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arXiv · 1007.3782

Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions

E. V. Ferapontov·A. V. Odesskii·N. M. Stoilov
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Abstract

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2+1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.

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E. V. Ferapontov, A. V. Odesskii, N. M. Stoilov. 2010-07-22. Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions. https://doi.org/10.1063/1.3602081

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