The nonlinear steepest descent approach to the singular asymptotics of the second Painlevé transcendent. (English) Zbl 1266.34142

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.


34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
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[1] Deift, P.A.; Zhou, X., A steepest descent method for oscillatory riemann – hilbert problems. asymptotics for the mkdv equation, Ann. of math., 137, 295-368, (1993) · Zbl 0771.35042
[2] Flaschka, H.; Newell, A.C., Monodromy- and spectrum-preserving deformations I, Comm. math. phys., 76, 65-116, (1980) · Zbl 0439.34005
[3] Fokas, A.; Its, A.; Kapaev, A.; Novokshenov, V., (), 553
[4] Bolibruch, A.A.; Its, A.R.; Kapaev, A.A., On the riemann – hilbert – birkhoff inverse monodromy problem and the painleve equations, Algebra anal., 16, 1, 121-162, (2004) · Zbl 1077.34089
[5] Kapaev, A.A., Global asymptotics of the second Painlevé transcendent, Phys. lett. A, 167, 356-362, (1992)
[6] Bleher, P.; Its, A., Semiclassical asymptotics of orthogonal polynomials, riemann – hilbert problem, and universality in the matrix models, Ann. of math., 150, 185-266, (1999) · Zbl 0956.42014
[7] Deift, P.A.; Zhou, X., Asymptotics for the Painlevé II equation, Comm. pure appl. math., 48, 3, 277-337, (1995) · Zbl 0869.34047
[8] Its, A.R.; Kapaev, A.A., The nonlinear steepest descent approach to the asymptotics of the second Painlevé transcendent in the complex domain, (), 273-311 · Zbl 1047.34104
[9] Bateman, H.; Erdelyi, A., Higher transcendental functions, (1953), McGraw-Hill New York
[10] Faddeev, L.D.; Takhtajan, L.A., Hamiltonian methods in the theory of solitons, (1987), Springer-Verlag Berlin, Heidelberg · Zbl 1327.39013
[11] Fokas, A.S.; Its, A.R.; Sung, A.Y., The nonlinear Schrödinger equation on the half-line, Nonlinearity, 18, 1771-1822, (2005) · Zbl 1181.37095
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