Frölicher–Nijenhuis geometry and integrable matrix PDE systems

Abstract Given two tensor fields of type (1, 1) on a smooth n -dimensional manifold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> </mml:mrow> </mml:math> , such that all their Frölicher–Nijenhuis brackets vanish, the algebra of differential forms on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> </mml:mrow> </mml:math> becomes a bi-differential graded algebra. As a consequence, there are systems of PDEs 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 equations, as well as corresponding generalizations to higher than four dimensions, and obtain new integrable non-autonomous nonlinear matrix PDEs.