# Painleve Equations and Discrete Dynamics (16w5027)

Arriving in Banff, Alberta Sunday, October 2 and departing Friday October 7, 2016

## Organizers

Nalini Joshi (University of Sydney)

Vladimir Dragovic (The University of Texas at Dallas)

## Objectives

The main objective of the event is to bring together researchers from several centers and schools, who work in topics related to Painlevé equations and discrete dynamics. We expect a very exciting meeting where leading experts will report on the newest development and which will intensify the exchange of experience, methods and ideas, and encourage collaboration among diverse groups in community.

Apart from purely theoretical importance in the classification of Ordinary Differential Equations of the second order, the Painlevé equations turn out to be related to various areas of modern mathematics and physics, to mention apart from integrable dynamical systems, the Frobenius manifolds (Dubrovin 1996), quantum cohomology (Manin 2005), self-dual Einstein metrics (Hitchin1995, Babich et al. 1998).

Nonlinear integrable systems and their solutions forms the core of modern mathematical physics and special function theory, arising in applications that are widespread and growing rapidly. Among such systems are the Painlevé equations. One of the areas of intense recent activity in which the Painlevé equations arise is the study of energy levels of heavy particles in atomic physics, which is connected intimately to the spectral properties of random matrices (Mehta 2004). This serendipitous connection has given rise to some of the most active and fruitful developments of mathematics in recent times. Applications of random matrix theory range widely from particle physics to the distribution of airline boarding times and the zeros of the Riemann zeta function on the critical line (Forrester et al. 1996).

Another notable instance where the Painlevé equations arise as universal limits is the study of neighbourhoods of a catastrophe (where the vector field vanishes) in Hamiltonian perturbations of hyperbolic equations. Dubrovin et al. 2009 have shown that integrable equations arise as universal limits in this context. In the elliptic region, the limiting solution is given by a specific solution, called the tritronquée solution, of the first Painlevé equation. There is keen interest in knowing its analytic properties over the entirety of its domain of asymptotic validity.

The most widely used method of finding information about the solutions of the Painlevé equations is the Riemann–Hilbert method (Fokas et al. 2006), related to the inverse scattering method of solving soliton equations, combined with the steepest descent method (Deift et al. 1993). It has been successful in finding the solutions of many connection problems, i.e., the problem of relating the asymptotic behaviour of a solution near a fixed singular point (or along one direction of approach to such a point) to the possible behaviours near another fixed point. It has also been used to prove the existence of bounded solutions (Claeys et al. 2007) along the real line under certain conditions, for example, when a vanishing lemma applies for the corresponding jump conditions.

However, Riemann–Hilbert theory has not been developed for discrete equations in the same way it has been for differential equations. The study of solutions of linear q-difference equations has been a focus of a significant body of research (Birkhoff 1913, Borodin 2004, Carmichael 1913, Ramis et al. 2006, Sauloy 2000). The study of linear elliptic difference (or ell-difference) equations has just begun (Krichever 2004). Neither of these fields yet furnishes information about analytic or asymptotic properties of the coefficients, which are solutions of associated nonlinear problems.

The workshop will extend our knowledge of the solutions of nonlinear systems, by unifying geometry, asymptotics and integrability theory and conformal field theory (Gamayun et al. 2012, Iorgov et al. 2014, Iorgov et al. 2013).

The broad impact of the topic is reflected in the fact that periodically there are international conferences devoted solely to the Painlevé equations. For example, in 2010: "Numerical solution ofthe Painlevé equations", ICMS, Edinburgh, UK; in 2011: "Painlevé equations and related topics", Euler Mathematical Institute, Saint-Petersburg, Russia; in 2013: "Recent progress in the domain of Painlevé equations: algebraic, asymptotic and topological aspects", Strasbourg, France. The topic is intensively developing in the last years, and interrelation with the discrete dynamical systems is one of the most important branches. According to MATHSCINET database, there are 4745 papers mentioning Painlevé, while 2784 of that number appear since 2000, and 994 since 2010.All this shows how timely the organization of such a conference is at the moment.

