Classical R-matrix theory for bi-Hamiltonian field systems

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.

review article, 39 pages

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• Post-Lie algebras and factorization theorems
• Hierarchies of Manakov-Santini Type by Means of Rota-Baxter and Other Identities
• On a Lagrangian reduction and a deformation of completely integrable systems
• An extension of the Kadomtsev-Petviashvili hierarchy and its Hamiltonian structures
• Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics
• Dispersionless (3+1)-dimensional integrable hierarchies
• Construction and separability of nonlinear soliton integrable couplings
• Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions
• The Lax Integrable Differential-Difference Dynamical Systems on Extended Phase Spaces