Transcript 申请者简介

Leading-order temporal asymptotics of
the Fokas-Lenells equation
----solitonless sector
Engui FAN
School of Mathematical Sciences,
Fudan University, PR China
Joint work with Jian Xu
近年来开展的工作

连续和离散方程族的代数几何解

发展方程初/初边值问题求解与长时间渐近分析

正交多项式与随机矩阵
 可积方程族的代数几何解 （二个博士生）
1.Algebro-geometric solutions for Degasperis-Procesi hierarchy, SIAM J Math
Anal, 45(2013), 1216–1266 (51 pages)
2.Algebro-geometric solutions for the Hunter-Saxton hierarchy， Z Angew
Math Phys , 2013, 1.1007 (35 pages）
3.Algebro-geometric solutions and their reductions for the Fokas-Lenells
hierarchy， J Nonl Math Phys, 20(2013), 355-393 (39 pages)
4.
Algebro-geometric initial value problem for Gerdjikov-Ivanov hierarchy
and quasi-periodic solutions, J Math Phys, 54(2013), 073515 (31 page)
5. An unfied construction of algebro-geometric solutions for the
Relativistic Lotka-Volterra hierarchy，submitted
6. Algebro-geometric solutions for the modified Camassa-Holm
Hierarchy, submitted
7.
Reality problems for the algebro-geometric solutions of Fokas-Lenell
hierarchy, Submitted
8. Algebro-geometric solutions for the Ruijsenaars-Toda Hierarchy
看在上帝的份上, 千万别放下工作! 这是你最好的药物。 ----达朗贝尔

可积发展方程初/初边值问题求解和长期行为 （二个博士生）
1. Long-time asymptotic for the derivative nonlinear Schrodinger equation with step-like initial
value， Math Phys Anal Geom, 2013 (38 pages)
2.
The Sasa-Satsuma equation on the half-line, Proc. Royal Soc. A, 2013 (26 page)
3.
Long-time asymptotic for the derivative nonlinear Schrödinger equation with decaying initial
value， J. Math Phys, accepted, arXiv:1209.4245
4. The derivative nonlinear Schrodinger equation on the interval， Acta Math Sin, Accepted ,
arXiv:1205.1559
5.
The unified method for the three-wave equation on the half-line, Phys Lett A, Accepted,
arXiv:1304.4339
6.
Leading-order temporal asymptotics of the Fokas-Lenells Equation without solitons, J Diff
Equ, Submitted arXiv: 1308.0755
 正交多项式与随机矩阵 （二个博士生）
关心问题：随机矩阵中的可积结构，大维数随机矩阵的渐近分析、系综、
全局、特征分布等, 获初步结果。
Riemann-Hilbert Problem
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
Honors
1. 1997, NSF Special Creativity Award
2. 1998, winner of the Pólya Prize.
3. 1998, an invited address at the International Congress of
Mathematicians in Berlin
4. 2003, Stelson Lecture at the Georgia Institute of Technology
5. 2006, a plenary address at the International Congress of
Mathematicians in Madrid
专著
1.Direct and inverse scattering on line, AMS,1988
2. Orthogonal polynomials and random martries: A
Riemann-Hilbert approach, AMS, 1998
3.Random matrix theory：Invariant ensembles and
Universality, AMS, 2009
代表论文
1.
2.
3.
4.
5.
P. Deift; X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert
problems. Asymptotics for the MKdV equation, Annals of Math. 137(1993), 295368
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
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
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
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”
代表论文
• Entanglement Entropy in Quantum Spin Chains with Finite Range
Interaction, Comm. Math. Phys. 284 (2008) : 117-185
• Integrable Fredholm operators and dual isomonodromic deformations
Comm. Math. Phys. 226 (2002) 497-530
• 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, 1997
• Temperature correlation of quantum spins, Phys Rev Lett 70 (1993) :
1704-1706
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
Books-Publications
1.
2.
3.
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
1.
2.
3.
4.
5.
6.
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
代表论文
1. The Davey-Stewartson on the Half-Plane, Comm. Math. Phys. 289, 957-993
(2009).
2.Integrable Nonlinear Evolution PDEs in 4+2 and 3+1 Dimensions, Phys. Rev.
Lett. 96, 190201 (2006)
3.A Generalised Dirichlet to Neumann Map for Certain Nonlinear Evolution PDEs,
Comm. Pure Appl. Math. LVIII, 639-670 (2005)
4.The long-time asymptotics of moving boundary problems using an Ehrenpreistype representation and its Riemann-Hilbert nonlinearisation, Comm. Pure Appl.
Maths 56, 517-548 (2002).
5.Integrable Nonlinear Evolution Equations on the Half-Line, Comm. Math. Phys.
230, 1-39 (2002).
6.Interaction of Lumps with a Line Soliton for the DSII Equation, Physica D 152153, 189-198 (2001).
Fokas-Lenells 方程的一些结果
The Fokas-Lenells equation
• Fokas (1995), bi-Hamiltonian method, Phys D
Lenells, (2009), pulse propagation in optical fibers, Stud Appl Math
 Lenells, (2009) , Soliton solution via RHP, Nonlinearity
 Lenells, Fokas, (2009) , Initial-boundary problem, Inverse Problem
 Matsuno, (20012) , Bright and dark soliton solutions, J Phys A
 He, Xu, (2012), Rogue waves, arXiv:1209.5540
 Zhao, Fan, (2013), Algebro-geometric solutions, J Nonl Math Phys
 Zhao, Fan, (2013), Real problem for algebro-geometric solutions , Submitted
 Xu, Fan, (2013) , Long-time asymptotic, Submitted
1、Lax pair and spectral function
We can use spectral analysis to express the solution of the Cauchy problem
of (2.9) in terms of the solutions of an appropriate Riemann-Hilbert problem.
2、Reimann-Hilbert problem
3、Long-time analysis
3.1. The first transformation: triangular factorization
3.2. The second transformation: along steepest descent line
3.1. The third transformation: From global to local
3.1. The fourth transformation: model RHP
H. F. Weber, Math. Ann. , 1 (1869) pp. 1–36
Thank You