# Research

## Differential Equations

### Wednesday, November 27, 2019

Title: D-bar method and its application to nonlinear evolution equations
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

Starting with a spectral problem, we will discuss how to generate some famous nonlinear evolution equations, such as NLS, mKdV, and DNLS. An example of how to derive pure (multi-)soliton solutions for DNLS will be presented, based on Jyh-Hao Lee's work on the global solvability of the DNLS (1989).

### Monday, November 25, 2019

Title: Multiple component mKdV systems and their classical reductions
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

In this presentation, I am going to talk about classical reductions of multiple component mKdV systems, and discuss about soliton solutions to the reduced systems, based on Riemannn-Hilbert problems.

### Wednesday, November 20, 2019

Title: Nonlocal PT-symmetric integrable equations and their Riemann-Hilbert problems, II
Speaker: Wen-Xiu Ma
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, November 18, 2019

Title: Nonlocal PT-symmetric integrable equations and their Riemann-Hilbert problems
Speaker: Wen-Xiu Ma
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

The talk aims to discuss how to construct nonlocal PT-symmetric integrable equations from nonlocal group reductions of matrix spectral problems. Such nonlocal equations can be classified into three types: reserve space, reverse time, and reverse spacetime, each of which can involve either the transpose or the Hermitian transpose. Notably there exist only five nonlocal PT-symmetric nonlinear Schrï¿½dinger and modified Korteweg-de Vires equations. Based on their associated spectral problems, a kind of Riemann-Hilbert problems is formulated and used to present the corresponding inverse scattering transforms. Soliton solutions are generated from solutions to special Riemann-Hilbert problems with the identity jump matrix.

### Wednesday, November 13, 2019

Title: Characteristic set method combined with Lie symmetry method for differential-difference systems
Speaker: Wenting Li, Heilongjiang University
China
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

This talk will give an introduction to the characteristic set method, focusing on the basic idea and combination with Lie symmetry method. An application to the Toda lattice equation will be given to show how the combined algorithm works.

### Wednesday, November 6, 2019

Title: Hirita bilinear forms for some nonlinear Schrödinger equations, II
Speaker: Yan Jiang, Beijing University of Posts and Telecommunications
China
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, November 4, 2019

Title: Hirita bilinear forms for some nonlinear Schrödinger equations
Speaker: Yan Jiang, Beijing University of Posts and Telecommunications
China
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

In this talk, I will first review some basic concepts of the bilinear method. Then, with the bilinear mehtod, I will discuss how to obtain Hirota bilinear forms and soliton solutions for some nonlinear Schrödinger (NLS) equations, including the standard NLS equation, the derivative NLS equation and the Sasa–Satsuma equation. Note that we need to make use of some extra formulas and introduce an auxiliary function for some NLS equations. Our discussions might be helpful for obtaining Hirota bilinear forms for other nonlinear evolution equations.

### Wednesday, October 23, 2019

Title: Riemann-Hilbert approach to a coupled nonisospectral Gross-Pitaevskii system and its multi-component generalization, II
Speaker: Xiaotong Chen, Zhejiang Normal University
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, October 21, 2019

Title: Riemann-Hilbert approach to a coupled nonisospectral Gross-Pitaevskii system and its multi-component generalization
Speaker: Xiaotong Chen, Zhejiang Normal University
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

In this talk, I focus on $$N$$-soliton solutions to a coupled nonisospectral Gross-Pitaevskii system by the Riemann-Hilbert approach. From the spectral problem of its Lax pair, a kind of Riemann-Hilbert problems is derived for the system, and its $$N$$-soliton solutions are then presented in the irregularity and reflectionless case. Moreover, a multi-component coupled nonisospectral Gross-Pitaevskii system will also be presented.

### Wednesday, October 16, 2019

Title: Inverse scattering for nonlocal nonlinear Schrödinger hierarchies, II
Speaker: Wen-Xiu Ma
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, October 14, 2019

Title: Inverse scattering for nonlocal nonlinear Schrödinger hierarchies
Speaker: Wen-Xiu Ma
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

The talk aims to discuss nonlocal nonlinear Schrödinger hierarchies and their inverse scattering transforms. The inverse scattering problems are formulated through Riemann-Hilbert problems and the closed systems for the Jost solutions are determined by the generalized Cauchy integral formula or the Sokhotski-Plemelj formula. The reflectionless transformations generate soliton solutions. A few illustrative examples will be presented.

