Computational methods and exploration of the multivalued painlevé transcendents, with special emphasis on PIII
Abstract
Paper 1: We extend the numerical pole field solver (B. Fornberg and J.A.C. Weideman,
J. Comput. Phys. 230:5957-5973, 2011) to enable the computation of the multivalued
Painleve transcendents, which are the solutions to the third, fifth and sixth Painleve
equations, on their Riemann surfaces. We display, for the first time, solutions to these equations on multiple Riemann sheets. We also provide numerical evidence for the existence
of solutions to the sixth Painleve equation that have pole-free sectors, known as
tronquee solutions.
Paper 2: The method recently developed by the authors for the computation of the
multivalued Painleve transcendents on their Riemann surfaces (J. Comput. Phys. 344:36-50, 2017) is used to explore families of solutions to the third Painleve equation that were
identied by McCoy, Tracy and Wu (J. Math. Phys. 18:1058-1092, 1977) and which
contain a pole-free sector. Limiting cases, in which the solutions are singular functions
of the parameters, are also investigated and it is shown that a particular set of limiting
solutions is expressible in terms of special functions. Solutions that are single-valued,
logarithmically (infinitely) branched and algebraically branched, with any number of
distinct sheets, are encountered. The algebraically branched solutions have multiple
pole-free sectors on their Riemann surfaces that are accounted for by using asymptotic
formulae and Backlund transformations.