2. Basics on Painlevé Equations and Quaternionic Description of Surfaces

Abstract

This chapter presents some basic facts of the theory of Painlevé equations and the description of surfaces and curves in Euclidean three-space in terms of 2 × 2 matrices. More details and complete proofs can be found for example in [IKSY], [ItN], [Bo2]. There are two natural ways of introducing the Painlevé equations; one could be called intrinsic, the other extrinsic. The intrinsic way presented in Section 2.1 was historically the first one. It is based on the analysis of the singularities of solutions (Painlevé property) and uses only the equations themselves. Alternatively the Painlevé equations can be introduced as equations of isomonodromic deformations of auxiliary linear systems of differential equations. This extrinsic characterization of the Painlevé equations is presented in Section 2.2. It provides us with an additional structure — the corresponding linear system, which is given in terms of 2 × 2 matrices and is called the Lax representation of the corresponding Painlevé equation. The Lax representation is important for geometric applications. A partial explanation of this fact is given in Section 2.4, where conformal immersions into Euclidean three-space are described using quaternions. Later in Chapters 3, 4, 5 we will identify the quaternionic frame equations of special surfaces and curves with the Lax representations of the Painlevé equations.