Skip to main content
SpringerLink
Log in

On Some Developments of the Stokes Phenomenon

  • Conference paper
  • First Online:
Geometric Methods in Physics XXXIX (WGMP 2022)

Part of the book series: Trends in Mathematics ((TM))

Included in the following conference series:

  • 202 Accesses

Abstract

This talk is a survey of our series of works. It focuses on the Stokes phenomenon of the meromorphic linear systems of Poincaré rank 1. In particular, it reviews the explicit expression of the Stokes matrices; asymptotics and connection formula for the associated nonlinear isomonodromy equation; the conjectural characterization of the equations with trivial Stokes matrices; the relation between the Stokes phenomenon and the representation theory of quantum groups; and so on.

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 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
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. Alekseev, A.Y., Malkin, A.: Symplectic structures associated to Lie-Poisson groups. Communications in Mathematical Physics 162(1), 147–173 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balser, W.: Explicit evaluation of the Stokes’ multipliers and central connection coefficients for certain systems of linear differential equations. Mathematische Nachrichten 138(1), 131–144 (1988). https://doi.org/10.1002/mana.19881380110

    Article  MathSciNet  MATH  Google Scholar 

  3. Boalch, P.: Stokes matrices, Poisson Lie groups and Frobenius manifolds. Inventiones mathematicae 146(3), 479–506 (2001)

    Article  MATH  Google Scholar 

  4. Boalch, P.: G-bundles, isomonodromy, and quantum Weyl groups. International Mathematics Research Notices 2002(22), 1129–1166 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boalch, P.: The fifty-two icosahedral solutions to Painlevé VI (2004). https://doi.org/10.48550/ARXIV.MATH/0406281

  6. Bridgeland, T.: Riemann–Hilbert problems from Donaldson–Thomas theory. Inventiones mathematicae 216(1), 69–124 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bridgeland, T., Toledano Laredo, V.: Stability conditions and Stokes factors. Inventiones mathematicae 187(1), 61–98 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dubrovin, B.: Geometry of 2D topological field theories. Integrable systems and quantum groups pp. 120–348 (1996). https://doi.org/10.1007/bfb0094793

  9. Feigin, B., Frenkel, E., Rybnikov, L.: Opers with irregular singularity and spectra of the shift of argument subalgebra. Duke Mathematical Journal 155(2), 337–363 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ginzburg, V.L., Weinstein, A.: Lie-Poisson structure on some Poisson Lie groups. Journal of the American Mathematical Society 5(2), 445–453 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Halacheva, I., Kamnitzer, J., Rybnikov, L., Weekes, A.: Crystals and monodromy of Bethe vectors. Duke Mathematical Journal 169(12), 2337–2419 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harnad, J.: Dual isomonodromic deformations and moment maps to loop algebras. Comm. Math. Phys. 166(2), 337–365 (1994). http://projecteuclid.org/euclid.cmp/1104271613

    Article  MathSciNet  MATH  Google Scholar 

  13. Jimbo, M.: Monodromy problem and the boundary condition for some Painlevé equations. Publications of the Research Institute for Mathematical Sciences 18(3), 1137–1161 (1982)

    Article  MATH  Google Scholar 

  14. Jimbo, M., Miwa, T., Môri, Y., Sato, M.: Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Physica D: Nonlinear Phenomena 1(1), 80–158 (1980). https://doi.org/10.1016/0167-2789(80)90006-8

    Article  MathSciNet  MATH  Google Scholar 

  15. Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function. Physica D: Nonlinear Phenomena 2(2), 306–352 (1981)

    Google Scholar 

  16. Kashiwara, M.: Crystalizing the q-analogue of universal enveloping algebras. Comm. Math. Phys. 133(2), 249–260 (1990). http://projecteuclid.org/euclid.cmp/1104201397

    Article  MathSciNet  MATH  Google Scholar 

  17. Kashiwara, M.: On crystal bases of the Q-analogue of universal enveloping algebras. Duke Math. J. 63(2), 465–516 (1991). https://doi.org/10.1215/S0012-7094-91-06321-0

    Article  MathSciNet  MATH  Google Scholar 

  18. Kirillov, A.N., Berenstein, A.D.: Groups generated by involutions, Gel′ fand-Tsetlin patterns, and combinatorics of Young tableaux. Algebra i Analiz 7(1), 92–152 (1995)

    MATH  Google Scholar 

  19. Loday-Richaud, M.: Divergent series, summability and resurgence II. Simple and multiple summability, Lecture Notes in Mathematics, vol. 2154. Springer, [Cham] (2016). https://doi.org/10.1007/978-3-319-29075-1

  20. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. Journal of the American Mathematical Society 3(2), 447–498 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Malgrange, B., Ramis, J.P.: Fonctions multisommables. Ann. Inst. Fourier (Grenoble) 42(1–2), 353–368 (1992). http://www.numdam.org/item?id=AIF_1992__42_1-2_353_0

  22. Miwa, T.: Painlevé property of monodromy preserving deformation equations and the analyticity of τ functions. Publications of the Research Institute for Mathematical Sciences 17(2), 703–721 (1981)

    Article  MATH  Google Scholar 

  23. Molev, A.I.: Gelfand-Tsetlin bases for classical Lie algebras (2002). https://doi.org/10.48550/ARXIV.MATH/0211289

  24. Reshetikhin, N.Y., Faddeev, L.D.: Hamiltonian structures for integrable models of field theory. In: Fifty Years of Mathematical Physics: Selected Works of Ludwig Faddeev, pp. 323–338. World Scientific (2016)

    Google Scholar 

  25. Schlesinger, L.: Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten. Crelle’s Journal (1912)

    Google Scholar 

  26. Tang, Q., Xu, X.: Stokes phenomenon and Yangians. In preparation.

    Google Scholar 

  27. Toledano-Laredo, V.: Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection (2016). https://doi.org/10.48550/ARXIV.1601.04076

  28. Toledano-Laredo, V., Xu, X.: Stokes phenomena, Poisson-Lie groups and quantum groups (2022). https://doi.org/10.48550/ARXIV.2202.10298

  29. Xu, X.: Closure of Stokes matrices I: caterpillar points and applications (2019). https://doi.org/10.48550/ARXIV.1912.07196

  30. Xu, X.: Representations of quantum groups arising from the Stokes phenomenon and applications (2020). https://doi.org/10.48550/ARXIV.2012.15673

  31. Xu, X.: On the connection formula of a higher rank analog of Painlevé VI (2022). https://doi.org/10.48550/ARXIV.2202.08054

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences & Beijing International Center for Mathematical Research, Peking University, Beijing, China

    Qian Tang & Xiaomeng Xu

Authors
  1. Qian Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaomeng Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaomeng Xu .

Editor information

Editors and Affiliations

  1. Departamento de FĂ­sica, CINVESTAV, Ciudad de MĂ©xico, Mexico

    Piotr Kielanowski

  2. Faculty of Mathematics, University of Białystok, Białystok, Poland

    Alina Dobrogowska

  3. Departments of Mathematics, Physics, Learning & Teaching, Rutgers University, New Brunswick, NJ, USA

    Gerald A. Goldin

  4. Faculty of Mathematics, University of Białystok, Białystok, Poland

    Tomasz Goliński

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Tang, Q., Xu, X. (2023). On Some Developments of the Stokes Phenomenon. In: Kielanowski, P., Dobrogowska, A., Goldin, G.A., Goliński, T. (eds) Geometric Methods in Physics XXXIX. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-30284-8_24

Download citation

Publish with us

Policies and ethics

Search

Navigation