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.