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

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

2008-02-13

arxiv.org/abs/0802.1818v1

Physics

Mathematical Physics

Scientific paper

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.

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