Intrinsic-geometric formulation and invariant reduction of local Poisson brackets for PDEs

Develop an intrinsic-geometric framework that defines Poisson brackets for partial differential equations and, within that framework, establish a direct invariant reduction procedure for local Poisson brackets that does not rely on presymplectic structures. This is needed to extend the paper’s invariant reduction mechanism—based on the intrinsic geometry of PDEs—to Poisson brackets themselves.

Background

The paper introduces a homological mechanism for invariant reduction of symmetry-invariant conservation laws, presymplectic structures, and internal Lagrangians for PDEs, relying on the intrinsic geometry of differential equations (e.g., Vinogradov’s C-spectral sequence). While presymplectic structures and variational principles are treated within this intrinsic framework, the authors note that a comparable intrinsic description for Poisson brackets is lacking.

Because the reduction mechanism is formulated intrinsically, a direct reduction of local Poisson brackets requires an intrinsic-geometric definition of Poisson brackets compatible with the PDE setting. The absence of such a description currently prevents extending the invariant reduction directly to Poisson brackets without passing through presymplectic structures.