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Riemann-Hilbert 方法 / Deift-Zhou 非线性速降法及其应用 范恩贵 复旦大学数学科学学院、数学研究所

Riemann-Hilbert 方法 / Deift-Zhou 非线性速降法及其应用 范恩贵 复旦大学数学科学学院、数学研究所. 报告内容 一、 Riemann-Hilbert 问题（ RHP) 二、可积系统与 RHP 1. 初值问题求解 --- 非标准的 RH 方法 2. 初边值求解 ---Fokas 方法 3. 长时间渐近行为 ---- RH 方法 / Deift-Zhou 非线性速降法。 三、正交多项式与 RHP ----RH 方法 /Deift-Zhou 非线性速降法 四、随机矩阵与 RHP

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Riemann-Hilbert 方法 / Deift-Zhou 非线性速降法及其应用 范恩贵 复旦大学数学科学学院、数学研究所

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  1. Riemann-Hilbert方法/Deift-Zhou非线性速降法及其应用范恩贵复旦大学数学科学学院、数学研究所Riemann-Hilbert方法/Deift-Zhou非线性速降法及其应用范恩贵复旦大学数学科学学院、数学研究所

  2. 报告内容 一、Riemann-Hilbert 问题（RHP) 二、可积系统与RHP 1. 初值问题求解---非标准的RH方法 2. 初边值求解 ---Fokas方法 3. 长时间渐近行为---- RH方法/ Deift-Zhou非线性速降法。 三、正交多项式与RHP ----RH方法/Deift-Zhou非线性速降法 四、随机矩阵与RHP ---- RH方法/ Deift-Zhou非线性速降法 五、聚焦NLS初值问题解的长时间渐近分析

  3. 一、Riemann-Hilbert问题/方法（modern version of IST) • Riemann-Hilbert问题与Riemann monodromy问题有关， 1851年，首先由Riemann引入 . • 1900年，在国际数学家大会上，Hilbert将Riemann monodromy归结为现今流行的Riemann-Hilbert问题, 即Hilbert所提23个问题的第21个问题：具有给定奇点和单值群的Fuchs类线性微分方程 • 解的存在性。 • 1908年，Plemelj将Riemann-Hilbert问题转化为奇异积分方程问题 • 1989年，Bolibrukh在解决Riemann-Hilbert问题获得突破进展，给出反例

  4. 同时，相对独立地RHP本身被发展成解决一大类纯粹和应用数学的强有力的分析工具---Riemann-Hilbert方法，或者称Deift-Zhou非线性速降法同时，相对独立地RHP本身被发展成解决一大类纯粹和应用数学的强有力的分析工具---Riemann-Hilbert方法，或者称Deift-Zhou非线性速降法 • Riemann-Hilbert方法在可积系统中的最大受益者：非线性方程初值问题解的整体渐近分析 • Riemann-Hilbert方法的最近发展和应用：正交多项式，随机矩阵和统计力学的渐近问题 • 粗略地讲，Riemann-Hilbert问题就是在复平面上寻找一个在给定曲线上具有特定跳跃形式的解析函数

  5. Beals-Cofiman理论：Riemann-Hilbert Problem可解性

  6. 二、Riemann-Hilbert与可积系统 1. 初值问题求解 • RPP可作为比反散射方法更一般的方法，用于求解可积系统快速衰减的初值问题。 Yang Jianke, 郭柏灵等 • 对于2×2阶谱问题，可以证明RHP等价于GLM积分方程

  7. 2. 初边值问题求解 • 对于可积系统，反散射方法可用于求解其快速衰减初值问题，代数几何方法用于求解其周期性初值问题。 • 90年代，Fokas发展利用RHP求解初边值问题的方法，对于2*2矩阵谱问题相关的1+1维和 2+1维方程获得巨大成功。 • 2013年，Lenells将这种方法推广到3*3矩阵的DP方程 • J. Lenells, The Degasperis-Procesi equation on the half-line, Nonlinear Anal. 76 (2013),122--139. • 最近，我们进一步推广到3*3矩阵的Sasa-Stsuma方程和三波方程 • Xu Jian, Fan Engui, The unified method for the Sasa-Satsuma equation on the half-line, Proc Royal Soc A, 469(2013) • Jian Xu, Engui Fan, The three-wave equation on the half-line, Physics Letters A 378 (2014)

  8. 代表人物 A S. Fokas Mathematican University of Cambridge 研究方向： Boundary Value Problems 、Inverse Problems 、 Asymptotics Analysis、Fluid Mechanics 、Complex Analysis 、 orthogonal polynomials and random matrices

