On realizations of the Witt algebra in $\mathbb{R}^3$

TL;DR

This paper classifies all inequivalent realizations of the Witt and Virasoro algebras as Lie vector fields in three-dimensional space, enabling the construction of new integrable PDEs with infinite-dimensional symmetries.

Contribution

It provides a complete classification of realizations of the Witt and Virasoro algebras in , leading to new integrable PDEs with infinite symmetries.

Findings

Classified all inequivalent realizations of Witt and Virasoro algebras in .

Constructed new classically integrable PDEs with infinite-dimensional symmetry algebras.

Identified all realizations of the direct sum of Witt algebras in .

Abstract

We obtain exhaustive classification of inequivalent realizations of the Witt and Virasoro algebras by Lie vector fields of differential operators in the space $\mathbb{R}^3$. Using this classification we describe all inequivalent realizations of the direct sum of the Witt algebras in $\mathbb{R}^3$. These results enable constructing all possible (1+1)-dimensional classically integrable equations that admit infinite dimensional symmetry algebra isomorphic to the Witt or the direct sum of Witt algebras. In this way the new classically integrable nonlinear PDE in one spatial dimension has been obtained. In addition, we construct a number of new nonlinear (1+1)-dimensional PDEs admitting infinite symmetries.