### Wednesday, October 9, 2019

Title: Damping and inhomogeneous exchange effects in the propagation of short waves in ferrites, II
Speaker: Xin-wei Jin, Zhejiang Normal Universitya
Time: 2:00pm‐3:00pm
Place: CMC 108

TBA

### Monday, October 7, 2019

Title: Damping and inhomogeneous exchange effects in the propagation of short waves in ferrites
Speaker: Xin-wei Jin, Zhejiang Normal University
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

The Kraenkelï¿½Mannaï¿½Merle (KMM) system is derived from the Maxwell's equations and Landauï¿½Lifshitzï¿½Gilbert equation to describe the propagation of short-waves in saturated ferromagnetic materials in an external field. By applying the consistent tanh expansion method and the truncated Painlevï¿½ analysis, new exact solutions including rogue waves to the KMM system are constructed, and numerical simulation is conducted to discuss the influence of damping and inhomogeneous exchange effects which exist in real circumstances.

### Wednesday, October 2, 2019

Title: Rogue wave solutions to the nonlocal nonlinear Schrödinger equation, II
Speaker: Rusuo Ye, Zhejiang Normal University
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, September 30, 2019

Title: Rogue wave solutions to the nonlocal nonlinear Schrödinger equation
Speaker: Rusuo Ye, Zhejiang Normal University
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

By using a matrix version of Darboux transformation with a spectral matrix of the Jordan block form, rogue wave solutions to the nonlocal nonlinear Schrödinger (NLS) equation are presented. General rogue wave solutions up to the fourth-order are also formulated. The structure of these solutions is essentially determined by the corresponding solutions of the Lyapunov equation. With appropriate choices of free parameters, dynamics and distribution patterns for the nonlocal NLS equation are shown to be much abundant than those for the local NLS equation.

### Wednesday, September 25, 2019

Title: General $$N$$-solitons to nonlocal nonlinear Schrödinger equations by Riemann-Hilbert problems, II
Speaker: Ahmed Ahmed
Time: 2:00pm‐3:00pm
Place: CMC 108

### Monday, September 23, 2019

Title: General $$N$$-solitons to nonlocal nonlinear Schrödinger equations by Riemann-Hilbert problems
Speaker: Ahmed Ahmed
Time: 2:00pm‐3:00pm
Place: PED 110

#### Abstract

General $$N$$-solitons to three nonlocal nonlinear Schrödinger equations are derived from Riemann-Hilbert problems of the AKNS hierarchy. The three nonlocal equations correspond to reverse-space, reverse-time and reverse-space-time symmetries, respectively. Their $$N$$-solitons are associated with the same Riemann-Hilbert problems, but their symmetry relations of scattering data are different.

### Wednesday, September 18, 2019

Title: Inverse scattering and $$N$$-soliton solution for the nonlocal nonlinear Schrödinger equation
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: CMC 108

#### Abstract

We will discuss the inverse scattering for the nonlocal nonlinear Schrödinger equation from a special reduction of the AKNS/ZS system. Analyticity of the Jost eigenfunctions, as well as the scattering data, will be discussed in detail. The inverse scattering will finally be transferred to an oscillatory Riemann-Hilbert problem. Assuming that the initial data is reflectionless, a general $$N$$-soliton solution will be presented.

### Monday, September 16, 2019

Title: Soliton solutions to the nonlocal nonlinear Schrödinger equation, II
Time: 1:30pm–2:30pm
Place: PED 105

### Wednesday, September 11, 2019

Title: Soliton solutions to the nonlocal nonlinear Schrödinger equation
Time: 1:30pm–2:30pm
Place: CMC 108

#### Abstract

In this seminar, I am going to discuss soliton solutions to the nonlocal nonlinear Schrödinger equation, based on the inverse scattering transform.

### Monday, September 9, 2019

Title: Inverse scattering transform for the nonlocal NLS equation, Part II
Speaker: Yeuhui Huang, North China Electric Power University
Time: 1:30pm–2:30pm
Place: CMC 209

### Wednesday, September 4, 2019

Title: Inverse scattering transform for the nonlocal NLS equation
Speaker: Yeuhui Huang, North China Electric Power University
Time: 1:30pm–2:30pm
Place: CMC 209

#### Abstract

In this seminar, we will discuss about the inverse scattering transform for the nonlocal nonlinear Schrödinger equation, and present its applications to soliton solutions.