The multi-dimensional Hamiltonian Structures in the Whitham method

TL;DR

This paper extends the theory of Hamiltonian structures in the Whitham method to multiple dimensions, constructing local Poisson brackets for multidimensional systems through averaging procedures and analyzing their properties.

Contribution

It introduces a direct construction method for multidimensional averaged Poisson brackets and discusses their canonical forms, advancing the understanding of Hamiltonian structures in higher-dimensional Whitham systems.

Findings

Constructed local Poisson brackets for multidimensional Whitham systems.

Identified differences in phase space structures compared to one-dimensional cases.

Discussed conditions for the applicability of the averaging scheme.

Abstract

In this paper we consider the averaging of local field-theoretic Poisson brackets in the multi-dimensional case. As a result, we construct a local Poisson bracket for the regular Whitham system in the multidimensional situation. The procedure is based on the procedure of averaging of local conservation laws and follows the Dubrovin - Novikov scheme of the bracket averaging suggested in one-dimensional case. However, the features of the phase space of modulated parameters in higher dimensions lead to a different natural class of the averaged brackets in comparison with the one-dimensional situation. Here we suggest a direct procedure of construction of the bracket for the Whitham system for $d > 1$ and discuss the conditions of applicability of the corresponding scheme. At the end, we discuss canonical forms of the averaged Poisson bracket in the multidimensional case.