Geodesic
Irregular singular point of the second Painlev\'e function and the nonlinear Stokes phenomenon
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 12, Tome 187 (1991), pp. 139-170

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On the base of Isomonodromy Deformation Method the Elliptical asymptotic in complex domain of argument of Boutroux type for the second Painlevé function is constructed. The equations for the modul which depends from only $\arg x$ of elliptical sine are written down. The phase of elliptical sine for any $\arg x$ except the Stokes rays is expressed in terms of Stokes multipliers of the associated linear system. The last are the first integrals of Painlevé functions. The nonlinear Stokes phenomenon for the second Painlevé equation is described. 
@article{ZNSL_1991_187_a8,
     author = {A. A. Kapaev},
     title = {Irregular singular point of the second {Painlev\'e} function and the nonlinear {Stokes} phenomenon},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {139--170},
     publisher = {mathdoc},
     volume = {187},
     year = {1991},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1991_187_a8/}
}
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A. A. Kapaev. Irregular singular point of the second Painlev\'e function and the nonlinear Stokes phenomenon. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 12, Tome 187 (1991), pp. 139-170. http://geodesic.mathdoc.fr/item/ZNSL_1991_187_a8/