Holomorphic solutions of soliton equations

We present a holomorphic version of the inverse scattering method for soliton equations of parabolic type in two-dimensional space-time. It enables one to construct examples of solutions holomorphic in both variables and study the properties of all such solutions. We show that every local holomorphic solution of any of these equations admits an analytic continuation to a globally meromorphic function of the spatial variable. We also discuss the role of the Riemann problem in the theory of integrable systems, solubility conditions for the Cauchy problem, the property of trivial monodromy for all solutions of the auxiliary linear system, and the Painlevé property for soliton equations.