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

TL;DR

This paper develops a systematic framework for reducing geometric structures like conservation laws and variational principles in PDEs under symmetry, ensuring properties like Noether's theorem are preserved in the reduced system.

Contribution

It introduces a general method to compute reduced geometric structures for PDEs with symmetries, extending the applicability of symmetry reduction techniques.

Findings

Framework applies to various PDE models, including complex and gauge systems.

Preserves key properties like Noether's theorem in reduction.

Demonstrated through detailed examples with point and higher symmetries.

Abstract

For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $\varphi$, invariant solutions satisfy the reduced system $F = \varphi = 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = \varphi = 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems.