On the Riemann--Hilbert--Birkhoff inverse monodromy problem and the Painlev\'e equations
Algebra i analiz, Tome 16 (2004) no. 1, pp. 121-162.

Voir la notice de l'article provenant de la source Math-Net.Ru

@article{AA_2004_16_1_a4,
     author = {A. A. Bolibruch and A. R. Its and A. A. Kapaev},
     title = {On the {Riemann--Hilbert--Birkhoff} inverse monodromy problem and the {Painlev\'e} equations},
     journal = {Algebra i analiz},
     pages = {121--162},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_2004_16_1_a4/}
}
TY  - JOUR
AU  - A. A. Bolibruch
AU  - A. R. Its
AU  - A. A. Kapaev
TI  - On the Riemann--Hilbert--Birkhoff inverse monodromy problem and the Painlev\'e equations
JO  - Algebra i analiz
PY  - 2004
SP  - 121
EP  - 162
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AA_2004_16_1_a4/
LA  - en
ID  - AA_2004_16_1_a4
ER  - 
%0 Journal Article
%A A. A. Bolibruch
%A A. R. Its
%A A. A. Kapaev
%T On the Riemann--Hilbert--Birkhoff inverse monodromy problem and the Painlev\'e equations
%J Algebra i analiz
%D 2004
%P 121-162
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AA_2004_16_1_a4/
%G en
%F AA_2004_16_1_a4
A. A. Bolibruch; A. R. Its; A. A. Kapaev. On the Riemann--Hilbert--Birkhoff inverse monodromy problem and the Painlev\'e equations. Algebra i analiz, Tome 16 (2004) no. 1, pp. 121-162. http://geodesic.mathdoc.fr/item/AA_2004_16_1_a4/

[1] Anosov D. V., Bolibruch A. A., The Riemann–Hilbert problem, Aspects Math., 22, Friedr. Vieweg, Braunschweig, 1994 | MR

[2] Balser W., “Meromorphic transformation to Birkhoff standard form In dimension three”, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 233–246 | MR | Zbl

[3] Balser W., Formal power series and linear systems of meromorphic ordinary differential equations, Springer-Verlag, New York, 2000 | MR | Zbl

[4] Bolibrukh A. A., “Problema Rimana–Gilberta”, Uspekhi mat. nauk, 45:2(272) (1990), 3–47 | MR | Zbl

[5] Bolibrukh A. A., “Problema Rimana–Gilberta na kompleksnoi proektivnoi pryamoi”, Mat. zametki, 46:3 (1989), 118–120 | MR | Zbl

[6] Bolibruch A. A., “On sufficient conditions for the existence of a Fuchsian equation with prescribed monodromy”, J. Dynam. Control Systems, 5 (1999), 453–472 | DOI | MR

[7] Bolibruch A. A., Inverse problem for linear differential equations with meromorphic coefficients (to appear)

[8] Beals R., Sattinger D. H., “Integrable systems and isomonodromy deformations”, Phys. D, 65 (1993), 17–47 | DOI | MR | Zbl

[9] Costin O., Correlation between pole location and asymptotic behavior for Painlevé I solutions, Preprint no. 094-97, Math. Sci. Res. Inst., Berkeley, CA, 1997 | MR

[10] Deift P. A., UPenn lecture notes on the Riemann–Hilbert method, 2000

[11] Deift P. A., Chastnoe soobschenie, (ne opublikovano)

[12] Deift P. A., Zhou X., “A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the mkdv equation”, Ann. of Math. (2), 137 (1993), 295–368 | DOI | MR | Zbl

[13] Deift P. A., Zhou X., “Asymptotics for the Painlevé II equation”, Comm. Pure Appl. Math., 48 (1995), 277–337 | DOI | MR | Zbl

[14] Flaschka H., Newell A., “Monodrom- and spectrum-preserving deformations. I”, Comm. Math. Phys., 76 (1980), 65–116 | DOI | MR | Zbl

[15] Fokas A. S., Its A. R., “The isomonodromy method and the Painlevé equations”, Important Developments in Soliton Theory, eds. A. S. Fokas, V. E. Zakharov, Springer, Berlin, 1993, 99–122 | MR | Zbl

[16] Fokas A. S., Zhou X., “On the solvability of Painlevé II and IV”, Comm. Math. Phys., 144 (1992), 601–622 | DOI | MR | Zbl

