A note on the Lax pairs for Painlevéequations

Abstract

For the classical Painleveequations, besides the method of similarity reduction of Lax pairs for integrable partial differential equations, two ways are known for Lax pair generation. The first is based on the confluence procedure in Fuchs' linear ODE with four regular singularities isomonodromy deformation which is governed by the sixth Painleveequation. The second method treats the hypergeometric equation and confluent hypergeometric equations as the isomonodromy deformation equations for the triangular systems of ODEs, in whose non-triangular extensions give rise to the Lax pairs for the Painleveequations. The theory of integrable integral operators suggests a new way of Lax pair generation for the classical Painleveequations. This method involves a special kind of gauge transformation that is applied to linear systems which are exactly solvable in terms of the classical special functions. Some of the Lax pairs we introduce are known, others are new. The question of gauge equivalence of different Lax pairs for the Painleveequations is considered as well.