Classical R-matrix theory for bi-Hamiltonian field systems

TL;DR

This paper reviews the R-matrix formalism for constructing integrable systems in (1+1) and (2+1) dimensions, applying it to various infinite-dimensional Lie algebras to develop both dispersionless and dispersive integrable field models.

Contribution

It provides a comprehensive, self-contained development of R-matrix theory for bi-Hamiltonian field systems in multiple dimensions, including proofs and applications to different algebraic structures.

Findings

Constructed integrable field systems using R-matrix formalism

Extended the theory to (2+1)-dimensional systems

Applied the framework to various infinite-dimensional Lie algebras

Abstract

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1+1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2+1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems.