The exotic conformal Galilei algebra and nonlinear partial differential equations

TL;DR

This paper explores the application of the exotic conformal Galilei algebra (ECGA) to construct and analyze nonlinear partial differential equations (PDEs) and systems that admit these symmetries, revealing new invariant classes and potential physical models.

Contribution

It introduces new classes of nonlinear PDEs and systems invariant under ECGA, including conditionally invariant equations and explicit examples with physical relevance.

Findings

No single second-order PDE is invariant under CGA.

Systems of PDEs can admit CGA invariance.

Wide classes of nonlinear PDEs are conditionally invariant under CGA.

Abstract

The conformal Galilei algebra (CGA) and the exotic conformal Galilei algebra (ECGA) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the CGA but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under CGA. It is further shown that there are systems of non-linear PDEs admitting ECGA with the realisation obtained very recently in [D. Martelli and Y. Tachikawa, arXiv:0903.5184v2 [hep-th] (2009)]. Moreover, wide classes of non-linear systems, invariant under two different 10-dimensional subalgebras of ECGA are explicitly constructed and an example with possible physical interpretation is presented.