Continuous limit of the difference second Painlevé equation and its asymptotic solutions
Shun SHIMOMURA
Source: J. Math. Soc. Japan Volume 64, Number 3
(2012), 733-781.
Abstract
The discrete second Painlevé equation dPII is mapped to the second Painlevé equation PII by its continuous limit, and then, as shown by Kajiwara et al., a rational solution of dPII also reduces to that of PII. In this paper, regarding dPII as a difference equation, we present a certain asymptotic solution that reduces to a triply-truncated solution of PII in this continuous limit. In a special case our solution corresponds to a rational one of dPII. Furthermore we show the existence of families of solutions having sequential limits to truncated solutions of PII.
First Page:
Show
Hide
Keywords: difference second Painlevé equation; second Painlevé equation; continuous limit; asymptotic solution
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1343133742
Digital Object Identifier: doi:10.2969/jmsj/06430733
Mathematical Reviews number (MathSciNet): MR2965426
References
P. M. Batchelder, An Introduction to Linear Difference Equations, Dover, New York, 1967.
Mathematical Reviews (MathSciNet): MR216187
A. S. Fokas, B. Grammaticos and A. Ramani, From continuous to discrete Painlevé equations, J. Math. Anal. Appl., 180 (1993), 342–360.
Mathematical Reviews (MathSciNet): MR1251864
Digital Object Identifier: doi:10.1006/jmaa.1993.1405
A. R. Its and A. A. Kapaev, Quasi-linear Stokes phenomenon for the second Painlevé transcendent, Nonlinearity, 16 (2003), 363–386.
Mathematical Reviews (MathSciNet): MR1950792
Digital Object Identifier: doi:10.1088/0951-7715/16/1/321
K. Kajiwara, K. Yamamoto and Y. Ohta, Rational solutions for the discrete Painlevé II equation, Phys. Lett. A, 232 (1997), 189–199.
Mathematical Reviews (MathSciNet): MR1469907
Digital Object Identifier: doi:10.1016/S0375-9601(97)00397-6
Y. Murata, Rational solutions of the second and the fourth Painlevé equations, Funkcial. Ekvac., 28 (1985), 1–32.
Mathematical Reviews (MathSciNet): MR803400
V. Periwal and D. Shevitz, Unitary matrix models as exactly solvable string theories, Phys. Rev. Lett., 64 (1990), 1326–1329.
A. Ramani and B. Grammaticos, Miura transforms for discrete Painlevé equations, J. Phys. A, 25 (1992), L633–L637.
Mathematical Reviews (MathSciNet): MR1167541
A. Ramani and B. Grammaticos, Discrete Painlevé equations: coalescences, limits and degeneracies, Phys. A, 228 (1996), 160–171.
Mathematical Reviews (MathSciNet): MR1399286
Digital Object Identifier: doi:10.1016/0378-4371(95)00439-4
A. Ramani, B. Grammaticos and J. Hietarinta, Discrete versions of the Painlevé equations, Phys. Rev. Lett., 67 (1991), 1829–1832.
Mathematical Reviews (MathSciNet): MR1125951
Digital Object Identifier: doi:10.1103/PhysRevLett.67.1829
J. Satsuma, K. Kajiwara, B. Grammaticos, J. Hietarinta and A. Ramani, Bilinear discrete Painlevé-II and its particular solutions, J. Phys. A, 28 (1995), 3541–3548.
Mathematical Reviews (MathSciNet): MR1344382
Digital Object Identifier: doi:10.1088/0305-4470/28/12/025
S. Shimomura, Meromorphic solutions of difference Painlevé equations, J. Phys. A, 42 (2009), 315213, 19 pp.
Mathematical Reviews (MathSciNet): MR2521315
Digital Object Identifier: doi:10.1088/1751-8113/42/31/315213
S. Shimomura, Rational solutions of difference Painlevé equations, Tokyo J. Math., 35 (2012).
Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation witha Polynomial Coefficient, North-Holland Mathematics Studies, 18, North-Holland, Amsterdam-Oxford; American Elsevier, New York, 1975.
Mathematical Reviews (MathSciNet): MR486867
Zentralblatt MATH: 0322.34006
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Pure Appl. Math., XIV, Interscience Publishers, New York, 1965.
Mathematical Reviews (MathSciNet): MR203188
Journal of the Mathematical Society of Japan
