Geodesic
Global asymptotic formulae for the~fourth Painleve transcendent
Sbornik. Mathematics, Tome 188 (1997) no. 12, pp. 1739-1760

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This paper is devoted to the study of the asymptotic and analytic properties of the fourth Painleve transcendent as the absolute value of the independent variable approaches infinity. The problem is solved using the WKB method, Whitham averaging, and monodromy preserving deformations. The corresponding modulation equation is deduced and the asymptotic distribution of the zeros of the fourth transcendent is calculated. The dominant term of the expansion for the solution of Painleve's fourth equation is written down in the form of an elliptic function with parameters satisfying the above-mentioned modulation equation. 
@article{SM_1997_188_12_a1,
     author = {V. L. Vereshchagin},
     title = {Global asymptotic formulae for the~fourth {Painleve} transcendent},
     journal = {Sbornik. Mathematics},
     pages = {1739--1760},
     publisher = {mathdoc},
     volume = {188},
     number = {12},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_12_a1/}
}
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V. L. Vereshchagin. Global asymptotic formulae for the~fourth Painleve transcendent. Sbornik. Mathematics, Tome 188 (1997) no. 12, pp. 1739-1760. http://geodesic.mathdoc.fr/item/SM_1997_188_12_a1/