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The variational bicomplex for hyperbolic second-order scalar partial differential equations in the plane. (English) Zbl 0881.35069

Studied are conservation laws for hyperbolic second-order scalar partial differential equations in two independent variables. The main tool is the variational bicomplex, restricted to the “equation manifold.” Conservation laws are frist horizontal degree cocycles in the bicomplex, classical if their contact degree is zero, and higher-degree or form-valued otherwise. The operator of universal linearization, whose formal adjoint determines the cohomology, is always linear and this made possible a nice and nontrivial application of the suitably adapted classical Laplace integration method for linear second-order hyperbolic scalar partial differential equations. Among the results are the following: If all Laplace invariants of the universal linearization are non-zero (both Laplace indices are \(\infty\)), then there are no nontrivial conservation laws of contact degree \(\geq 3\). Darboux integrable equations must have finite Laplace indices. Every Darboux integrable equation has an infinite number of conservation laws of contact degree \(s\), for every \(s \geq 0\).
Reviewer: M.Marvan (Opava)

MSC:

35L65 Hyperbolic conservation laws
58J10 Differential complexes
58J45 Hyperbolic equations on manifolds
35L10 Second-order hyperbolic equations
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