Looped cotangent Virasoro algebra and non-linear integrable systems in dimension 2+1

TL;DR

This paper introduces a generalized Virasoro algebra in two spatial dimensions, studies its coadjoint representation, and derives an integrable non-linear PDE analogous to the KP equation, expanding the understanding of higher-dimensional integrable systems.

Contribution

It presents a new Lie algebra generalizing the Virasoro algebra to two dimensions and derives an associated integrable PDE, advancing the theory of multi-dimensional integrable systems.

Findings

Derived a bi-Hamiltonian system leading to an integrable PDE

Established an analogue of the KP equation in 2+1 dimensions

Analyzed the coadjoint representation of the generalized algebra

Abstract

We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev--Petviashvili (of type B) equation.