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

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. 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 be inherited by the corresponding dispersive counterpart. This procedure leads to a complete list of integrable third-order equations, some of which are apparently new.