Computational methods and exploration of the multivalued painlevé transcendents, with special emphasis on PIII

[摘要] 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 multivaluedPainleve transcendents, which are the solutions to the third, fifth and sixth Painleveequations, 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 existenceof solutions to the sixth Painleve equation that have pole-free sectors, known astronquee solutions.Paper 2: The method recently developed by the authors for the computation of themultivalued 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 wereidentied by McCoy, Tracy and Wu (J. Math. Phys. 18:1058-1092, 1977) and whichcontain a pole-free sector. Limiting cases, in which the solutions are singular functionsof the parameters, are also investigated and it is shown that a particular set of limitingsolutions is expressible in terms of special functions. Solutions that are single-valued,logarithmically (infinitely) branched and algebraically branched, with any number ofdistinct sheets, are encountered. The algebraically branched solutions have multiplepole-free sectors on their Riemann surfaces that are accounted for by using asymptoticformulae and Backlund transformations.

[发布机构] University of the Free State