For the current meeting, we plan to invite some of the leading experts from all around the world, to mention Deift (USA), Dubrovin (Italy), Hitchin (UK), Mulase (USA), Sakai (Japan), Its (USA), Bobenko (Berlin), Suris (Berlin), Previato (USA), Korotkin (Canada), Ramani (France), Grammaticos (France), Shramchenko (Canada), Bertola (Canada), Harnad (Canada), Lisovyy (France), Mazzocco (UK), Grava (Italy), Radnovic (Australia), Guzzetti (Italy), Olver (Australia), Kajiwara (Japan), Noumi (Japan).

As an important part of the broad impact of the conference, let us stress that among the invited speakers, there are several female scientists: Previato, Grava, Mazzocco, Radnovic, Shramchenko. The meeting will have a strong Canadian component as well, since Canadian mathematicians are very active in this field.

References:

[1] Babich, M., Korotkin, D., Self-dual SU(2)-invariant Einstein metrics and modular dependence of theta functions. Lett. Math. Phys. 46, no. 4, 323337 (1998)

[2] G. D. Birkhoff, Equivalent singular points of ordinary linear differential equations. Math. Ann. 74(1913), no. 1, 134-139.

[3] A. Borodin, Isomonodromy transformations of linear systems of difference equations. Ann. of Math. (2) 160 (2004), no. 3, 1141-1182.

[4] R. D. Carmichael, On the Theory of Linear Difference Equations. Amer. J. Math. 35 (1913), no. 2, 163-182.

[5] T. Claeys, M. Vanlessen, The existence of a real pole-free solution of the fourth order analogue of the Painlevé I equation. Nonlinearity 20 (2007), no. 5, 1163-1184.

[6] P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. (2) 137 (1993), no. 2, 295-368.

[7] B. Dubrovin, Geometry of 2D topological field theories, Integrable Systems and Quantum Groups, Montecatini Terme (1993), Lecture Notes in Math. 1620, Springer, Berlin (1996)

[8] B. Dubrovin, T. Grava, C. Klein, On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation, J. Nonlinear Sci. 19 (2009), no. 1, 57-94.

[9] A. S. Fokas, A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, Painlevé transcendents. The Riemann-Hilbert approach. Mathematical Surveys and Monographs, 128. American Mathematical Society, Providence, RI, 2006.

[10] P. J. Forrester, A. M. Odlyzko, Gaussian unitary ensemble eigenvalues and Riemann zeta function zeros: a nonlinear equation for a new statistic. Phys. Rev. E (3) 54 (1996), no. 5, R4493–R4495.

[11] O. Gamayun, N. Iorgov, O. Lisovyy, Conformal field theory of Painlevé VI, Journal of High Energy Physics 38 (2012).

[12] N. Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations, J. Diff. Geom. 3, 52-134 (1995)

[13] N. Iorgov, O. Lisovyy, A. Shchechkin, Y. Tykhyy, Painlevé functions and conformal blocks. Constr. Approx. 39 (2014), no. 1, 255-272

[14] N. Iorgov, O. Lisovyy, Yu. Tykhyy, Painlevé VI connection problem and monodromy of $c=1$ conformal blocks. J. High Energy Phys. 2013, no. 12, 029

[15] I. M. Krichever, Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem. Uspekhi Mat. Nauk 59 (2004), no. 6(360), 111-150

[16] Yu. Manin, Rational curves, elliptic curves, and the Painlevé equation, Surveys in Modern Mathematics, Cambridge University Press (2005)

[17] M. L. Mehta, Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142.Elsevier/Academic Press, Amsterdam, 2004.

[18] J.-P. Ramis, J. Sauloy, C. Zhang, Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences. C. R. Math. Acad. Sci. Paris 342 (2006), no. 7, 515-518.

