Geodesic
On connection formulas for the second Painleve transcendent. Proof of the Miles conjecture, and a quantization rule
Izvestiya. Mathematics , Tome 42 (1994) no. 3, pp. 501-560.

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The method of isomonodromy deformations is used to prove connection formulas for the second Painleve transcendent, which is exponentially decreasing on one side of a turning point and has a Kuzmak–Luke–Whitham decomposition on the other. The phase advance turns out to be equal to $\pi/2$ ($\operatorname{mod}\pi$). These connection formulas lead to the determination of the asymptotics of the eigenvalues for the Sturm–Liouville equation with a cubic nonlinearity. 
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M. V. Karasev; A. V. Pereskokov. On connection formulas for the second Painleve transcendent. Proof of the Miles conjecture, and a quantization rule. Izvestiya. Mathematics , Tome 42 (1994) no. 3, pp. 501-560. http://geodesic.mathdoc.fr/item/IM2_1994_42_3_a2/

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