Existence and uniqueness of Tronquée solutions of the fourth-order Jimbo-Miwa second Painlevé equation

Joshi, Nalini; Morrison, Tegan

doi:10.1090/S0002-9939-09-09819-0

Existence and uniqueness of Tronquée solutions of the fourth-order Jimbo-Miwa second Painlevé equation
HTML articles powered by AMS MathViewer

by Nalini Joshi and Tegan Morrison PDF

Proc. Amer. Math. Soc. 137 (2009), 2005-2014 Request permission

Abstract:

We consider the asymptotic limit as the independent variable approaches infinity, of the fourth-order second Painlevé equation obtained from a hierarchy based on the Jimbo-Miwa Lax pair. We prove that there exist two families of algebraic formal power series solutions and that there exist true solutions with these behaviours in sectors $\sigma$ of the complex plane. Given $\sigma$ we also prove that there exists a wider sector $\Sigma \supset \sigma$ in which there exists a unique solution in each family. These provide the analogue of Boutroux’s tri-tronquée solutions for the classical second Painlevé equation. Surprisingly, they also extend beyond the tri-tronquée solutions in the sense that we find penta-, hepta-, ennea-, and hendeca-tronquée solutions.

References

P. Boutroux, Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre, Ann. Sci. École Norm. Sup. (3) 30 (1913), 255–375 (French). MR 1509163, DOI 10.24033/asens.661
P. Boutroux, Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre (suite), Ann. Sci. École Norm. Sup. (3) 31 (1914), 99–159 (French). MR 1509174, DOI 10.24033/asens.672
B. Dubrovin, T. Grava, and C. Klein, On universality of critical behaviour in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tri-tronquée solution to the Painlevé-I equation (2007), arXiv:0704.0501. To appear in J. Nonlinear Sci.
Hermann Flaschka and Alan C. Newell, Monodromy- and spectrum-preserving deformations. I, Comm. Math. Phys. 76 (1980), no. 1, 65–116. MR 588248, DOI 10.1007/BF01197110
Pilar R. Gordoa, Nalini Joshi, and Andrew Pickering, On a generalized $2+1$ dispersive water wave hierarchy, Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, 327–347. MR 1855426, DOI 10.2977/prims/1145477227
Pilar R. Gordoa, Nalini Joshi, and Andrew Pickering, Second and fourth Painlevé hierarchies and Jimbo-Miwa linear problems, J. Math. Phys. 47 (2006), no. 7, 073504, 16. MR 2250304, DOI 10.1063/1.2217647
E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
A. R. Its and A. A. Kapaev, Quasi-linear Stokes phenomenon for the second Painlevé transcendent, Nonlinearity 16 (2003), no. 1, 363–386. MR 1950792, DOI 10.1088/0951-7715/16/1/321
Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), no. 2, 306–352. MR 630674, DOI 10.1016/0167-2789(81)90013-0
N. Joshi and M. Mazzocco, Existence and uniqueness of tri-tronquée solutions of the second Painlevé hierarchy, Nonlinearity 16 (2003), no. 2, 427–439. MR 1958609, DOI 10.1088/0951-7715/16/2/304
Nalini Joshi, The second Painlevé hierarchy and the stationary KdV hierarchy, Publ. Res. Inst. Math. Sci. 40 (2004), no. 3, 1039–1061. MR 2074710, DOI 10.2977/prims/1145475502
Nalini Joshi and Martin D. Kruskal, The Painlevé connection problem: an asymptotic approach. I, Stud. Appl. Math. 86 (1992), no. 4, 315–376. MR 1160153, DOI 10.1002/sapm1992864315
A. V. Kitaev, Caustics in $1+1$ integrable systems, J. Math. Phys. 35 (1994), no. 6, 2934–2954. MR 1275481, DOI 10.1063/1.530495
Rupert C. Littlewood, Hyperelliptic asymptotics of Painlevé-type equations, Nonlinearity 12 (1999), no. 6, 1629–1641. MR 1726668, DOI 10.1088/0951-7715/12/6/312
Gregory Moore, Geometry of the string equations, Comm. Math. Phys. 133 (1990), no. 2, 261–304. MR 1090426, DOI 10.1007/BF02097368
Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Dover Publications, Inc., New York, 1987. Reprint of the 1976 edition. MR 919406