Abstract
The isomonodromy deformation method is applied to the scaling limits in the linear N × N matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves that describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated with the n-large asymptotics in the sequence of the biorthogonal polynomials with cubic potentials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. L. Ince, Ordinary Differential Equations [in Russian], DNTVU, Kharkov (1939); English transl., Dover, New York (1956).
A. A. Kapaev, CRM Proc. Lect. Notes, 32, 157–179 (2002); nlin.SI/0105002 (2001).
M. Jimbo, T. Miwa, and K. Ueno, Phys. D, 2, 306–352 (1981).
H. Flaschka and A. C. Newell, Comm. Math. Phys., 76, 65–116 (1980).
A. R. Its and V. Yu. Novokshenov, The Isomonodromy Deformation Method in the Theory of Painlevé Equations (Lect. Notes Math., Vol. 1191), Springer, Berlin (1986).
M. L. Mehta, Random Matrices, Acad. Press, New York (1967).
P. Nevai, J. Approx. Theory, 48, 3–167 (1986).
A. R. Its, A. V. Kitaev, and A. S. Fokas, Russ. Math. Surveys, 45, 155–157 (1990); Comm. Math. Phys., 142, 313–344 (1991); 147, 395–430 (1992).
N. M. Ercolani and K. T.-R. McLaughlin, Phys. D, 152–153, 232–268 (2001).
M. Bertola, B. Eynard, and J. Harnad, Comm. Math. Phys., 229, 73–120 (2002); nlin.SI/0108049 (2001).
P. Zinn-Justin, Nucl. Phys. B, 634, 417–432 (2002); math-ph/0202045 (2002).
M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations [in Russian], Nauka, Moscow (1983); English transl., Springer, Berlin (1993).
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Interscience, New York (1965).
M. V. Fedoryuk, Differential Equations, 22, 675–680 (1986).
M. Jimbo and T. Miwa, Phys. D, 2, 407–448 (1981); 4, 26–46 (1981).
R. Garnier, Ann. Sci. Ecole Norm. Super., 34, 239–253 (1917); H. Flaschka and A. C. Newell, Phys. D, 3, 203–221 (1981).
A. A. Kapaev, J. Math. Sci., New York, 83, No. 1, 38–61 (1997); “Scaling limits in the fourth Painlevé transcendent,” PDMI Preprint 15/1996, PDMI, St. Petersburg (1996); “Discriminant set for the scaling limits in the third Painlevé transcendent,” PDMI Preprint 21/1997, PDMI, St. Petersburg (1997).
A. A. Kapaev, J. Phys. A, 36, 4629–4640 (2003); nlin.SI/0207036 (2002).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kapaev, A.A. Monodromy Approach to the Scaling Limits in Isomonodromy Systems. Theoretical and Mathematical Physics 137, 1691–1702 (2003). https://doi.org/10.1023/B:TAMP.0000007917.73394.24
Issue Date:
DOI: https://doi.org/10.1023/B:TAMP.0000007917.73394.24