Whitham's Method and Dubrovin-Novikov Bracket in Single-Phase and Multiphase Cases

TL;DR

This paper analyzes the averaging procedure of local Poisson brackets in Whitham's method, focusing on its justification, applicability, and behavior in single-phase and multiphase cases, including resonance effects.

Contribution

It provides a detailed examination of the Dubrovin-Novikov averaging procedure, highlighting its robustness and conditions for application in complex multi-phase scenarios.

Findings

The averaging procedure is justified under broad conditions.

Resonances do not affect the averaging process significantly.

Distinct features of single-phase and multiphase cases are clarified.

Abstract

In this paper we examine in detail the procedure of averaging of the local field-theoretic Poisson brackets proposed by B.A. Dubrovin and S.P. Novikov for the method of Whitham. The main attention is paid to the questions of justification and the conditions of applicability of the Dubrovin-Novikov procedure. Separate consideration is given to special features of single-phase and multiphase cases. In particular, one of the main results is the insensitivity of the procedure of bracket averaging to the appearance of "resonances" which can arise in the multi-phase situation.