On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

TL;DR

This paper develops a geometric framework for Clairin's conditional symmetries in higher-order PDEs, introducing new methods for symmetry analysis and solution derivation, with applications to physical equations.

Contribution

It provides a geometric formulation of Clairin's theory, removes previous assumptions, and introduces PDE Lie systems for analyzing and solving higher-order PDEs.

Findings

New geometric methods for conditional symmetry analysis

Derivation of solutions using PDE Lie systems

Applications to nonlinear wave and minimal surface equations

Abstract

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are removed. As a consequence, new insights into other types of conditional symmetries arise. We then apply the so-called PDE Lie systems to the derivation and analysis of Lie algebras of conditional symmetries. In particular, we develop a method for obtaining solutions of a higher-order system of PDEs via the solutions and geometric properties of a PDE Lie system, whose form gives a Lie algebra of conditional symmetries of the Clairin type. Our methods are illustrated with physically relevant examples such as nonlinear wave equations, the Gauss--Codazzi equations for minimal soliton surfaces, and generalised Liouville equations.