An obstruction to the integrability of a class of non-linear wave equations by 1-stable cartan characteristics

Fackerell, E. D. and Hartley, D. H. and Tucker, Robin (1995) An obstruction to the integrability of a class of non-linear wave equations by 1-stable cartan characteristics. Journal of Differential Equations, 115 (1). pp. 153-165. ISSN 0022-0396

Abstract

We examine in detail the Cauchy problem for a class of non-linear hyperbolic equations in two independent variables. This class is motivated by the analysis of the dynamics of a line of non-linearly coupled particles by Fermi, Pasta, and Ulam and extends the recent investigation of this problem by Gardner and Kamran. We find conditions for the existence of a 1-stable Cartan characteristic of a Pfaffian exterior differential system whose integral curves provide a solution to the Cauchy problem. The same obstruction to involution is exposed in Darboux′s method of integration and the two approaches are compared. A class of particular solutions to the obstruction is constructed.