A computational exploration of the McCoy–Tracy–Wu solutions of the third Painlevé equation

The method recently developed by the authors for the computation of the multivalued Painlevé transcendents on their Riemann surfaces (Fasondini et al., 2017) is used to explore families of solutions to the third Painlevé equation that were identified by McCoy et al. (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 Bäcklund transformations.