Fr\"olicher-Nijenhuis geometry and integrable matrix PDE systems

TL;DR

This paper explores the geometric conditions under which certain tensor fields lead to integrable matrix PDE systems, revealing new classes of nonlinear equations and their solution-generating transformations.

Contribution

It introduces a geometric framework based on Fr"olicher-Nijenhuis brackets that yields new integrable matrix PDEs and generalizes known models like self-dual Yang-Mills.

Findings

Derived new integrable nonlinear matrix PDEs.

Established a geometric criterion for PDE integrability.

Connected geometric structures to solution-generating transformations.

Abstract

Given two tensor fields of type (1,1) on a smooth n-dimensional manifold M, such that all their Fr\"olicher-Nijenhuis brackets vanish, the algebra of differential forms on M becomes a bi-differential graded algebra. As a consequence, there are partial differential equation (PDE) systems associated with it, which arise as the integrability condition of a system of linear equations and possess a binary Darboux transformation to generate exact solutions. We recover chiral models and potential forms of the self-dual Yang-Mills, as well as corresponding generalizations to higher than four dimensions, and obtain new integrable non-autonomous nonlinear matrix PDEs and corresponding systems.