Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 087, 34 pages      arXiv:1804.05688      https://doi.org/10.3842/SIGMA.2018.087
Contribution to the Special Issue on Painlevé Equations and Applications in Memory of Andrei Kapaev

Notes on Non-Generic Isomonodromy Deformations

Davide Guzzetti
SISSA, Via Bonomea 265 - 34136 Trieste, Italy

Received April 17, 2018, in final form August 14, 2018; Published online August 21, 2018

Abstract
Some of the main results of [Cotti G., Dubrovin B., Guzzetti D., Duke Math. J., to appear, arXiv:1706.04808], concerning non-generic isomonodromy deformations of a certain linear differential system with irregular singularity and coalescing eigenvalues, are reviewed from the point of view of Pfaffian systems, making a distinction between weak and strong isomonodromic deformations. Such distinction has a counterpart in the case of Fuchsian systems, which is well known as Schlesinger and non-Schlesinger deformations, reviewed in Appendix A.

Key words: isomonodromy deformations; Stokes phenomenon; Pfaffian system; coalescing eigenvalues; Schlesinger deformations.

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References

  1. Anosov D.V., Bolibruch A.A., The Riemann-Hilbert problem, Aspects of Mathematics, Vol. 22, Friedr. Vieweg & Sohn, Braunschweig, 1994.
  2. Balser W., Jurkat W.B., Lutz D.A., Birkhoff invariants and Stokes' multipliers for meromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), 48-94.
  3. Balser W., Jurkat W.B., Lutz D.A., A general theory of invariants for meromorphic differential equations. I. Formal invariants, Funkcial. Ekvac. 22 (1979), 197-221.
  4. Balser W., Jurkat W.B., Lutz D.A., A general theory of invariants for meromorphic differential equations. II. Proper invariants, Funkcial. Ekvac. 22 (1979), 257-283.
  5. Balser W., Jurkat W.B., Lutz D.A., On the reduction of connection problems for differential equations with an irregular singular point to ones with only regular singularities. I, SIAM J. Math. Anal. 12 (1981), 691-721.
  6. Bibilo Yu., Filipuk G., Middle convolution and non-Schlesinger deformations, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 66-69.
  7. Bolibruch A.A., The fundamental matrix of a Pfaffian system of Fuchs type, Math. USSR Izv. 11 (1977), 1031–-1054.
  8. Bolibruch A.A., On isomonodromic deformations of Fuchsian systems, J. Dynam. Control Systems 3 (1997), 589-604.
  9. Bolibruch A.A., On isomonodromic confluences of Fuchsian singularities, Proc. Steklov Inst. Math. 221 (1998), 117-132.
  10. Cotti G., Coalescence phenomenon of quantum cohomology of Grassmannians and the distribution of prime numbers, arXiv:1608.06868.
  11. Cotti G., Dubrovin B., Guzzetti D., Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, Duke Math. J., to appear, arXiv:1706.04808.
  12. Cotti G., Dubrovin B., Guzzetti D., Local moduli of semisimple frobenius coalescent structures, arXiv:1712.08575.
  13. Cotti G., Dubrovin B., Guzzetti D., Helix structures in quantum cohomology of Fano varieties, in preparation.
  14. Cotti G., Guzzetti D., Analytic geometry of semisimple coalescent Frobenius structures, Random Matrices Theory Appl. 6 (2017), 1740004, 36 pages.
  15. Cotti G., Guzzetti D., Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity, Random Matrices Theory Appl., to appear.
  16. Dubrovin B., Geometry of $2$D topological field theories, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120-348, hep-th/9407018.
  17. Dubrovin B., Geometry and analytic theory of Frobenius manifolds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. (1998), Extra Vol. II, 315-326, math.AG/9807034.
  18. Dubrovin B., Painlevé transcendents in two-dimensional topological field theory, in The Painlevé Property, Editor R. Conte, CRM Ser. Math. Phys., Springer, New York, 1999, 287-412, math.AG/9803107.
  19. Dubrovin B., On almost duality for Frobenius manifolds, in Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 212, Amer. Math. Soc., Providence, RI, 2004, 75-132, math.DG/0307374.
  20. Galkin S., Golyshev V., Iritani H., Gamma classes and quantum cohomology of Fano manifolds: gamma conjectures, Duke Math. J. 165 (2016), 2005-2077, arXiv:1404.6407.
  21. Gantmacher F.R., The theory of matrices, Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998.
  22. Guzzetti D., Deformations with a resonant irregular singularity, in Proceedings of the Workshop Formal and Analytic Solutions of Differential Equations (Alcalá, September 4-8, 2017), to appear.
  23. Guzzetti D., On Stokes matrices in terms of connection coefficients, Funkcial. Ekvac. 59 (2016), 383-433, arXiv:1407.1206.
  24. Haraoka Y., Linear differential equations in the complex domain, Suugaku Shobou, Tokyo, 2015 (in Japanese).
  25. Hsieh P.-F., Sibuya Y., Note on regular perturbations of linear ordinary differential equations at irregular singular points, Funkcial. Ekvac. 8 (1966), 99-108.
  26. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. 16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  27. Jimbo M., Miwa T., Ueno K., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and $\tau$-function, Phys. D 2 (1981), 306-352.
  28. Katsnelson V., Volok D., Deformations of Fuchsian systems of linear differential equations and the Schlesinger system, Math. Phys. Anal. Geom. 9 (2006), 135-186, math.CA/0506328.
  29. Kitaev A.V., Non-Schlesinger deformations of ordinary differential equations with rational coefficients, J. Phys. A: Math. Gen. 34 (2001), 2259-2272, nlin.SI/0102019.
  30. Poberezhnyi V.A., General linear problem of the isomonodromic deformation of Fuchsian systems, Math. Notes 81 (2007), 529-542.
  31. Schäfke R., Confluence of several regular singular points into an irregular singular one, J. Dynam. Control Systems 4 (1998), 401-424.
  32. Sibuya Y., Simplification of a system of linear ordinary differential equations about a singular point, Funkcial. Ekvac. 4 (1962), 29-56.
  33. Sibuya Y., Perturbation of linear ordinary differential equations at irregular singular points, Funkcial. Ekvac. 11 (1968), 235-246.
  34. Yoshida M., Takano K., On a linear system of Pfaffian equations with regular singular points, Funkcial. Ekvac. 19 (1976), 175-189.

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