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

TL;DR

This paper demonstrates how the exotic cohomology of symmetry algebras leads to new integrable multi-dimensional PDEs, extending the self-dual Yang-Mills equation.

Contribution

It introduces a novel connection between exotic cohomology and integrable hierarchies for nonlinear PDEs.

Findings

Non-trivial second exotic cohomology implies integrable generalizations.

Constructed a hierarchy of multi-dimensional nonlinear PDEs.

Explicitly derived the first three equations of the hierarchy.

Abstract

We consider the four-dimensional reduced quasi-classical self-dual Yang--Mills equation and show that non-triviality of the second exotic cohomology group of its symmetry algebra implies existence of a two-component integrable generalization of this equation. The sequence of natural extensions of this symmetry algebra generate an integrable hierarchy of multi-dimensional nonlinear PDEs. We write out the first three elements of this hierarchy.