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\section*{Stokes phenomenon}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{references_and_related_entries}{References and related entries}\dotfill \pageref*{references_and_related_entries} \linebreak


\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

Matching asymptotic 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 equation|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.''.

\hypertarget{references_and_related_entries}{}\subsection*{{References and related entries}}\label{references_and_related_entries}

\begin{itemize}%
\item [[Airy function]], [[semiclassical approximation]], [[wall crossing]].

\end{itemize}
The historical work of G. Stokes is

\begin{itemize}%
\item G. G. Stokes, \emph{On the numerical calculation of a class of definite integrals and infinite series}, Trans. Camb. Phil. Soc., 9 (1847), 379-407


\item \href{http://www.phy.bris.ac.uk/people/berry_mv/research.html}{M. V. Berry}, \emph{Stokes' phenomenon; smoothing a Victorian discontinuity}, Publ. Math. de l'IH\'E{}S, 68 (1988), p. 211-221 (1988), \href{http://www.numdam.org/numdam-bin/item?id=PMIHES_1988__68__211_0}{numdam}.



\end{itemize}
Sibuya's has a textbook chapter on Stokes phenomena

\begin{itemize}%
\item Yasutaka Sibuya, \emph{Linear differential equations in the complex domain: Problems of analytic continuation}, Transl. of Math. Monographs \textbf{82}, Amer. Math. Soc. 1990 (Japanese original Kinokuniya, Tokyo, 1976, 1980)


\item C. Sabbah, \emph{Introduction to Stokes structures}, \href{http://arxiv.org/abs/0912.2762}{http://arxiv.org/abs/0912.2762}


\item Claude Sabbah, \emph{Isomonodromic deformations and Frobenius manifolds}, Springer 2007, \href{http://dx.doi.org/10.1007/978-1-84800-054-4}{doi}


\item C. Sabbah, slides of a talk at Rennes, \href{http://www.math.polytechnique.fr/cmat/sabbah/sabbah_rennes091027-pro.pdf}{pdf}; \emph{Wild geometry}, summary of Oberwolfach, \href{http://www.math.polytechnique.fr/cmat/sabbah/sabbah_oberwolfach090922.pdf}{pdf}


\item Y. Sibuya, \emph{Stokes phenomena}, Bull. Amer. Math. Soc. \textbf{83} (1977), 1075-1077 \href{http://www.ams.org/mathscinet-getitem?mr=0442337}{MR0442337} \href{http://dx.doi.org/10.1090/S0002-9904-1977-14391-7}{doi} \href{http://projecteuclid.org/euclid.bams/1183539486}{euclid}


\item T. Bridgeland, V. Toledano-Laredo, \emph{Stability conditions and Stokes factors}, \href{http://arxiv.org/abs/0801.3974}{arxiv/0801.3974}; \emph{Stokes factors and multilogarithms} \href{http://arxiv.org/abs/1006.4623}{arxiv/1006.4623}


\item [[Valerio Toledano Laredo]], \emph{Stability conditions and Stokes factors}, audio of the talk at UCSB, \href{http://doug-pc.itp.ucsb.edu/online/duality09/toledanolaredo1}{link}; slides \href{http://doug-pc.itp.ucsb.edu/online/duality09/toledanolaredo1/pdf/Toledanolaredo1_String_Duality_KITP.pdf}{pdf}


\item [[L. Katzarkov]], [[M. Kontsevich]], [[T. Pantev]], \emph{Hodge theoretic aspects of mirror symmetry}, \href{http://arxiv.org/abs/0806.0107}{arxiv/0806.0107}


\item M. Loday-Richaud, \emph{Stokes cocycle and differential Galois groups}, J. Math. Sci. \textbf{124}, No. 5, 2004, \href{http://www.springerlink.com/content/h42h3vk6w59q0820/fulltext.pdf}{pdf}


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


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


\item A. Voros, \emph{The return of the quartic oscillator. The complex WKB method}, Annales de l'I.H.P. Physique théorique 39:3 (1983) 211-338 \href{http://eudml.org/doc/76217}{eudml}


\item E Delabaere, H Dillinger, F Pham, \emph{Résurgence de Voros et périodes des courbes hyperelliptiques}, Annales de l'institut Fourier, 1993 \href{https://doi.org/10.5802/aif.1326}{doi}; \emph{Exact semiclassical expansions for one-dimensional quantum oscillators} J. Math. Physics \textbf{38} (1997) 6126 \href{https://doi.org/10.1063/1.532206}{doi}


\item Eric Delabaere, Frédéric Pham, \emph{Unfolding the quartic oscillator} Annals of Physics \textbf{261}:2 (1997) 180-218 \href{https://doi.org/10.1006/aphy.1997.5737}{doi}


\item R.B. Paris, A.D. Wood, \emph{Stokes phenomenon demystified}, Bull. Inst. Math. Appl. \textbf{31} (1995) 21-28.


