From Osher Doctorow
Peter A. Clarkson (U. Kent at Canterbury, U.K.) in his online "The
Painleve equations - nonlinear special functions,"
www.ima.edu/talks/workshops/7-22-8-2-2002/clarkson/SF02L.pdf, 87 pages,
gives the most comprehensive and clear presentation of the solutions,
nature, and physical applications of the Painleve equations that I have
seen. Most of the Painleve Equations (there are 6 types) involve
(df/dx)(dg/dx) expressions with f = g. He lists 21 types of physics
application, including GR, QG (quantum gravity) and QFT, plasma
physics, superconductivity, Bose-Einstein condensation, nonlinear
optics and fibre optics, polyelectrolytes and electrolysis, resonant
oscillations in shallow water, statistical mechanics, nonlinear
evolution equation asymptotics, TQFT, etc.
He also intensively compares the Riccati and Painleve equations and
makes use of the former to solve the latter for important scenarios.
Osher Doctorow
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