Journal of Computational Physics
We extend the numerical pole field solver (Fornberg and Weideman (2011) ) to enable the computation of the multivalued Painlevé transcendents, which are the solutions to the third, fifth and sixth Painlevé 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 Painlevé equation that have pole-free sectors, known as tronquée solutions.
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Fasondini, Marco; Fornberg, Bengt; and Weideman, J.A.C., "Methods for the computation of the multivalued Painleve transcendents on their Riemann surfaces" (2017). Applied Mathematics Faculty Contributions. 36.