Towards a Theory of Homotopy Structures for Differential Equations: First Definitions and Examples

We work within the framework of the variational bicomplex: we define A<inf>∞</inf>-algebra structures on horizontal and vertical cohomologies of (formally integrable) partial differential equations with the help of Merkulov s theorem. Since higher order A<inf>∞</inf>-algebra operations are related to Massey products, our observation implies the existence of invariants for differential equations that go beyond conservation laws. We also propose notions of formality for PDEs, and we present examples of formal equations. © 2024 Elsevier Inc.