  9. Books-Publications • Ablowitz and A S Fokas, Introduction and Applications of Complex Variables, Cambridge University Press, (2003). • Fokas, A R Its, A A Kapaev and V Yu Novokshenov, Painlevé Transcendents: A Riemman-Hilbert Approach, AMS (2006). • A Unified Approach to Boundary Value Problems, CBMS-SIAM (2008). • Member of the Editorial Board • Proceedings of the Royal Society (Series A) • Selecta Mathematica • Journal of Mathematical Physics • Nonlinearity • Studies in Applied Mathematics • Journal of Nonlinear Science

  10. 代表论文 The Davey-Stewartson on the Half-Plane, Comm. Math. Phys. 289, 957-993 (2009). Integrable Nonlinear Evolution PDEs in 4+2 and 3+1 Dimensions, Phys. Rev. Lett. 96, 190201 (2006) A Generalised Dirichlet to Neumann Map for Certain Nonlinear Evolution PDEs, Comm. Pure Appl. Math. LVIII, 639-670 (2005) The long-time asymptotics of moving boundary problems using an Ehrenpreis-type representation and its Riemann-Hilbert nonlinearisation, Comm. Pure Appl. Maths 56, 517-548 (2002). Integrable Nonlinear Evolution Equations on the Half-Line, Comm. Math. Phys. 230, 1-39 (2002). Interaction of Lumps with a Line Soliton for the DSII Equation, Physica D 152-153, 189-198 (2001).

  11. 3. 初边值问题解的长时间分析： • 由于非线性PDE的复杂性和对初值的敏感性，关于PDE初（边）值问题解的长期行为研究一直是非困难问题。在PDE理论，关于PDE解的存在性和长时间行为分析,一般在Sobolev弱空间中，在小初值问题下,获得解的局部或整体存在性，解的渐进估计也是某一范数下,　很少有精确结果,　例如

  12. 70年代之后，由于反散射方法的发展，对非线性可积系统初边值问题解长期行为研究获得很大发展。充分利用可积性质，不仅对大初值, 而且解的渐进估计也可用精确的公式给出。例如

  13. 70年代末，Manakov, Shabat, Zakharov等借助反散射方法首先考虑可积发展方程的长时间性质。 • 1981年，Its首先将 stationary phase 思想用于RHP研究, 将这种RHP转化为稳态点上的简单RHP，其可由形变理论中的技巧进行求解的。 • 1993年，受经典速降法和Its工作的启发，Deift, Zhou直接考虑RHP，通过形变路径, 获取可积系统严格精确解的long-time性质。 • Deift; X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Annals of Math. 137(1993), 295-368 (74 pages)

  14. 1992年，Fokas ,Its首次建立了正交多项式与RHP的联系，为RH方法研究正交多项式提供可行性（CMP,1992)。 • 1999年，结合Fokas, Its正交多项式的RHP刻画，Deift, Bleher, McLaughlin等开创RH方法分析正交多项式、随机矩阵一致渐进性的新途径，形成了近十年来正交多项式渐近分析的全新研究方法 (CPAM, 1999)。 • 2000年代， Baik, Deift, Johansson将Riemann-Hilbert方法应用于组合学中随机排列等研究 • 2006-2007年， Deift在国际会议提出有关可积系统和随机矩阵的20多个公开问题. • 　（1）Universality for mathematical and physical systems, arxiv: 0603038v2 • 　（2）some open problems in random matrix theory and the theory of integrale systems, arXiv:0712.0849v1

  15. 代表人物： Percy A. Deift, September, 1945, Mathematician. Courant Institute，New York University Member of the U.S. National Academy of Sciences 研究方向： spectral theory, integrable systems, random matrix theory

  16. Honors 1997, NSFSpecial Creativity Award 1998, winner of the Pólya Prize. 1998, an invited address at the International Congress of Mathematicians in Berlin 2003, Stelson Lecture at the Georgia Institute of Technology 2006, a plenary address at the International Congress of Mathematicians in Madrid

  17. 专著 Direct and inverse scattering on line, AMS,1988 Orthogonal polynomials and random martries: A Riemann-Hilbert approach, AMS, 1998 Random matrix theory：Invariant ensembles and Universality, AMS, 2009

  18. 代表论文 Deift Percy; Its Alexander; Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results, Comm Pure Appl MathVol. LXVI, 1360–1438 (2013) (88 pages) Deift Percy; Its Alexander; Krasovsky Icor, Asymptotics of Toeplitz, Hankel, and Toeplitz plus Hankel determinants with Fisher-Hartwig singularities Annals of Math. 174 (2011): 1243-1299   Deift PA; Its AR; Zhou X, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics，Annals of Math. 146(1997), 149-235 (88 pages)  P. Deift; X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Annals of Math. 137(1993), 295-368   Jinho Baik, Percy Deift, Kurt JohanssonReviewed, On the distribution of the length of the longest increasing subsequence of random permutations ,J. Am. Math Soc. 12(1999): 1119-1178 Deift P; Kriecherbauer T; McLaughlin KTR , Strong asymptotics of orthogonal polynomials with respect to exponential weights, Commun. Math. Phys. 52(1999:1491-1552  

