Skip to main content
SpringerLink
Log in

Deformations with a Resonant Irregular Singularity

  • Conference paper
  • First Online:
Book cover Formal and Analytic Solutions of Diff. Equations (FASdiff 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 256))

Abstract

I review topics of my talk in Alcalá, inspired by the paper [1]. An isomonodromic system with irregular singularity at \(z=\infty \) (and Fuchsian at \(z=0\)) is considered, such that \(z=\infty \) becomes resonant for some values of the deformation parameters. Namely, the eigenvalues of the leading matrix at \(z=\infty \) coalesce along a locus in the space of deformation parameters. I give a complete extension of the isomonodromy deformation theory in this case.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Cotti, G., Dubrovin, B.A., Guzzetti, D.: Isomonodromy deformations at an irregular singularity with coalescing eigenvalues (2017). arXiv:1706.04808

  2. Dubrovin, B.: Geometry of 2D topological field theories. Lect. Notes Math. 1620, 120–348 (1996)

    Article  Google Scholar 

  3. Dubrovin, B.: Geometry and analytic theory of Frobenius manifolds. In: Proceedings of ICM98, vol. 2, pp. 315–326 (1998)

    Google Scholar 

  4. Dubrovin, B.: Painlevé trascendents in two-dimensional topological field theory. In: Conte, R. (ed.) The Painlevé Property, One Century later. Springer, Berlin (1999)

    Chapter  Google Scholar 

  5. Guzzetti, D.: Inverse problem and monodromy data for three-dimensional Frobenius manifolds. Math. Phys. Anal. Geom. 4, 245–291 (2001)

    Article  MathSciNet  Google Scholar 

  6. Cotti, G.: Coalescence phenomenon of quantum cohomology of grassmannians and the distribution of prime numbers (2016). arXiv:1608.06868

  7. Cotti, G., Dubrovin, B.A., Guzzetti, D.: Local moduli of semisimple Frobenius coalescent structures (2017). arXiv:1712.08575

  8. Mazzocco, M.: Painlevé sixth equation as isomonodromic deformations equation of an irregular system. CRM Proc. Lect. Notes, Am. Math. Soc. 32, 219–238 (2002)

    Google Scholar 

  9. Fokas, A., Its, A., Kapaev, A., Novokshenov, V.: Painlevé Transcendents: The Riemann-Hilbert Approach. AMS (2006)

    Google Scholar 

  10. Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformations of linear ordinary differential equations with rational coefficients (I). Physics D2, 306–352 (1981)

    MathSciNet  MATH  Google Scholar 

  11. Guzzetti, D.: Stokes matrices and monodromy of the quantum cohomology of projective spaces. Commun. Math. Phys. 207(2), 341–383 (1999)

    Article  MathSciNet  Google Scholar 

  12. Cotti, G., Dubrovin, B.A., Guzzetti, D.: Helix structures in quantum cohomology of Fano manifolds. Ready to appear

    Google Scholar 

  13. Cotti, G., Guzzetti, D.: Results on the extension of isomonodromy deformations to the case of a resonant irregular singularity. In: Proceedings for the Workshop at CRM. Pisa 13–17 February 2017. Random Matrices: Theory and Applications (2018). https://doi.org/10.1142/S2010326318400038

  14. Miwa, T.: Painlevé Property of monodromy preserving deformation equations and the analyticity of \(\tau \) functions. Publ. RIMS, Kyoto Univ. 17, 703–721 (1981)

    Article  Google Scholar 

  15. Hsieh, P.-F., Sibuya, Y.: Note on regular perturbation of linear ordinary differential equations at irregular singular points. Funkcial. Ekvac 8, 99–108 (1966)

    MathSciNet  MATH  Google Scholar 

  16. Balser, W., Jurkat, W.B., Lutz, D.A.: Birkhoff invariants and stokes’ multipliers for meromorphic linear differential equations. J. Math. Anal. Appl. 71, 48–94 (1979)

    Article  MathSciNet  Google Scholar 

  17. Balser, W., Jurkat, W.B., Lutz, D.A.: A general theory of invariants for meromorphic differential equations; Part I. Form. Invariants. Funkcialaj Evacioj 22, 197–221 (1979)

    MATH  Google Scholar 

  18. Cotti, G., Guzzetti, D.: Analytic geometry of semisimple coalescent structures. Random Matrices Theory Appl. 6(4), 1740004, 36, 53 (2017)

    Google Scholar 

  19. Dubrovin, B., Mazzocco, M.: Monodromy of certain Painlevé transcendents and reflection groups. Invent. Math. 141, 55–147 (2000)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SISSA, Via Bonomea 265, Trieste, Italy

    Davide Guzzetti

Authors
  1. Davide Guzzetti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davide Guzzetti .

Editor information

Editors and Affiliations

  1. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Galina Filipuk

  2. Departamento de Física y Matemáticas, University of Alcalá, Alcalá de Henares, Spain

    Alberto Lastra

  3. Faculty of Mathematics and Natural Sciences, College of Science, Cardinal Stefan Wyszynski University, Warsaw, Poland

    Sławomir Michalik

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Guzzetti, D. (2018). Deformations with a Resonant Irregular Singularity. In: Filipuk, G., Lastra, A., Michalik, S. (eds) Formal and Analytic Solutions of Diff. Equations . FASdiff 2017. Springer Proceedings in Mathematics & Statistics, vol 256. Springer, Cham. https://doi.org/10.1007/978-3-319-99148-1_14

Download citation

Publish with us

Policies and ethics

Search

Navigation