[19] J. Sauloy, Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie. Ann. Inst. Fourier (Grenoble) 50 (2000), no. 4, 1021-1071.

Apart from purely theoretical importance in the classification of Ordinary Differential Equations of the second order, the Painlevé equations turn out to be related to various areas of modern mathematics and physics, to mention apart from integrable dynamical systems, the Frobenius manifolds (Dubrovin 1996), quantum cohomology (Manin 2005), self-dual Einstein metrics (Hitchin1995, Babich et al. 1998).

Nonlinear integrable systems and their solutions forms the core of modern mathematical physics and special function theory, arising in applications that are widespread and growing rapidly. Among such systems are the Painlevé equations. One of the areas of intense recent activity in which the Painlevé equations arise is the study of energy levels of heavy particles in atomic physics, which is connected intimately to the spectral properties of random matrices (Mehta 2004). This serendipitous connection has given rise to some of the most active and fruitful developments of mathematics in recent times. Applications of random matrix theory range widely from particle physics to the distribution of airline boarding times and the zeros of the Riemann zeta function on the critical line (Forrester et al. 1996).

Another notable instance where the Painlevé equations arise as universal limits is the study of neighbourhoods of a catastrophe (where the vector field vanishes) in Hamiltonian perturbations of hyperbolic equations. Dubrovin et al. 2009 have shown that integrable equations arise as universal limits in this context. In the elliptic region, the limiting solution is given by a specific solution, called the tritronquée solution, of the first Painlevé equation. There is keen interest in knowing its analytic properties over the entirety of its domain of asymptotic validity.

The most widely used method of finding information about the solutions of the Painlevé equations is the Riemann–Hilbert method (Fokas et al. 2006), related to the inverse scattering method of solving soliton equations, combined with the steepest descent method (Deift et al. 1993). It has been successful in finding the solutions of many connection problems, i.e., the problem of relating the asymptotic behaviour of a solution near a fixed singular point (or along one direction of approach to such a point) to the possible behaviours near another fixed point. It has also been used to prove the existence of bounded solutions (Claeys et al. 2007) along the real line under certain conditions, for example, when a vanishing lemma applies for the corresponding jump conditions.

However, Riemann–Hilbert theory has not been developed for discrete equations in the same way it has been for differential equations. The study of solutions of linear q-difference equations has been a focus of a significant body of research (Birkhoff 1913, Borodin 2004, Carmichael 1913, Ramis et al. 2006, Sauloy 2000). The study of linear elliptic difference (or ell-difference) equations has just begun (Krichever 2004). Neither of these fields yet furnishes information about analytic or asymptotic properties of the coefficients, which are solutions of associated nonlinear problems.

The workshop will extend our knowledge of the solutions of nonlinear systems, by unifying geometry, asymptotics and integrability theory and conformal field theory (Gamayun et al. 2012, Iorgov et al. 2014, Iorgov et al. 2013).

The broad impact of the topic is reflected in the fact that periodically there are international conferences devoted solely to the Painlevé equations. For example, in 2010: "Numerical solution ofthe Painlevé equations", ICMS, Edinburgh, UK; in 2011: "Painlevé equations and related topics", Euler Mathematical Institute, Saint-Petersburg, Russia; in 2013: "Recent progress in the domain of Painlevé equations: algebraic, asymptotic and topological aspects", Strasbourg, France. The topic is intensively developing in the last years, and interrelation with the discrete dynamical systems is one of the most important branches. According to MATHSCINET database, there are 4745 papers mentioning Painlevé, while 2784 of that number appear since 2000, and 994 since 2010.All this shows how timely the organization of such a conference is at the moment.

For the current meeting, we plan to invite some of the leading experts from all around the world, to mention Deift (USA), Dubrovin (Italy), Hitchin (UK), Mulase (USA), Sakai (Japan), Its (USA), Bobenko (Berlin), Suris (Berlin), Previato (USA), Korotkin (Canada), Ramani (France), Grammaticos (France), Shramchenko (Canada), Bertola (Canada), Harnad (Canada), Lisovyy (France), Mazzocco (UK), Grava (Italy), Radnovic (Australia), Guzzetti (Italy), Olver (Australia), Kajiwara (Japan), Noumi (Japan).

