ABSTRACT
This chapter introduces a wider class of solutions of the Painlevé equations and related equations: the multiphase similarity solutions of integrable evolution equations. These are supposed to be to the standard similarity solutions–what multisoliton or finite-gap periodic solutions are to solitons and cnoidal waves. The study of multiphase similarity solutions originated in attempts to understand the work on the scaling limit of the n-point correlation functions of the Ising model. The chapter shows that many-phase Painlevé functions are somehow analogous to the Theta-functions which afford formulas for finite-gap solutions of integrable systems, and that correspondingly the concept of similarity should have a natural generalization to many phases.
