Ferapontov E.V., Novikov V.S., Odeskii A.V., Stoilov N. Hamiltonian systems of hydrodynamic type in 2+1 dimensions and their dispersive deformations, talk Workshop Geom. of PDEs and Integrability, 14-18 Oct 2013, Teplice nad Becvou, Czech Rep. (abstract)
Speaker: Nikola Stoilov
Title: Hamiltonian systems of hydrodynamic type in 2+1 dimensions and their dispersive deformations (joint work with E.V. Ferapontov, V.S. Novikov, and A.V. Odeskii)
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. 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. Furthermore, we develop a theory of integrable dispersive deformations of these Hamiltonian systems following a scheme similar to that proposed by Dubrovin and his collaborators in 1 + 1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.