nLab Stokes phenomenon




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



Matching asymptotic series expansions of certain functions like Airy functions and more generally of solutions to the wave equations (say in short wave approximation or solutions to non-Fuchsian meromorphic ODEs) comes with sharp changes (disconintuities) at certain phase angles (Stokes lines) described by certain factors, Stokes matrices.

G. G. Stokes has discovered these phenomena – now also known more generally as wall crossing phenomena – in the study of geometric optics, more specifically the study of Airy function which is an example of a solution to a linear meromorphic ordinary differential equation which has an irregular singular point, hence the equation is not Fuchsian. The Stokes phenomenon does not happen to Fuchsian equations. Their formal meromorphic solutions are automatically convergent. Exactly the presence of the irregular singularities makes the appearance of formal solutions with zero radius of convergence. Now look around the origin. One can try to prove that there are asymptotic expressions at best in some regions of argument. There are jumps at certain slopes. In fact there is Stokes sheaf and the first nonabelian cohomology of the Stokes sheaf measures the obstruction for a formal meromomorphic expansion to be build up of sectorial true meromorphic expansions.

Stokes factors (multipliers, matrices) appear also in the study of stability in the geometry of moduli spaces of sheaves. According to Toledano Laredo “Joyce’s wall-crossing formulae for invariants counting semistable objects in an abelian category A may be understood as Stokes phenomena for a connection on the Riemann sphere taking value in the Ringel-Hall Lie algebra of A.”.



The historical work of G. Stokes is

  • G. G. Stokes, On the numerical calculation of a class of definite integrals and infinite series, Trans. Camb. Phil. Soc., 9 (1847), 379-407

  • M. V. Berry, Stokes’ phenomenon; smoothing a Victorian discontinuity, Publ. Math. de l’IHÉS, 68 (1988), p. 211-221 (1988), numdam.

Sibuya’s has a textbook chapter on Stokes phenomena

  • Yasutaka Sibuya, Linear differential equations in the complex domain: Problems of analytic continuation, Transl. of Math. Monographs 82, Amer. Math. Soc. 1990 (Japanese original Kinokuniya, Tokyo, 1976, 1980)

  • C. Sabbah, Introduction to Stokes structures,

  • Claude Sabbah, Isomonodromic deformations and Frobenius manifolds, Springer 2007, doi

  • C. Sabbah, slides of a talk at Rennes, pdf; Wild geometry, summary of Oberwolfach, pdf

  • Y. Sibuya, Stokes phenomena, Bull. Amer. Math. Soc. 83 (1977), 1075-1077 MR0442337 doi euclid

  • T. Bridgeland, V. Toledano-Laredo, Stability conditions and Stokes factors, arxiv/0801.3974; Stokes factors and multilogarithms arxiv/1006.4623

  • Valerio Toledano Laredo, Stability conditions and Stokes factors, audio of the talk at UCSB, link; slides pdf

  • L. Katzarkov, M. Kontsevich, T. Pantev, Hodge theoretic aspects of mirror symmetry, arxiv/0806.0107

  • M. Loday-Richaud, Stokes cocycle and differential Galois groups, J. Math. Sci. 124, No. 5, 2004, pdf

  • T. Aoki, T. Kawai, Y. Takei, On the exact steepest descent method: A new method for the description of Stokes curves, J. Math. Phys. 42 (2001), 3691-3713.

  • R. Balian, G. Parisi, A. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev. Lett. 41 (1978), 1141-1144.

  • A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’I.H.P. Physique théorique 39:3 (1983) 211-338 eudml

  • E Delabaere, H Dillinger, F Pham, Résurgence de Voros et périodes des courbes hyperelliptiques, Annales de l’institut Fourier, 1993 doi; Exact semiclassical expansions for one-dimensional quantum oscillators J. Math. Physics 38 (1997) 6126 doi

  • Eric Delabaere, Frédéric Pham, Unfolding the quartic oscillator Annals of Physics 261:2 (1997) 180-218 doi

  • R.B. Paris, A.D. Wood, Stokes phenomenon demystified, Bull. Inst. Math. Appl. 31 (1995) 21-28.

