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The Haantjes tensor and double waves for multi-dimensional systems of hydrodynamic type: a necessary condition for integrability. (English) Zbl 1149.37322

Summary: An invariant differential-geometric approach to the integrability of (2+1)-dimensional systems of hydrodynamic type, \(ut+A(u)u_{x}+B(u)u_{y}=0\), is developed. We prove that the existence of special solutions known as ‘double waves’is equivalent to the diagonalizability of an arbitrary matrix of the two-parameter family \((kE+A)_{-1}(lE+B)\). Since the diagonalizability can be effectively verified by calculating the Haantjes tensor, this provides a simple necessary condition for integrability.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q58 Other completely integrable PDE (MSC2000)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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