Existence of Conservation Laws in Nilpotent Case

Using the Spencer-Goldschmidt version of the Cartan-Kähler theorem, we prove the local existence of conservation laws for analytical quasi-linear systems of two independent variables in the nilpotent and 2-cyclic case. Introduction A conservation law for a (1-1) tensor field h on a manifold M , dimM = n, is a 1-form θ which satisfies dθ = 0 and dh∗θ = 0, where h∗ is the transpose of h : h∗θ := θ ◦ h. Conservation laws arise, for example, in the following classical problem. Consider a system of n quasi-linear equations in two independent variables: (∗) ∂x i ∂u + hj(x) ∂x ∂v = 0 (i, j = 1, . . . , n). If θ := λi(x)dx is a conservation law with respect to the (1-1) tensor field h 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 + λihj(x) ∂x ∂v = ∂f ∂xj ∂x ∂u + ∂g ∂xj ∂x ∂v = 0. 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 ([6]). Locally, giving a conservation law is equivalent to giving a function f such that (dh∗d)(f) = 0. Thus the study of the local existence of conservation laws is equivalent (in an analytic context) to the study of the formal integrability of the differential operator dh∗d. Received May 19, 1999; revised September 14, 1999. 1980 Mathematics Subject Classification (1991 Revision). Primary 35G20, 35N10; Secondary 58F07, 58G30.