Lecture On Integrable System
time: 2021-03-29

Title 1 : From integrable spatial discrete hierarchy to integrable nonlinear PDE hierarchy

Speaker:  Zuonong Zhu ( Shanghai Jiao Tong University )

Time: Fri, Apr.26, 2021, PM:20:00-21:00

Location: Tencent Conference

Meeting Number: 260 177 127



Title 2 : The kink solutions of the SIdV equation and the associated surfaces

Speaker:  Prof. Jingsong He ( Institute for Advanced Study, Shenzhen University )

Time: Sun, Apr.28, 2021, PM:14:30-15:30

Location: Tencent Conference

Meeting Number: 627 396 862


Title 3: Matrix integral solutions to the related Leznov lattice equations

Speaker:  Prof. Guofu Yu ( Shanghai Jiao Tong University )

Time: Sun, Apr.28, 2021, PM:15:30-16:30

Location: Tencent Conference

Meeting Number: 627 396 862


Title 4: Local waves and integrable deep learning algorithms

Speaker:  Prof. Yong Chen ( East China Normal University )

Time: Wed, Apr.31, 2021, PM:15:00-16:00

Location: Tencent Conference

Meeting Number: 574 117 344


Inviter: Dr. Liming Ling


Abstract1:

      In this talk, we will address the topic that from integrable spatial discrete hierarchy to integrable nonlinear PDE hierarchy. We will review that how to get the integrability theory of the KdV hierarchy from the integrability of spatial discrete KdV hierarchy.


Abstract2:

     In this talk, we study a new non-linear integrable equation, , which is invariant under scaling of dependent variable and was called the SIdV equation, see [Commun. Nonlinear Sci. Numeric. Simulat. 17 (2012) 4155.] The order-n kink solution  of the SIdV equation, which is associated with the n-soliton solution of the Korteweg-de Vries equation, is constructed by  using the n-fold Darboux transformation (DT) from zero “seed” solution. Moreover,  we also proivde the evolution scenarios of surfaces of revolution associated with the kink-type solutions of the SIdV, where the kink-type solutions play the role of a metric. We put forward two kinds of evolution scenarios for surfaces of revolution associated with two types of kink-type metric (solution) and we study the key properties of these surfaces.


Abstract3:

       Matrix integrals used in random matrix theory for the study of eigenvalues of matrix ensembles have been shown to provide τ-functions for several hierarchies of integrable equations. In this talk, we construct the matrix integral solutions to the Leznov lattice equation, the related discrete Leznov lattice and the Pfaffianized  lattice systems, respectively. We demonstrate that the partition function of Jacobi unitary ensemble is a solution to the semi-discrete Leznov lattice and the partition function of Jacobi orthogonal/symplectic ensemble gives solutions of the Pfaffianized Leznov lattice.