Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions

Abstract

We develop a theory of integrable dispersive deformations of 2 + 1 dimensional Hamiltonian systems of hydrodynamic type following the scheme 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 the triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.MSC: 35L40, 37K05, 37K10, 37K55.

On Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

“…To our knowledge, there do not exist similar results even in dimension two. The research work by E. Ferapontov and collaborators has produced a big outcome in the direction of classifying the integrable Hamiltonian equations of hydrodynamic type with d spatial variables and their deformations (for instance, in [17,16]); analogous results in the direction of deformations of the Poisson structure itself are not available. One of the main reason of this fact is that the required computations are very cumbersome, and their complexity increases dramatically with the order of the operators..…”