The solutions of Painlevé equations define comprehensible new transcendental special functions which are applicable to non-linear differential equations, to non-linear optics, to the calculation of two-point correlations of some statistical models such as the Ising models, or the Bose-Einstein condensation problems etc.), to random matrix theory, to combinatorics, or the distribution of zeros of the Riemann zeta function, etc..

 

Painlevé II is used as an example in this talk to expose the complete integrability of Painlevé equations. The contents are listed as

(a) Backlünd transformations, tau-functions and the  Weyl group symmetry.

(b) Derect monodromy and the monodromy data.

(c) Riemann-Hilbert Problems.

(d) Inverse monodromy and the complete integrability of PII.

(e) Some special PII transcendentals: the Hasting-McLeod functions, Tracy-Widom distribution, Flaschka-Newell function, tri-tronquee solutions, etc..

 

This talk attempts to give a comprehensive mini-survey of this fascinating subject. But remind you that the contents are at least 20 to 25 years old. Newer developments make use of more general Lie groups or quantum groups.