  • A R Its, A A Kapaev, Quasi-linear Stokes phenomenon for the second Painlevé transcendent, Nonlinearity 16 (2003) 363–386, doi

  • M. Hukuhara, Sur les points singuliers des équations différentielles linéaires, II, Jour.Fac. Sci. Hokkaido Univ. 5 (1937), 123-166; Sur les points singuliers des équations différentielles linéaires, III , Jour. Fac. Sci. Kyushu Univ. 2 (1942), 125-137.

  • D. G. Babbitt, V. S. Varadarajan, Local moduli for meromorphic differential equations, Bull. Amer. Math. Soc. (N.S.) 12, N. 1 (1985), 95-98.

  • P. Boalch, Symplectic manifolds and isomonodromic deformations, link; Geometry and braiding of Stokes data; fission and wild character varieties, arXiv:1111.6228; P. P. Boalch, Stokes matrices, Poisson Lie groups and Frobenius manifolds, Invent. Math. 146 (2001), no. 3, 479–506. MR1869848 doi

  • A. A. Kapaev, Quasi-linear Stokes phenomenon for the Painlevé first equation, J. Phys. A: Math. Gen. 37, 11149 (2004) doi

  • Kiran S. Kedlaya, Good formal structures for flat meromorphic connections, I: Surfaces, Duke Math. J. 154, n. 2 (2010), 343-418, MR2682186, euclid

  • Marco Gualtieri, Songhao Li, Brent Pym, The Stokes groupoids, arxiv/1305.7288

We construct and describe a family of groupoids over complex curves which serve as the universal domains of definition for solutions to linear ordinary differential equations with singularities. As a consequence, we obtain a direct, functorial method for resumming formal solutions to such equations.

Interconnection between exact WKB method, Stokes data and cluster algebras has been studied in

  • Kohei Iwaki, Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) 474009 arxiv/1401.7094; Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv:1409.4641

  • Xiaomeng Xu, Stokes phenomenon and Yang-Baxter equations, arxiv/1808.07654, doi; Stokes phenomenon, dynamical r-matrices and Poisson geometry, doi; Frobenius manifolds and quantum groups arxiv/1801.00123; Stokes phenomenon, Gelfand-Zeitlin systems and relative Ginzburg-Weinstein linearization arxiv/1701.08113

  • Kenzo Ishikawa, Kenji Nishiwaki, Kin-ya Oda, New effect in wave-packet scatterings of quantum fields: Saddle points, Lefschetz thimbles, and Stokes phenomenon (arXiv:2102.12032)

We find a new contribution in wave-packet scatterings, which has been overlooked in the standard formulation of S-matrix. As a concrete example, we consider a two-to-two scattering of light scalars ϕ by another intermediate heavy scalar Φ, in the Gaussian wave-packet formalism: ϕϕ→Φ→ϕϕ. This contribution can be interpreted as an “in-time-boundary effect” of Φ for the corresponding Φ→ϕϕ decay, proposed by Ishikawa et al., with a newly found modification that would cure the previously observed ultraviolet divergence. We show that such an effect can be understood as a Stokes phenomenon in an integral over complex energy plane: The number of relevant saddle points and Lefschetz thimbles (steepest descent paths) discretely changes depending on the configurations of initial and final states in the scattering.

Proceedings volumes:

  • B L J Braaksma, G K Immink, M van der Put, eds, The Stokes Phenomenon and Hilbert’s 16th Problem, World Sci. 1996

  • B L J Braaksma, G K Immink, M van der Put, J Top, eds, Differential equation and the Stokes phenomenon, World Sci. 2002

In string theory

On non-perturbative effects and Stokes phenomena in topological string amplitudes:

  • Murad Alim, Arpan Saha, Jörg Teschner, Iván Tulli, Mathematical structures of non-perturbative topological string theory: from GW to DT invariants (arXiv:2109.06878)

Last revised on September 16, 2021 at 03:27:50. See the history of this page for a list of all contributions to it.