The nonlinear steepest descent approach to the singular asymptotics of the second Painlevé transcendent
The authors fill the methodological gap in the implementation of the Deift-Zhou nonlinear steepest descent asymptotic analysis of the Riemann-Hilbert (RH) problems associated with the Painlevé equations. Namely, in the case of the spectral curves with the double branch points, the conventional version of the Deift-Zhou method works well under particular restrictions imposed on the set of the jump matrices. For the second Painlevé equation on the real line, \(u_{xx}=xu+2u^3\), these restrictions correspond to the regular asymptotic solutions as \(x\to+\infty\) or \(x\to-\infty\). The authors extend the nonlinear steepest descent approach to the RH problems corresponding to the solutions of the Painlevé II equation with real poles accumulating at infinity. Technically, the Deift-Zhou method is based on a construction of a set of parametrices (local approximate solutions) to the initial RH problem and the following formulation of the small jump RH problem for a correction function. It occurs that, if the jump matrices of the original RH problem satisfy certain restrictions, then the local parametrices match well enough, the initial RH problem is solvable, and the asymptotics of the corresponding Painlevé function can be extracted from the parametrix at infinity. In the opposite case, some of the parametrices do not match well, and the jumps in the RH problem for the correction function are not small. The authors observe that, in the Painlevé II equation case, the ``bad'' jump matrices factorize in a particular way and allow one to replace the RH problem for the piece-wise holomorphic correction function by an RH problem for the function singular at the node points of the jump graph. The latter RH problem is solvable using a ``dressing'' procedure borrowed from the theory of integrable systems.
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