As an important part of the broad impact of the conference, let us stress that among the invited speakers, there are several female scientists: Previato, Grava, Mazzocco, Radnovic, Shramchenko. The meeting will have a strong Canadian component as well, since Canadian mathematicians are very active in this field.

References:

[1] Babich, M., Korotkin, D., Self-dual SU(2)-invariant Einstein metrics and modular dependence of theta functions. Lett. Math. Phys. 46, no. 4, 323337 (1998)

[2] G. D. Birkhoff, Equivalent singular points of ordinary linear differential equations. Math. Ann. 74(1913), no. 1, 134-139.

[3] A. Borodin, Isomonodromy transformations of linear systems of difference equations. Ann. of Math. (2) 160 (2004), no. 3, 1141-1182.

[4] R. D. Carmichael, On the Theory of Linear Difference Equations. Amer. J. Math. 35 (1913), no. 2, 163-182.

[5] T. Claeys, M. Vanlessen, The existence of a real pole-free solution of the fourth order analogue of the Painlevé I equation. Nonlinearity 20 (2007), no. 5, 1163-1184.

[6] P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation. Ann. of Math. (2) 137 (1993), no. 2, 295-368.

[7] B. Dubrovin, Geometry of 2D topological field theories, Integrable Systems and Quantum Groups, Montecatini Terme (1993), Lecture Notes in Math. 1620, Springer, Berlin (1996)

[8] B. Dubrovin, T. Grava, C. Klein, On universality of critical behavior in the focusing nonlinear Schrödinger equation, elliptic umbilic catastrophe and the tritronquée solution to the Painlevé-I equation, J. Nonlinear Sci. 19 (2009), no. 1, 57-94.

[9] A. S. Fokas, A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, Painlevé transcendents. The Riemann-Hilbert approach. Mathematical Surveys and Monographs, 128. American Mathematical Society, Providence, RI, 2006.

[10] P. J. Forrester, A. M. Odlyzko, Gaussian unitary ensemble eigenvalues and Riemann zeta function zeros: a nonlinear equation for a new statistic. Phys. Rev. E (3) 54 (1996), no. 5, R4493–R4495.

[11] O. Gamayun, N. Iorgov, O. Lisovyy, Conformal field theory of Painlevé VI, Journal of High Energy Physics 38 (2012).

[12] N. Hitchin, Twistor spaces, Einstein metrics and isomonodromic deformations, J. Diff. Geom. 3, 52-134 (1995)

[13] N. Iorgov, O. Lisovyy, A. Shchechkin, Y. Tykhyy, Painlevé functions and conformal blocks. Constr. Approx. 39 (2014), no. 1, 255-272

[14] N. Iorgov, O. Lisovyy, Yu. Tykhyy, Painlevé VI connection problem and monodromy of $c=1$ conformal blocks. J. High Energy Phys. 2013, no. 12, 029

[15] I. M. Krichever, Analytic theory of difference equations with rational and elliptic coefficients and the Riemann-Hilbert problem. Uspekhi Mat. Nauk 59 (2004), no. 6(360), 111-150

[16] Yu. Manin, Rational curves, elliptic curves, and the Painlevé equation, Surveys in Modern Mathematics, Cambridge University Press (2005)

[17] M. L. Mehta, Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142.Elsevier/Academic Press, Amsterdam, 2004.

[18] J.-P. Ramis, J. Sauloy, C. Zhang, Développement asymptotique et sommabilité des solutions des équations linéaires aux q-différences. C. R. Math. Acad. Sci. Paris 342 (2006), no. 7, 515-518.

[19] J. Sauloy, Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie. Ann. Inst. Fourier (Grenoble) 50 (2000), no. 4, 1021-1071.