Speaker : 1.Prof. Jenhsu Chang (National Defense University) 2.Prof. Ting-Jung Kuo (Taiwan Normal University) 3.Prof. Chun-Kong Law (National Sun Yat-sen University)
Title : 1.The resonant structure of Kink-Solitons in the Modified KP Equation 2.Monodromy aspect of generalized Lame equation with eliiptic KdV potential 3.Plancherel-Rotach asymptotics for -orthogonal polynomials
Time : 2018-12-15 (Sat) 14:00 - 17:00
Place : Seminar Room 638, Institute of Mathematics (NTU Campus)
Abstract: 1.We study resonant theory in the Modified KP equation (MKP) using the totally nonnegative Grassmannian based on Kodama’s group investigations. One constructs the multi-kinks solitons solutions of MKP equation using the $\tau$ function and the Binet-Cauchy formula, and then can study the interactions between kink solitons and line solitons. Thus, we can get terraces shape solutions. Especially, the diffraction of kink fronts, Y-type kink soliton resonance, O-type and the P-type interactions of X-shape with kink solitons are investigated. Also, the amplitudes of the intersection of the line solitons and kink fronts are computed. It is found that the resonant structure of the soliton graph is obtained by the superimposition of two corresponding soliton graphs of the two Le-Diagrams given an irreducible Schubert cell in a totally non-negative Grassmannian . In addition, one makes a comparison with the KPII equation.

2.In this talk, I will review some basic results of KdV theory and then introduce a second order complex ODE so called the generalized Lame equation (GLE) with elliptic KdV potential. Firstly, I will talk about the monodromy representation of this GLE. For this GLE, there associates a hyperelliptic curve to this GLE. I will focus on the geometry of this hyperelliptic curve from monodromy aspect and introduce a relevant problem which plays a role in our study. As an application, we can give a criterion to the existence or non-existence problem to a special nonlinear PDE. This is a joint work with Z. Chen and C. S. Lin.

3.It is well known that the Hermite polynomials have Plancherel-Rotach asymptotics about its largest zero, as well as outside it. However a rigorous proof seems to be missing in the literature. We shall fill it up and try to use this approach to prove the Plancherel-Rotach asymptotics around the largest zero of $q$-orthogonal polynomials. Our approach only involves the second order difference operator, and seems to be new for $q$-series problems. This is joint work with Mourad Ismail.