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

Ben Gormley,Eugene V Ferapontov,Vladimir S Novikov

doi:10.1098/rspa.2021.0047

Abstract

We classify integrable Hamiltonian equations of the form ut=∂x(δHδu),H=∫h(u,w) dxdy, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality . 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) ew. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

Highlights

We summarize three existing approaches to integrability of equations of type (1.2), namely, the method of hydrodynamic reductions, the geometric approach based on integrable conformal geometry (Einstein–Weyl geometry) and the method of dispersionless Lax pairs
The equations for a and b are consistent modulo integrability conditions (1.9). This proves the existence of commuting flows (1.10)
It still remains a challenging problem to construct dispersive deformations of all Hamiltonian systems (1.2) obtained in this paper

Summary

Introduction

The system of integrability conditions (1.9) is involutive, and modulo natural equivalence transformations its solutions can be reduced to one of the six canonical forms. The classification of solutions will be performed modulo equivalence transformations leaving system (1.2) form-invariant (and preserving the integrability conditions). These include x = x − 2at, y = y − 2bt, h = h + au2 + buw + mu + nw + p,. A(u) = σ (u; 0, g3) eαu+β and modulo equivalence transformations (2.9) we obtain the last case of our classification This finishes the proof of theorem 1.2. The paper [12] gives a classification of integrable two-component Hamiltonian systems of the form

Ut Wt

Note that the degeneration

Setting z