  This is an introductory course on nonlinear integrable partial differential
and differential-difference equ\-a\-ti\-ons based on lectures given for
students of Moscow Institute of Physics and Technology and Higher School of
Economics. The typical examples of Korteweg-de Vries (KdV),
Kadomtsev-Petviashvili (KP) and Toda lattice equations are studied in detail.
We give a detailed description of the Lax representation of these equations and
their hierarchies in terms of pseudo-differential or pseudo-difference
operators and also of different classes of the solutions including famous
soliton solutions. The formulation in terms of tau-function and Hirota bilinear
differential and difference equations is also discussed. Finally, we give a
representation of tau-functions as vacuum expectation values of certain
operators composed of free fermions.
