Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor

2005/05/04 by E. V. Ferapontov, Ferapontov, E. V., D. G. Marshall +1 · 3 citations
Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #nlin.SI

paper · pdf · doi:10.48550/arxiv.nlin/0505013

36 pages, the classification results and proofs are refined. A section on generating functions is added

arxiv created 2007/01/22 · arxiv updated 2009/12/01

Abstract

The integrability of an m-component system of hydrodynamic type, u_t=V(u)u_x, by the generalized hodograph method requires the diagonalizability of the mxm matrix V(u). This condition is known to be equivalent to the vanishing of the corresponding Haantjes tensor. We generalize this approach to hydrodynamic chains -- infinite-component systems of hydrodynamic type for which the infinite matrix V(u) is `sufficiently sparse'. For such systems the Haantjes tensor is well-defined, and the calculation of its components involves finite summations only. We illustrate our approach by classifying broad classes of conservative and Hamiltonian hydrodynamic chains with the zero Haantjes tensor. We prove that the vanishing of the Haantjes tensor is a necessary condition for a hydrodynamic chain to possess an infinity of semi-Hamiltonian hydrodynamic reductions, thus providing an easy-to-verify necessary condition for the integrability.

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• Jacobi-Haantjes manifolds, integrability and dissipative mechanical systems
• Haantjes torsion and integrability: a proof of Bolsinov-Konyaev-Matveev's conjecture

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