Extensions of the symmetry algebra and Lax representations for the two-dimensional Euler equation

TL;DR

This paper extends the symmetry algebra of the 2D Euler equation in vorticity form, constructs new Lax representations, and generalizes these results using Lie--Rinehart algebras, introducing a family of representations with spectral parameters.

Contribution

It introduces twisted symmetry algebra extensions and a generalized family of Lax representations for the 2D Euler equation, incorporating functional parameters and spectral parameters.

Findings

New Lax representations for the 2D Euler equation

Family of representations depending on functional parameters

Inclusion of a non-removable spectral parameter

Abstract

We find the twisted extensions of the symmetry algebra of the 2D Euler equation in the vorticity form and use them to construct new Lax representation for this equation. Then we generalize this result by considering the transformation Lie--Rinehart algebras generated by finite-dimensional subalgebras of the symmetry algebra and derive a family of Lax representations for the Euler equation. The family depends on functional parameters and contains a non-removable spectral parameter.