Riemann-Hilbert Methods and Frontiers of Computer Mathematics

Riemann-Hilbert Methods and Frontiers of Computer Mathematics

Content

Forum on Integrable Systems, Mathematical Physics and Related Problems

(2017/12/09-2017/12/11, Bei jing)

2017/12/09 (Saturday): Liaoning Hotel

2017/12/10 (Sunday) Morning Session                 Place: N210   

08: 30-08: 45 Registration   Place: N210

Chair: Changzheng Qu

08: 45-09: 00 Opening: Shikun Wang, Ke Wu

Chair: Youjin Zhang

09: 00-09: 40 Zhangju Liu, Double Principal Bundles

09: 40-10: 20 Changzheng Qu, Nonlocal symmetries and conservation laws to Camassa-Holm type equations

10: 20-10: 40                   Tea Break                                        

Chair: Zhangju Liu

10: 40-11: 20 Youjin Zhang  Bi-Jacobian structures and their classifications

11: 20-12: 00 Qingping Liu  Supersymmetric Sawada-Kotera equation

12: 00-13: 00 Lunch  (4th floor, Wuke Hotel)

2017/12/10 (Sunday) Afternoon Session                   Place: N210   

Chair: Qingping Liu

14: 30-15: 10 Lixin Tian, Wave propagation in SIR epidemic models

15: 10-15: 50 Siqi Liu, The Loop equation for integrable hierarchies and their applications

15: 50-16: 10                    Tea Break                                     

Chair: Lixin Tian 

16: 10-16: 50 Xingbiao Hu, A numerical study of the 3-periodic wave solutions to KdV-type and coupled KdV-type equations

16: 50-17: 30 Jinsong He, Double degeneration of the multi-breathers and the optical rogue waves

18: 00-      Dinner (3rd floor, Wuke Hotel)

2017/12/11 (Monday) Morning Session                  Place: N202                          

Chair: Xingbiao Hu

08: 30-09: 10 Engui Fan, Hankel determinants for a perturbed Laguerre weight and Painleve-V equation

09: 10-09: 50 Yong Chen, Nonlinear Local waves

09: 50-10: 10                  Group Photo, Tea Break                                            

Chair: Changzheng Qu

10: 10-12: 00  Discussion

12: 00-13: 00  Lunch  (4th floor, Wuke Hotel)

 

 

 

 

Abstract:

2017/12/10 Morning      Place: N210

Talk1: Double Principal Bundles

Prof. Zhangju Liu (Beijing University)

2017/12/10 9: 00-9: 40,  N210

Abstract: In this talk, we give a brief introduction  to the notion of double principal bundles , for which the frame bundle of a double vector bundle, double Lie groups and double homogeneous spaces are basic examples.  It is shown that a double vector bundle can be realized as the associated bundle of its frame bundle. Also dual structures, gauge transformations and connections  are investigated. This is a joint work with H.L. Lang and Y.P. Li.

 

Talk2: Nonlocal symmetries and conservation laws to Camassa-Holm type equations

Prof. Changzheng Qu (Ningbo University)

2017/12/10 9: 40-10: 20,  N210

Abstract: Camassa-Holm type equations have attracted much attention in recent years because of their nontrivial properties. In this talk, we investigate the nonlocal symmetries and conserved laws to Camassa-Holm equations. It is shown that certain Camassa-Holm type equations admit nonlocal symmetries. Conservation laws and invariant properties to Camassa-Holm type equations will also be studied.

 

Talk3: Bi-Jacobian structures and their classifications

Prof. Youjin Zhang  (Tsinghua University)

2017/12/10 10: 40-11: 20,  N210

Abstract: 双雅可比结构是非线性可积系统理论中出现的双哈密顿结构的一类非局部推广,它们在有关非线性偏微分方程的可积性及其代数和几何结构的研究中具有重要应用。我们将介绍双雅可比结构的概念、关于它们的分类问题及已有结果。

 

Talk4: Supersymmetric Sawada-Kotera equation

Prof. Qingping Liu  (China University of Mining and Technology)

2017/12/10 11: 20-12: 00,  N210

Abstract: Sawada-Kotera equation is one of the integrable equations and has been studied extensively. In this talk, we introduce a supersymmetric version for the Sawada-Kotera equation and demenstrate that this system is integrable. Also, a Darboux transformation is constructed for the sSK system.

