Dispersive deformations of Hamiltonian systems of hydrodynamic type in   2+1 dimensions
  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. 
E.V. Ferapontov, V.S. Novikov and N.M. Stoilov
nlin.SI math-ph math.MP