Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions
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.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- July 2011
- DOI:
- 10.1063/1.3602081
- arXiv:
- arXiv:1007.3782
- Bibcode:
- 2011JMP....52g3505F
- Keywords:
-
- 02.30.Jr;
- 02.40.-k;
- Partial differential equations;
- Geometry differential geometry and topology;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- Mathematics - Analysis of PDEs;
- 35L40;
- 35L65;
- 37K10
- E-Print:
- Latex, 34 pages