 

2017/12/10 Afternoon     Place: N210

 

Talk5: Wave propagation in SIR epidemic models

Prof. Lixin Tian (Nanjing Normal University)

2017/12/10 14: 30-15: 10,  N210

Abstract: In this study, two types SIR epidemic models are proposed. By introducing auxiliary systems and upper-lower solution method together with Schauder’s fixed point theorem, the existence of traveling waves and time periodic traveling waves for two models are established, respectively. Utilizing two-sided Laplace transform, the non-existence of traveling waves for one of the models are obtained, accordingly. Our theoretic results can indicate the inflection spread or not.

 

Talk6: The Loop equation for integrable hierarchies and their applications

Prof. Siqi Liu (Tsinghua University)

2017/12/10 15: 10-15: 50,  N210

Abstract: Many integrable hierarchies possess linearizable Virasoro symmetries. By considering the generating function of all Virasoro symmetries, one can obtain the loop equation of an integrable hierarchy, which determines the corresponding free energy uniquely. I will take the KdV hierarchy as an example to explain the basic idea of loop equation, then give some applications of this equation in the Gromov-Witten invariants and Hodge integrals.

 

Talk7: A numerical study of the 3-periodic wave solutions to KdV-type and coupled KdV-type equations

Prof. Xingbiao Hu (AMSS)

2017/12/10 16: 10-16: 50,  N210

Abstract: In the talk, I will report some results on numerical study of the 3-periodic wave solutions to KdV-type and coupled KdV-type equations. This is joint work with Sun Jianqing and Zhang Yingnan.

 

Talk8: Double degeneration of the multi-breathers and the optical rogue waves

Prof. Jinsong He  (Ningbo University)

2017/12/10 16: 50-17: 30,  N210

Abstract: It is quite tricky and challenging to generate the higher-order rogue waves in optical system. In this talk, the double degeneration of the multi-breather is demonstrated visually. Furthermore, we introduce a special kind of breather solution of the nonlinear Schrodinger (NLS) equation, the so-called breather-positon (b-positon for short), which can be obtained by taking the limit lambda_j->lambda_1 in the order-n breather, which is generated by the n-fold Darboux transformation from a special “seed” solution–plane wave.  Further, an order-n b-positon gives an order-n rogue wave under a limit lambda_j->lambda_0. We also suggest a new way to observe higher-order rogue waves generation in an optical fiber, namely, measure the patterns at the central region of the higher-order b-positon generated by ideal initial pulses when lambda_1 is very close to the lambda_0.  Four animations demonstrate clearly the tendency from a breather to a b-positon, and then to a rogue wave.

 

2017/12/11  Morning   Place: N202

 

Talk9: Hankel determinants for a perturbed Laguerre weight and Painleve V equation

Prof. Engui Fan (Fudan University)

2017/12/11 08: 30-09: 10,  N202

Abstract: We adopt the double scaling scheme and the Deift-Zhou nonlinear steepest descent method to obtain the uniform asymptotic approximations of Hankel determinants in terms of a solution of a third-order nonlinear differential equation, which is equivalent to a particular Painleve-V equation. In fact, this Painleve-V equation is equivalent to a general Painleve-III equation. The asymptotic approximations of the leading coefficients and the recurrence coefficients for the corresponding orthogonal polynomials also involve the Painleve-V equation.

 

Talk10: Nonlinear Local waves

Prof. Yong Chen  (East China Normal University)

2017/12/11 09: 10-09: 50,  N202

Abstract: 首先介绍局域波的发展概况,然后介绍我们团队近年来关于局域波开展的研究工作。主要在以下三个方面: 达布变换;KP约化;双线性算法。以及相应的程序算法进展情况。首先介绍局域波的发展概况,然后介绍我们团队近年来关于局域波开展的研究工作。主要在以下三个方面: 达布变换;KP约化;最新进展。