One Space Variable

Abstract

The family of simple quasilinear systems in one space variable is partitioned into classes of commuting flows, i.e., symmetry classes. The systems in a symmetry class have the same zeroth order conserved densities and the same Hamiltonian structure. The zeroth and first order conservation laws and the Hamiltonian structure of the systems in a complete symmetry class are described. If such a system has a degenerate characteristic speed, then it has conservation laws of arbitrarily high order. Symmetry classes of 2-component hyperbolic systems correspond to coframes on the plane. The invariants of 2-component Hamiltonian hyperbolic symmetry classes are given. An exact symmetry class of 2-component hyperbolic systems is characterized by its canonical representative, and the first order conservation laws of the canonical system correspond to the infinitesimal automorphisms of the coframe. The normal forms of the rank 0 and rank 1 exact classes are listed. A simple symmetry class is tri-Hamiltonian if and only if the metric of its coframe has constant curvature. The normal forms of the tri-Hamiltonian simple classes are listed.