Towards the classification of integrable differential–difference equations in 2 + 1 dimensions

We address the problem of classification of integrable differential–difference equations in 2 + 1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalization to dispersive equations as proposed in Ferapontov et al (2009 J. Phys. A: Math. Theor. 42 035211, 2009 J. Phys. A: Math. Theor. 42 345205). We obtain a number of classification results of scalar integrable equations including that of the intermediate long wave and Toda type.