[17] Hartman P., Ordinary differential equations, John Wiley and Sons, Inc., New York etc., 1964 | MR | Zbl

[18] Its A. R., “Connection formulae for the Painlevé transcendents”, The Stokes Phenomenon and Hilbert's 16th Problem (Groningen, 1995), eds. B. L. J. Braaksma, G. K. Immink, and M. van der Put, World Sci. Publishing, River Edge, NJ, 1996, 139–165 | MR | Zbl

[19] Its A. R., Novokshenov V. Yu., The isomonodromic deformation method in the theory of Painlevé equations, Lecture Notes in Math., 1191, Springer-Verlag, Berlin–New York, 1986 | MR | Zbl

[20] Its A. P., Novokshenov V. Yu., “Ob effektivnykh dostatochnykh usloviyakh razreshimosti obratnoi zadachi teorii monodromii dlya sistem lineinykh obyknovennykh differentsialnykh uravnenii”, Funkts. anal. i ego pril., 22:3 (1988), 25–36 | MR

[21] Its A. R., Fokas A. S., Kapaev A. A., “On the asymptotic analysis of the Painlevé equations via the isomonodromy method”, Nonlinearity, 7 (1994), 1291–1325 | DOI | MR | Zbl

[22] Kitaev A. V., “Obosnovanie asimptoticheskikh formul, poluchaemykh metodom izomonodromnykh deformatsii”, Zap. nauch. semin. LOMI, 179, 1989, 101–109 | MR

[23] Joshi N., Kruskal M. D., “An asymptotic approach to the connection problem for the first and the second Painlevé equations”, Phys. Lett. A, 130 (1988), 129–137 ; A simple proof that Painlevé equations have no movable essential singularities, Preprint no. CMA-R06-90, Centre for Math. Anal., Australian Nat. Univ., 1990 | DOI | MR

[24] Jimbo M., Miwa T., Ueno K., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I”, Phys. D, 2 (1981), 306–352 | DOI | MR

[25] Jurkat W. B., Lutz D. A., Peyerimhoff A., “Birkhoff invariants and effective calculations for meromorphic linear differential equations. I, II”, Math. Anal. Appl., 53 (1976), 438–470 | DOI | MR | Zbl

[26] Kostov V. P., “Fuchsian linear systems on $CP^1$ and the Riemann–Hilbert problem”, C. R. Acad. Sci. Paris Ser. I Math., 315:2 (1992), 143–148 | MR | Zbl

[27] Malgrange B., “La classification des connexions irrégulières à une variable”, Mathematics and Physics (Paris, 1979/1982), Progr. Math., 37, eds. L. Boutet de Monvel, A. Douady, J.-L. Verdier, Birkhäuser, Boston, MA, 1983, 381–399 ; ibid. ; ibid. | MR | MR | Zbl | MR | Zbl

[28] Mason L. J., Singer M. A., Woodhouse N. M. J., Tau functions and the twistor theory of integrable systems, Preprint, Univ. of Oxford, 1999 | MR | Zbl

[29] Miwa T., “Painlevé property of monodromy preserving deformation equations and the analyticity of $\tau$-functions”, Publ. Res. Inst. Math. Sci., 17 (1981), 703–712 | DOI | MR

[30] Palmer J., “Zeros of the Jimbo, Miwa, Ueno tau function”, J. Math. Phys., 40 (1999), 6638–6681 | DOI | MR | Zbl

[31] Sibuya Y., Linear differential equations in the complex domain: problems of analytic continuation, Transl. Math. Monogr., 82, Amer. Math. Soc., Providence, RI, 1990 | MR | Zbl

[32] Varadarajan V. S., “Linear meromorphic differential equations: a modern point of view”, Bull. Amer. Math. Soc., 33 (1996), 1–42 | DOI | MR | Zbl

[33] Wasow W., Asymptotic expansions for ordinary differential equations, Robert E. Krieger Publishing Co., Huntington, NY, 1976 | MR

[34] Zhou X., “The Riemann–Hilbert problem and inverse scattering”, SIAM J. Math. Anal., 20 (1989), 966–986 | DOI | MR | Zbl

[35] Kapaev A. A., “Lax pairs for Painlevé equations”, Isomonodromic Deformations and Applications in Physics (Montréal, QC, 2000), CRM Proc. Lecture Notes, 31, Amer. Math. Soc., Providence, RI, 2002, 37–48 | MR | Zbl