Linearizing integral transform for the multicomponent lattice KP

We present a generalization of the direct linearization method for the Kadomtsev-Petviashvili (KP) equation and its lattice analogues such as the two-dimensional Toda equation. The generalization consists of a Riemann-Hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem. By dimensional reduction we obtain the integral transform for the Korteweg-de Vries (KdV) equation and its lattice analogues. The integral transform is well suited for the investigation of the finite-matrix generalizations of the above mentioned equations. Such finite-matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two-dimensional Ising model.