Existence of Conservation Laws and Characterization of Recursion Operators for Completely Integrable Systems

Using the Spencer-Goldschmidt version of the Cartan-Kähler theorem, we give conditions for (local) existence of conservation laws for analytical quasi-linear systems of two independent variables. This result is applied to characterize the recursion operator (in the sense of Magri) of completely integrable systems. Introduction A conservation law for a (1-1) tensor field h on a manifold M is a 1-form θ which satisfies dθ = 0 and dh∗θ = 0, where h∗ is the transpose of h : (h∗θ)(X) := θ(hX). Conservation laws arise, for example, in the following classical problem. Consider a system of n quasi-linear equations in two independent variables : ∂x ∂u + hj(x) ∂x ∂v = 0 (i = 1, ..., n) (∗) (repeated indices being summed from 1 to n). If θ := λi(x) dx i is a conservation law with respect to the (1-1) tensor field defined by the matrix hj , there exist locally two functions f and g so that θ = df and h∗θ = dg, i.e. λi = ∂f ∂xi and h i jλi = ∂g ∂xj , and we have 0 = λi ∂x ∂u + λih i j ∂x ∂v = ∂f ∂xj ∂x ∂u + ∂g ∂xj ∂x ∂v . Then for any solution x(u, v) of the system (*), we have ∂f(x(u, v)) ∂u + ∂g(x(u, v)) ∂v = 0, and it contains a conservation law in the sense of Lax [9]. Many interesting properties have been developed for systems of partial differential equations which contain conservation laws, and in particular for systems which can be expressed entirely in terms of conservation laws. It is therefore of interest to know when these conditions are satisfied. More recently, conservation laws have been employed by Magri in his classical paper concerning Hamiltonian completely integrable systems [11]: h is the recursion Received by the editors November 28, 1994 and, in revised form, April 3, 1996. 1991 Mathematics Subject Classification. Primary 35G20, 35N10; Secondary 58F07, 58G30.