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On an exact WKB approach to Ablowitz-Segur's connection problem for the second Painlevé equation

Published online by Cambridge University Press:  17 February 2009

Yoshitsugu Takei
Yoshitsugu Takei
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan; e-mail: takei@kurims.kyoto-u.ac.jp.
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Abstract

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We discuss Ablowitz-Segur's connection problem for the second Painlevé equation from the viewpoint of WKB analysis of Painlevé transcendents with a large parameter. The formula they first discovered is rederived from a suitable combination of connection formulas for the first Painlevé equation.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 111 - 119
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Ablowitz, M. J. and Segur, H., “Asymptotic solution of the Korteweg-de Vries equation”, Stud. Appl. Math. 57 (1977) 13–44.CrossRefGoogle Scholar
[2]Aoki, T., “Stokes geometry of Painlevé equations with a large parameter”, RIMS Kôkyzûroku 1088 (1999) 39–54.Google Scholar
[3]Aoki, T., Kawai, T. and Takei, Y., “WKB analysis of Painlevée transcendents with a large parameter. II”, in Structure of Solutions of Differential Equations (eds. Morimoto, M. and Kawai, T.), (World Scientific, 1996) 1–49.Google Scholar
[4]Bassom, A. P., Clarkson, P. A., Law, C. K. and McLeod, J. B., “Application of uniform asymptotics to the second Painlevé transcendent”, Arch. Rat. Mech. Anal. 143 (1998) 241–271.CrossRefGoogle Scholar
[5]Candelpergher, B., Nosmas, J. C. and Pham, F., Approche de la résurgence (Hermann, 1993).Google Scholar
[6]Clarkson, P. A. and McLeod, J. B., “A connection formula for the second Painlevé transcendent”, Arch. Rat. Mech. Anal. 103 (1988) 97–138.CrossRefGoogle Scholar
[7]Deift, P. and Zhou, X., “A steepest descent method for oscillatory Riemann-Hilbert problems”, Ann. Math. 137 (1993) 295–368.CrossRefGoogle Scholar
[8]Deift, P. and Zhou, X., “Asymptotics for the Painlevé II equation”, Comm. Pure Appl. Math. 48 (1995) 277–337.CrossRefGoogle Scholar
[9]Delabaere, E. and Pham, F., “Resurgent methods in semi-classical asymptotics”, Ann. Inst. H. Poincaré 71 (1999) 1–94.Google Scholar
[10]Joshi, N., “True solutions asymptotic to formal WKB solutions of the second Painlevé equation with large parameter”, in Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear (eds. Howls, C. J., Kawai, T. and Takei, Y.), (Kyoto Univ. Press, 2000) 223–230.Google Scholar
[11]Karasev, M. V. and Pereskokov, A. V., “On connection formulas for the second Painlevé transcendent”, Russian Acad. Sci. Izv. Math. 42 (1994) 501–560.Google Scholar
[12]Kawai, T. and Takei, Y., “WKB analysis of Painlevé transcendents with a large parameter. I”, Adv. in Math. 118 (1996) 1–33.CrossRefGoogle Scholar
[13]Kawai, T. and Takei, Y., Algebraic Analysis of Singular Perturbations (Iwanami, 1998), (In Japanese. English translation will be published by Amer. Math. Soc.).Google Scholar
[14]Kawai, T. and Takei, Y., “WKB analysis of Painlevé transcendents with a large parameter. III”, Adv. in Math. 134 (1998) 178–218.CrossRefGoogle Scholar
[15]Segur, H. and Ablowitz, M. J., “Asymptotic solutions of nonlinear evolution equations and a Painlevé transcendent”, Physica D 3(1981) 165–184.CrossRefGoogle Scholar
[16]Takei, Y., “Singular-perturbative reduction to Birkhoff normal form and instanton-type formal solutions of Hamiltonian systems”, Publ. RIMS, Kyoto Univ. 34(1998) 601–627.CrossRefGoogle Scholar
[17]Takei, Y., “An explicit description of the connection formula for the first Painlevé equation”, in Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear (eds. Howls, C. J., Kawai, T. and Takei, Y.),(Kyoto Univ. Press, 2000) 271–296.Google Scholar
[18]Takei, Y., “The space of initial conditions of Painlevé equations and WKE analysis”, RIMS Kôkyûroku 1133 (2000) 104–116, (In Japanese).Google Scholar
[19]Voros, A., “The return of the quartic oscillator. The complex WKB method”, Ann. Inst. H. Poincaré 39 (1983) 211–338.Google Scholar