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)

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

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

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

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

代表人物：

1. 1997, NSF Special Creativity Award

2. 1998, winner of the Pólya Prize.

3. 1998, an invited address at the International Congress of Mat

hematicians in Berlin

4. 2003, Stelson Lecture at the Georgia Institute of Technology

5. 2006, a plenary address at the International Congress of Math

ematicians in Madrid

Honors

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. 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

2. Deift Percy; Its Alexander; Krasovsky Icor, Asymptotics of Toeplitz, Hankel, and

Toeplitz plus Hankel determinants with Fisher-Hartwig singularities Annals of Mat

h. 174 (2011): 1243-1299  

3. 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   

4. 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

5. Deift P; Kriecherbauer T; McLaughlin KTR , Strong asymptotics of orthogonal

polynomials with respect to exponential weights, Commun. Math. Phys.

52(1999:1491-1552  

研究方向：Algebro-geometric solutions 、 differential operators 、

special functions, orthogonal polynomials and random matrices

Alexander R. Its

Mathematican

Indiana University-Purdue

University

代表人物

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

研究方向：Boundary Value Problems 、 Inverse Problems 、 Asymptotics

Analysis 、 Fluid Mechanics 、 Complex Analysis 、 orthogonal

polynomials and random matrices

A S. Fokas

Mathematican

University of Cambridge

代表人物

Books-Publications1. Ablowitz and A S Fokas, Introduction and Applications of Complex

Variables, Cambridge University Press, (2003).

2. Fokas, A R Its, A A Kapaev and V Yu Novokshenov, Painlevé

Transcendents: A Riemman-Hilbert Approach, AMS (2006).

3. A Unified Approach to Boundary Value Problems, CBMS-SIAM (2008).

Member of the Editorial Board1. Proceedings of the Royal Society (Series A) 2. Selecta Mathematica 3. Journal of Mathematical Physics4. Nonlinearity 5. Studies in Applied Mathematics 6. 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 Ehrenpreis-

type 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 152-

153, 189-198 (2001).

The Fokas-Lenells equation

Fokas-Lenells 方程的一些结果

• 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

3.1. The first transformation: triangular factorization

3 、 Long-time analysis

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

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