Tangential Characteristic Symmetries and First Order Hyperbolic Systems

Abstract

Hyperbolic systems of first order partial differential equations in two dependent and two independent variables are studied from the point of view of their local geometry. We illustrate an earlier result on such systems, which derived a complete set of local invariants for the class of systems which are (2,2)-Darboux integrable on the 1-jets, by explicitly computing the tangential characteristic symmetries of a certain Fermi–Pasta–Ulam equation. As a consequence we reduce its associated intrinsic hyperbolic structure to Vessiot normal form and prove that this equation may be regarded as an ordinary differential equation on ℜ2. This result has a far reaching generalisation in which the well known local linearisation of this equation by a hodograph transformation is shown to be a special case of a theorem due to Vessiot. This is further illustrated by studying the Born-Infeld system of Arik, Neyzi, Nutku, Olver and Verosky and the s = 0 Liouville system of Bryant, Griffiths and Hsu. It is shown that the latter may be regarded as an ordinary differential equation on the Lie group of affine transformations of the real line.