为加强国内外同行间的学术交流，展现数学物理方程与可积系统方向的研究成果，并促进华中科技大学数学与统计学院和国内外相关领域数学学者的密切交流，华中科技大学数学与统计学院将于2019年1月3日至5日举办“可积系统与孤立子理论研讨会”。

会议日程表

1月3日

注册报到（地点：华中科技大学国际学术交流中心八号楼大厅).

1月4日上午

8：45-9：00：开幕式（地点：科技楼702）

主持人：李骥
报告人	时 间	报告题目
乔志军	9：00-9：45	Integrable   systems with peakons and cuspons
9：45-10：15：会间休息
张玉峰	10：15-11：00	Some similarity   solutions and numerical solutions to the time-fractional Burgers system
11：00-11：30   交流讨论

1月4日下午

主持人：张光辉
报告人	时 间	报告题目
贺劲松	14：30-15：15	Introduction to optical rogue waves
15：15-15：45 会间休息
李骥	15：45-16：30	Introduction to invariant manifold theory
16:30-17:00交流讨论

1月5日上午

主持人：李骥
报告人	时 间	报告题目
乔志军	8：30-9：15	Integrable system with peakon, weak kink, and   kink-peakon interactional solutions
李淑霞	9：15-10：00	Lax Algebraic  Representation for an Integrable   Hierarchy
10：00-10：30 会间休息
10：30-11：00 交流讨论

1月5日下午

会议结束，离会

Integrable systems with peakons and cuspons

乔志军

Abstract：In my talk, I will introduce integrable peakon and cuspon equations and present a basic approach how to get peakon solutions. Those equations include the well-known Camassa-Holm (CH), the Degasperis-Procesi (DP), and other new peakon equations with M/W-shape solutions. I take the CH case as a typical example to explain the details. My presentation is based on my previous work (Communications in Mathematical Physics 239, 309-341). I will show that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierarchy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. In particular, the CH peakon equation is extended to the FORQ and other higher order peakon models with peakon and weak-kink solutions. In the end of my talk, some open problems are also addressed for discussion.

Some similarity solutions and numerical solutions to the time-fractional Burgers system

张玉峰

Abstract：In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the time-fractional derivative is the Riemann-Liouville type. The FODS can be approximated by some integer-order ordinary differential equations, here we present three such integer-order ordinary differential equations (called IODE-one, IODE-two and IODE-three, respectively). For the IODE-one, we obtain its similarity solutions and its numerical solutions, which are the approximated similarity solutions and the approximated numerical solutions of the TFBS. Secondly, with the aid  of a scalar Lie transformation, we transform the TFBS to another FODS under the case where the  time-fractional derivative is the Caputo type, whose similarity solutions are expressed by the Beta function. Finally, we apply the numerical method to obtain the numerical solutions of the IODE-two and the IODE-three.

Introduction to optical rogue wave

贺劲松

Abstract：The optical rogue waves will be discussed based on two integrable models which are two well-known modeling equations in optical systems, i.e.., nonlinear Schrodinger equation and derivative nonlinear Schrodinger equation. We shall summary the experiment observations of the rogue waves. The constructions of the rogue waves of the nonlinear Schrodinger equations and the derivative nonlinear Schrodinger equations will be discussed by the Darboux transformation and the determinant representations.  

Introduction to invariant manifold theory

李骥

Abstract：Invanriant manifolds are used to describ the long term behavior of a dynamical system. The existence of invariant manifold for general system is hard to prove unless the system is close to some simple well hehaved one. We discuss the persistence of invariant manifold under perturbation. Then we introduce the this theory could be used to get the geometric singular perturbation theory. We also show how these theory could be applied and extended.

Integrable system with peakon, weak kink, and kink-peakon interactional solution

乔志军

Abstract：In this work, we study an integrable system with both quadratic and cubic nonlinearity. This model is kind of a cubic generalization of the Camassa-Holm (CH) equation. The equation is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. In the case of no linear term, the peaked soliton (peakon) and multi-peakon solutions are presented. In particular, the two-peakon dynamical system is explicitly presented and their collisions are investigated in details. In some special case, the weak kink and kink-peakon interactional solutions are found. Significant difference from the CH equation is analyzed through a comparison. In the paper, we also study all possible smooth one-soliton solutions for the system. This work is joint with Dr. Baoqiang Xia and Professor Jibin Li.

Lax Algebraic  Representation for an Integrable Hierarchy 

李淑霞

Abstract: Using the functional gradient approach of eigenvalues, this talk presents a pair of Lenard’s operators for the Levi’s vector fields and establishes commutator representations for hierarchies of Levi’s equations. The relationship between potential and stationary Levi’s system is discussed in the end.