Abstract
The asymptotics of the generic second Painlevé transcendent \(u(x) as |x| \to \infty , \arg x \in \left( {\tfrac{\pi }{3}k,\tfrac{\pi }{3}(k + 1)} \right), k = 0, 1, \ldots ,5\) is found and justified via the direct asymptotic analysis of the associated Riemann-Hilbert problem based on the Deift-Zhou nonlinear steepest descent method. The asymptotics is proved of the Boutroux type, i.e., it is expressed in terms of the elliptic functions. Kapaev-Novokshenov’s explicit connection formulae between the asymptotic phases in the different sectors are obtained as well.
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Its, A.R., Kapaev, A.A. (2002). The Nonlinear Steepest Descent Approach to the Asymptotics of the Second Painlevé Transcendent in the Complex Domain. In: Kashiwara, M., Miwa, T. (eds) MathPhys Odyssey 2001. Progress in Mathematical Physics, vol 23. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0087-1_10
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