Abstract
The initial value problem associated with the second Painlevé Transcendent is linearized via a matrix, discontinuous, homogeneous Riemann-Hilbert (RH) problem defined on a complicated contour (six rays intersecting at the origin). This problem is mapped through a series of transformations to three different simple Riemann-Hilbert problems, each of which can be solved via a system of two Fredholm integral equations. The connection of these results with the inverse scattering transform in one and two dimensions is also pointed out.
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Fokas, A.S., Ablowitz, M.J. On the initial value problem of the second Painlevé Transcendent. Commun.Math. Phys. 91, 381–403 (1983). https://doi.org/10.1007/BF01208781
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DOI: https://doi.org/10.1007/BF01208781