Invariant Reduction for Partial Differential Equations. II: The General Mechanism

TL;DR

The paper develops a general invariant reduction framework for PDEs with local symmetries, casting symmetry-invariant conservation laws, presymplectic structures, and variational principles in a cohomological setting via Vinogradov's $\mathcal{C}$-spectral sequence. It provides a Noether-type theorem for PDEs satisfied by symmetry-invariant solutions and proposes algorithmic procedures to reduce conservation laws and presymplectic structures for evolution systems, including $(1+1)$-D cases. Through detailed examples, the authors demonstrate how invariant reductions yield reduced invariants, potentials, and even Liouville-type integrability in the reduced systems, while noting limitations and the potential need for multi-step reductions. The work offers a global, intrinsic framework for deriving and analyzing symmetry-induced reductions, with implications for qualitative analysis and the discovery of integrable reductions in nonlinear PDEs.

Abstract

A mechanism of reduction of symmetry-invariant conservation laws, presymplectic structures, and variational principles of partial differential equations (PDEs) is proposed. The mechanism applies for an arbitrary PDE system that admits a local (point, contact, or higher) symmetry, and relates symmetry-invariant conservation laws, as well as presymplectic structures, variational principles, etc., to their analogs for systems that describe the corresponding invariant solutions. A version of Noether's theorem for the PDE system satisfied by symmetry-invariant solutions is presented. Several detailed examples, including cases of point and higher symmetry invariance, are considered.