Deformations of infinite-dimensional Lie algebras, exotic cohomology, and integrable nonlinear partial differential equations

Abstract

The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such conditions in internal terms of the PDEs under the study. In this paper we consider certain examples of infinite-dimensional Lie algebras with nontrivial second exotic cohomology groups and show that the Maurer-Cartan forms of the associated extensions of these Lie algebras generate Lax representations for integrable systems, both known and new ones.

References in corpus (3)

• Integrability of the Manakov--Santini hierarchy
• A Bäcklund transformation between 4D Mart\'ınez Alonso -- Shabat and Ferapontov -- Khusnutdinova equations
• Modified dispersionless Veselov--Novikov equations and corresponding hydrodynamic chains

Cited by in corpus (4)

• Lax representations with non-removable parameters and integrable hierarchies of PDEs via exotic cohomology of symmetry algebras
• Extensions of the symmetry algebra and Lax representations for the two-dimensional Euler equation
• Integrable partial differential equations and Lie--Rinehart algebras
• Six-dimensional heavenly equation. Dressing scheme and the hierarchy