Mathematical Surveys and Monographs 128. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3651-X/hbk). xii, 553 p. (2006).
In the Introduction (which contains an extensive survey of the known results), the authors recall the definition of the six Painlevé differential equations of second order and of the Painlevé trascendents, i.e., of their general solutions. Then, they consider Fuchsian systems of linear differential equations, and they examine in detail the case when there are three singularities and the size of the system is $2$. They explain the link between this case and (via the method of isomonodromy deformations) the theory of Painlevé VI (PVI) and PII equations. They recall the direct and inverse monodromy problem. They give a complete asymptotic description of the purely imaginary solutions of the PII equation followed by some applications to physics including the link between the three-dimensional wave collapse in the nonlinear Schrödinger equation and PII, the link between bound states of the elliptic sine-Gordon equation and PIII, the one between two-dimensional quantum gravity and PI, and also between Painlevé equations and random matrices and random permutations. Further, in the book most of these topics are considered in more detail. In Part 1, the authors expose the analytic theory of linear systems of ODEs including the analytic presentation, the monodromy and (at irregular singular points) the Stokes phenomenon; also, the inverse monodromy problem and the Riemann-Hilbert factorization, the isomonodromy deformations and Schlesinger’s equations, the isomonodromy method and the Bäcklund transformations. For each of these topics, they develop in detail the theory in the case of $2\times 2$-systems and describe the link with some special functions and with the Painlevé equations. Part 2 is devoted to a detailed study of the asymptotic behaviour of the solutions of PII. This includes the Boutroux elliptic ansatz, WKB-analysis for the $\Psi$-function, the Deift-Zhou method, the model Baker-Akhiezer Riemann-Hilbert problem, asymptotics on the canonical six-rays, the study of the quasi-linear Stokes phenomenon and the Hastings-McLeod solution. Part 3 deals with the asymptotics of PIII. It includes a study of its algebraic and rational solutions, the sine-Gordon reduction of PIII (SG-PIII) and the direct and inverse monodromy problems for it, the canonical four-rays, real-valued and singular solutions and asymptotics in the complex plane of the SG-PIII (including WKB approximation to the $\Psi$-function). An appendix treats the Birkhoff-Grothendieck theorem with a parameter.