Bi-Hamiltonian nature of the equation $u_{tx}=u_{xy} u_y-u_{yy} u_x$

TL;DR

This paper investigates the bi-Hamiltonian structure of a specific nonlinear integrable PDE related to a generalized Virasoro algebra, revealing connections to well-known equations like KdV and Camassa-Holm.

Contribution

It introduces the bi-Hamiltonian nature of a new class of integrable PDEs associated with the looped cotangent Virasoro algebra, expanding understanding of their algebraic structure.

Findings

Identification of the bi-Hamiltonian structure of the PDE

Establishing the relation to classical integrable equations like KdV and Camassa-Holm

Highlighting the role of the looped cotangent Virasoro algebra in integrability

Abstract

We study non-linear integrable partial differential equations naturally arising as bi-Hamiltonian Euler equations related to the looped cotangent Virasoro algebra. This infinite-dimensional Lie algebra (constructed in \cite{OR}) is a generalization of the classical Virasoro algebra to the case of two space variables. Two main examples of integrable equations we obtain are quite well known. We show that the relation between these two equations is similar to that between the Korteweg-de Vries and Camassa-Holm equations.