\item A R Its, A A Kapaev, \emph{Quasi-linear Stokes phenomenon for the second Painlev\'e{} transcendent}, Nonlinearity \textbf{16} (2003) 363--386, \href{http://dx.doi.org/10.1088/0951-7715/16/1/321}{doi}


\item M. Hukuhara, \emph{Sur les points singuliers des \'e{}quations diff\'e{}rentielles lin\'e{}aires, II}, Jour.Fac. Sci. Hokkaido Univ. 5 (1937), 123-166; \emph{Sur les points singuliers des \'e{}quations diff\'e{}rentielles lin\'e{}aires, III} , Jour. Fac. Sci. Kyushu Univ. 2 (1942), 125-137.


\item D. G. Babbitt, V. S. Varadarajan, \emph{Local moduli for meromorphic differential equations}, Bull. Amer. Math. Soc. (N.S.) \textbf{12}, N. 1 (1985), 95-98.


\item P. Boalch, \emph{Symplectic manifolds and isomonodromic deformations}, \href{http://www.dma.ens.fr/~boalch/files/smid.pdf}{link}; \emph{Geometry and braiding of Stokes data; fission and wild character varieties}, \href{http://arxiv.org/abs/1111.6228}{arXiv:1111.6228}; P. P. Boalch, \emph{Stokes matrices, Poisson Lie groups and Frobenius manifolds}, Invent. Math. 146 (2001), no. 3, 479--506. \href{http://www.ams.org/mathscinet-getitem?mr=1869848}{MR1869848} \href{http://dx.doi.org/10.1007/s002220100170}{doi}


\item A. A. Kapaev, \emph{Quasi-linear Stokes phenomenon for the Painlev\'e{} first equation}, J. Phys. A: Math. Gen. \textbf{37}, 11149 (2004) \href{http://dx.doi.org/10.1088/0305-4470/37/46/005}{doi}


\item Kiran S. Kedlaya, \emph{Good formal structures for flat meromorphic connections, I: Surfaces}, Duke Math. J. \textbf{154}, n. 2 (2010), 343-418, \href{http://www.ams.org/mathscinet-getitem?mr=2682186}{MR2682186}, \href{http://projecteuclid.org/euclid.dmj/1281963652}{euclid}


\item Marco Gualtieri, Songhao Li, Brent Pym, \emph{The Stokes groupoids}, \href{http://arxiv.org/abs/1305.7288}{arxiv/1305.7288}



\end{itemize}
\begin{quote}%
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.


\end{quote}
Interconnection between exact WKB method, Stokes data and cluster algebras has been studied in

\begin{itemize}%
\item Kohei Iwaki, Tomoki Nakanishi, \emph{Exact WKB analysis and cluster algebras}, J. Phys. A 47 (2014) 474009 \href{http://arxiv.org/abs/1401.7094}{arxiv/1401.7094}; \emph{Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras}, \href{http://arxiv.org/abs/1409.4641}{arXiv:1409.4641}


\item Xiaomeng Xu, \emph{Stokes phenomenon and Yang-Baxter equations}, \href{https://arxiv.org/abs/1808.07654}{arxiv/1808.07654}, \href{https://doi.org/10.1007/s00220-019-03565-7}{doi}; \emph{Stokes phenomenon, dynamical r-matrices and Poisson geometry}, \href{https://doi.org/10.13097/archive-ouverte/unige:84496}{doi}; \emph{Frobenius manifolds and quantum groups} \href{https://arxiv.org/abs/1801.00123}{arxiv/1801.00123}; \emph{Stokes phenomenon, Gelfand-Zeitlin systems and relative Ginzburg-Weinstein linearization} \href{https://arxiv.org/abs/1701.08113}{arxiv/1701.08113}



\end{itemize}
Proceedings volumes:

\begin{itemize}%
\item B L J Braaksma, G K Immink, M van der Put, eds, \emph{The Stokes Phenomenon and Hilbert's 16th Problem}, World Sci. 1996


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



\end{itemize}


\end{document}