  19. 代表人物 Alexander R. Its Mathematican Indiana University-Purdue University 研究方向： Algebro-geometric solutions、differential operators、 special functions, orthogonal polynomials and random matrices

  20. Honors： • 1976 Moscow Mathematical Society Prize for Young Mathematicians • 1981 Leningrad Mathematical Society Prize • 2002 Hardy Fellow of the London Mathematical Society • 2010, invited speakers，the International Congress of Mathematicians, Hyderabad, India • presentation “Asymptotic analysis of the Toeplitz and Hankel determinants via the Riemann-Hilbert method”

  21. 代表论文 • A Riemann-Hilbert Approach to Asymptotic Problems Arising in the Theory of Random Matrix Models, and Also in the Theory of Integrable Statistical Mechanics, Annals of Math. 146, 149-235 (1997). • Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem,and universality in the matrix model, Annals of Math, 150 (1999), 185-266 • Toeplitz Matrices and Toeplitz Determinants under the Impetus of the Ising Model: Some History and Some Recent Results, Comm Pure Appl MathVol. LXVI, 1360–1438 (2013) • Higher order analogues of the Tracy-Widom distribution and the Painleve II hierarchy, Comm. Pure. Appl. Math. LXIII （2010）, 362-412 • Double Scaling Limit in the Random Matrix Model: the Riemann-Hilbert Approach, Comm Pure Appl Math， Vol. LVI, (2003) 433 – 516 • Asymptotics of the Airy-kernel determinant, Comm. Math. Phy. ， 278 (2008), 643-678. • Entanglement entropy in quantum spin chains with finite range interaction, Comm. Math. Phy., 284 (2008), 117-18. • Temperature correlation of quantum spins, Phys Rev Lett 70 (1993) : 1704-1706

  22. 我们的工作: • Xu Jian, Fan Engui, Chen Yong, Long-time asymptotic for the derivative nonlinear Schrodinger equation with step-like initial value, Math Phys Anal Geom, 16 (2013), 253-288 • Xu Jian, Fan Engui, Leading-order temporal asymptotic of the Fokas-Lenells equation without solitons. J Diff Equs, accepted • Xu Jian, Fan Engui, Long-time asymptotic for the derivative nonlinear Schrodinger equation with decaying initial value, submitted • Xiao Yu, Fan Engui, A Riemann-Hilbert approach to the Harry-Dym equation on the line, submitted

  23. 三、正交多项式与RHP • 1835年，Murphy首先给出经典正交函数定义。 • 1855年, Chebyshev受Fourier级数和逼近论影响，研究了二类Chebyshev多项式，开启了正交多项式研究大门。 • 19-20世纪是正交多项式理论和应用发展的繁荣时期，大量主要专著。 • 近20年，人们发现概率统计、随机矩阵理论中的许多模型可以用正交多项式刻画、归结为正交多项式问题。 • 正交多项式很多分支密切联系，如，算子理论、解析函数、插值理论、逼近论、数论、组合学、扩散过程、可积系统、凝聚态物理、统计物理等

  24. 代表人物 Barry Simon, April 1946 Mathematical Physicist Princeton University California Institute of Technology

  25. 研究方向：Mathematics and Theoretical Physics • Spectral theory • Functional analysis • Schrödinger operators, • Orthogonal polynomials. • Quantum mechanics,quantum field theory, • Electricand magnetic fields, • Atomic and molecular physics, • Statistical mechanics, • Brownian motion

  26. 著作: • Methods of Modern Mathematical Physics, I-IV: Academic Press, 1977 • Functional integration and quantum physics, Academic Press, 1979 • Schrodinger Operators: with Application to Quantum Mechanics and Global Geometry, Springer-Verlag, 1987 • Representation of finite and compact group, AMS, 1991 • Statistical mechanicsand lattice, Princeton University Press, 1993 • Trace ideal and applications, AMS, 2005, • Convexity：an analytic viewpoint, Cambridge University Press, 2011 • Szegö's Theorem and its Descendants: Spectral Theory for L2 Perturbations of Orthogonal Polynomials, Princeton University Press, 2010

