报告题目:可积系统理论、计算及其应用研讨会

报告人：胡星标（中国科学院数学与系统科学研究院）、田凯（中国矿业大学（北京））、常向科（中国科学院数学与系统科学研究院）、张英楠（南京师范大学）、虞国富（上海交通大学）、李世豪（四川大学）、陈晓敏 （北京工业大学）

报告时间：2022年11月4日 下午13:00-15:40，晚上19:15-21:15

报告地点：线上腾讯会议，腾讯号：771-622-527 会议密码: 2250

会议邀请人：胡娟，沈守枫

会议特邀报告人：

报告人1：胡星标 （中国科学院数学与系统科学研究院）

报告题目：双线性方法的历史

摘要：如所周知，KdV方程、sine-Gordon方程、NLS方程和Toda方程是数学物理中最重要的四个1+1维可积模型。本报告将沿着有关这四个著名方程的研究发展轨迹，简要回顾Hirota双线性方法的历史。

报告人2：田凯（中国矿业大学（北京））

报告题目：On a supersymmetric nonlinear Schr"{o}dinger equation

摘要：For a supersymmetric nonlinear Schrödinger (NLS) equation, the Wahlquist-Estabrook prolongation algebra L is calculated, and successfully embedded in the Lie superalgebra A(1,1). A non-trivial matrix realization of L is eastablished. Consequently, a proper linear spectral problem is constructed for the supersymmetric NLS equation. We also derive its integrable properties, including a recursion operator, Hamiltonian structure. This is a joint work with Prof. Q. P. Liu and Dr. Liming Zang.

报告人3：常向科 （中国科学院数学与系统科学研究院）

报告题目：On non-smooth solitons of Camassa-Holm type equations

摘要： As non-smooth solitons, peakons are special weak solutions of a class of nonlinear partial differential equations (PDEs) modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. One natural concept of peakons is dictated by the distributional compatibility of the Lax pairs. Due to the Lax integrability, the inverse spectral method is a powerful tool for the constructions of these solutions. In this talk, I will introduce some related results on explicit constructions of peakons for the well-known Camassa-Holm equation and its modifications, as well as our recent progress.

报告人4：张英楠 （南京师范大学）

报告题目：Computation of quasi-periodic solutions to the nonlinear Schrodinger equation

摘要：In this talk, I will show our recent work on the computation of  the quasi-periodic solutions to the nonlinear Schrodinger equation. By use of Hirota's bilinear method and theta function, we transform the problem of solving this kind of solutions into an over-determined nonlinear algebraic system, which can be formulated as a nonlinear least square problem and solved by the Levenberg-Marquardt method. The numerical experiments show that, in some cases the quasi-periodic solutions behave like breathers while in some other cases, they may act like periodic bright solitons and homoclinic solutions which can be taken as rogue waves when their amplitudes is much larger than their backgrounds. Besides, the quasi-periodic solutions may have singularities in some cases. Such kind of singular quasi-periodic solutions can be used to describe the phenomena of wave collapse.

报告人5：虞国富 （上海交通大学）

报告题目：Moment  matrix  and  C-Toda lattice hierarchy

摘要： In this talk, we focus on the reduction of the 2-dimensional Toda hierarchy. Based on the Gauss-Borel decomposition of the moment matrix and the time evolution,  we construct the Lax representation. Symmetric moment matrices are considered that lead to the differential relations between symmetric tau functions and 2d-Toda’s tau function. Motivated by the Cauchy two-matrix model, we study the rank one shift condition in the symmetric case, from which the C-Toda lattice hierarchy is found with the special Lax matrices and intergrable structure. This is a joint work with Shi-Hao Li.

报告人6：李世豪 （四川大学）

报告题目：Skew orthogonal polynomials: generalizations and applications

摘要： Orthogonal polynomials play an important role in modern mathematics and physics. In this talk, I will start with standard orthogonal polynomials, and then talk about skew orthogonal polynomials with generalizations and applications.

报告人7：陈晓敏 （北京工业大学）

报告题目：Extensions of some integrable systems and their molecule solutions

摘要：In this talk, I will introduce some works on extensions of certain integrable systems, including nonisospectral extensions of the semi-discrete Volterra lattices and Schur flow, a generalized discrete Lotka-Volterra equation, a generalized epsilon algorithm and an extension of the hungry Lotka-Volterra lattice to nonzero boundaries. We also present their corresponding molecule solutions expressed explicitly by determinant. Moreover, we investigate the relations between these integrable systems and orthogonal polynomials. All these results are obtained by Hirota’s bilinear method and determinant techniques.