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

TL;DR

This paper classifies a specific class of integrable Hamiltonian equations in 2+1 dimensions, revealing that most integrable densities are expressed via elliptic functions and exploring their associated structures.

Contribution

It introduces a classification of integrable Hamiltonian equations with a specific non-local Hamiltonian operator using hydrodynamic reductions and elliptic functions.

Findings

Most integrable densities are expressed in terms of Weierstrass elliptic functions.

The paper derives integrability conditions as an involutive PDE system.

It discusses dispersionless Lax pairs, commuting flows, and dispersive deformations.

Abstract

We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. 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 elliptic functions. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.