  27. 代表论文 • Operators with singular continuous spectrum: I. General operators, Annals of Math. 141 (1995), 131-145 • Semiclassical analysis of low lying eigenvalues, II. Tunneling, Annals of Math. 120 (1984), 89-118 • The P(φ)2 quantum theory as classical statistical mechanics, Annals of Math. 101 (1975), 111-259 • Spectral analysis of multiparticle Schrödinger operators, Annals of Math. 114 (1981), 519-567 • Brownian motion and Harnack's inequality for Schrödinger operatorsCommun. Pure Appl. Math 35 (1982), 209-273 • Infrared bounds, phase transitions and continuous symmetry breaking, Commun. Math. Phys. 50 (1976), 79-85 • The Thomas-Fermi theory of atoms, molecules and solids, Advances Math. 23 (1977), 22-116 • Holonomy, the quantum adiabatic theorem and Berry's phase, Phys. Rev. Lett. 51 (1983), 2167–2170

  28. 正交多项式与RHP联系

  29. 三、随机矩阵与RHP • 随机矩阵，即取矩阵值的随机变量，随机矩阵理论之所以能够成为许多领域研究的有力工具，除了它的灵活性、可预测性，最为重要的是随机矩阵在简化复杂模型的同时，保留了物理模型中最本质的特征。 • 1928年，Wishart最早引入随机矩阵，用于大量样本的统计分布。该领域的突破进展始于50年代美国物理学家Wigner（63年诺贝尔奖的得主）发现随机矩阵可以用来模拟某些重原子的谱分布。随机矩阵理论来自物理和数学大量问题。如 • 在统计学中，经典的极限理论在分析大批量和高维数据中存在严重不足 • 生物科学中，一个DNA序列有几十亿链 • 金融研究中，不同股票的数量可能达到几十万个 • 无线电通讯中，用户可能几百万个 • 这些领域大数据的处理挑战了经典的统计学，因此，从概率论发展到随机矩阵理论是非常自然的。随机矩阵已经广泛应用于：统计物理、信息科学等。

  30. 代表人物 Pavel Bleher, Mathematican Indiana University-Purdue University statistical physics, quantum integrable systems, theory of random polynomials, and random matrix models

  31. 代表论文 • Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1999), no. 1, 185–266. • Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142 (2000), no. 2, 351–395. • Double scaling limit in the random matrix model: the Riemann-Hilbert approach. Comm. Pure Appl. Math. 56 (2003), no. 4, 433–516. • Exact solution of the six-vertex model with domain wall boundary conditions: antiferroelectric phase. Comm. Pure Appl. Math. 63 (2010), no. 6, 779–829. • Random matrix model with external source and a constrained vector equilibrium problem. Comm. Pure Appl. Math. 64 (2011), no. 1, 116–160. • Orthogonal polynomials in the normal matrix model with a cubic potential . Adv. Math. 230 (2012), 1272–1321 • Large n limit of Gaussian random matrices with external source I, Comm. Math. Phys. 252 (2004), no. 1-3, 43–76.

  32. 代表人物 M. Adler, New York University, Ph.D. Mathematician Brandeis University，Professor 研究方向： Matrix integrals、Probability、 Random matrix theory 、Completely integrable systems

  33. 代表作 M Adler, Hermitian, symmetric and symplectic random ensembles: PDE's for the distribution of the spectrum, Annals of Math, 153(2001), 149--189, M Adler, The spectrum of coupled random matrices, Annals of Math, 149(1999), 921--976 M Adler, The complex geometry of the Kowalewski-Painleve� analysis. Invent. Math. 97 (1989), 3—51 M Adler, The Toda lattice, Dynkin diagrams, singularities and abelian varieties. Invent. Math. 103 (1991), 223—278 I M Adler, ntegrals over classical groups, random permutations, Toda and Toeplitz lattices. Comm. Pure Appl. Math., 54(2000), 153--205, M Adler, Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials. Duke Math. J., 80(1995), 863-911 M Adler, Moment matrices and multi-component KP, with applications to random matrix theory, Comm. Math. Phys. 2009

  34. 随机矩阵与正交多项式的联系

  35. 我们的工作: • Xu Jian, Fan Engui, Random matrices with external source and multiple orythogonal polynomials, Submitted • Xu Jian, Fan Engui, Large n-limit for random matrices with external source with three distinct eigenvalues, submitted • Huang Lin, Fan Engui, T-R McLaughlin, Asymptotic formulas of statistics of eigenvalue in micro-interval for Hermitian random matrices via Riemann-Hilbert approach, preprint

  36. Long-time asymptotic for the Schrodinger equation with decaying initial value Engui FANSchool of Mathematics, Fudan University

  37. ① 振荡RHP ② 分解 ③ ① ② ④ ⑥ ⑤ 局域化 ② ② 零延拓、压缩 ③ ① ③ ① ④ ⑥ ④ ⑤ 拉回 标准RHP ① · 转化 Weber方程： ②

  38. 1. Lax pairs and eigenfunctions

  39. 2. Analytic and symmetries

  40. 3. Riemann-Hilbert problem

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