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

TL;DR

This paper extends the theory of integrable dispersive deformations from 1+1 to 2+1 dimensional Hamiltonian systems of hydrodynamic type, revealing increased rigidity and classifying nontrivial second-order deformations.

Contribution

It develops a framework for dispersive deformations in higher dimensions and classifies Hamiltonians with nontrivial second-order deformations.

Findings

Multi-dimensional Hamiltonian systems are more rigid than 1+1 dimensional ones.

First order deformations are trivial for the studied class.

Certain Hamiltonians admit nontrivial second-order deformations.

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.