On a class of integrable Hamiltonian equations in 2+1 dimensions

Abstract

We classify integrable Hamiltonian equations of the form $u_t={\partial }_x\left(\frac{\delta H}{\delta u}\right),H=\int h(u,w)\text{d}x\text{d}y,$ where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality $w={\partial }_x^{-1}{\partial }_{y}u$ . Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h). We show that the generic integrable density is expressed in terms of the Weierstrass σ-function: h(u, w) = σ(u) e ^w . Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

Keywords: Hamiltonian PDEs; commuting flows; dispersionless Lax pairs; dispersive deformations; hydrodynamic reductions.