Invariant reduction for PDEs. IV: Symmetries that rescale geometric structures

Abstract: For a system of differential equations and a symmetry, the framework of invariant reduction systematically computes how invariant geometric structures are inherited by the subsystem governing invariant solutions. In this setting, the reduction of structures invariant under a two-dimensional Lie algebra requires its commutativity. We extend this mechanism to the case where geometric structures are invariant under one symmetry $X$ and are rescaled, by a factor of $-a$, by another symmetry $X_s$ satisfying $[X_s, X] = aX$. As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

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Geometry of differential equations seminar