Integrable equations in 2+1-dimensions: deformations of dispersionless limits

TL;DR

This paper classifies integrable third order equations in 2+1 dimensions by analyzing their dispersionless limits and hydrodynamic reductions, leading to a comprehensive list of known and new integrable equations.

Contribution

It introduces a novel perturbative approach using hydrodynamic reductions to classify and reconstruct dispersive deformations of integrable equations in 2+1 dimensions.

Findings

Complete list of integrable third order equations in 2+1 dimensions.

Identification of new integrable equations.

Method applicable to classify dispersionless limits and their deformations.

Abstract

We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of integrable systems in 2+1 dimensions possess infinitely many multi-phase solutions coming from the so-called hydrodynamic reductions. %Conversely, the requirement of the existence of hydrodynamic reductions proves to be an efficient classification criterion. In this paper we adopt a novel perturbative approach to the classification problem. Based on the method of hydrodynamic reductions, we first classify integrable quasilinear systems which may (potentially) occur as dispersionless limits of soliton equations in 2+1 dimensions. To reconstruct dispersive deformations, we require that all hydrodynamic reductions of the dispersionless limit are inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third order equations, some of which are apparently new.