# Research

## Differential Equations

### Tuesday, March 10, 2020

Title: Inverse scattering for a nonlocal reverse-time six-component AKNS system of fourth-order and its exact solutions
Time: 12:30pm‐1:45pm
Place: CMC 108

#### Abstract

In this talk, I am going to talk about solvability of an integrable nonlocal reverse-time six-component AKNS system of fourth-order obtained from a reduced coupled system in the reverse-time AKNS hierarchy. An inverse scattering transform based on the Riemann-Hilbert problems will be formulated in order to derive exact solutions associated with a specific jump identity matrix.

### Tuesday, March 3, 2020

Title: Binary Darboux transformation for nonlocal integrable equations
Speaker: Wen-Xiu Ma
Time: 12:30pm‐1:45pm
Place: CMC 108

#### Abstract

We will talk about the Darboux transformation in soliton theory. A binary Darboux transformation is formulated for nonlocal integrable equations through eigenfunctions and adjoint eigenfunctions, and its general $$N$$-fold decomposition is shown explicitly. An application with zero potentials generates soliton solutions, and illustrative examples are about nonlocal integrable nonlinear Schrödinger equations.

### Tuesday, February 25, 2020

Title: From the inverse scattering transform to the Darboux transformation
Speaker: Yehui Huang, North China Electric Power University
Time: 12:30pm‐1:45pm
Place: CMC 108

#### Abstract

There are various methods for solving integrable systems. Starting from matrix spectral problems, the Darboux transformation can be applied to construction of solutions of nonlinear equations. The aim of the talk is to discuss applications of the Darboux transformation and its relation with the inverse scattering transform.

### Tuesday, February 18, 2020

Title: Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero background
Speaker: Yong Zhang, Shandong University of Science and Technology
Time: 12:30pm‐1:45pm
Place: CMC 108

#### Abstract

We will discuss about a modified variant of the standard inverse scattering transform for the focusing nonlinear Schrödinger equation formulated in the presence of a nonzero background field. The aim of the talk is to solve Riemann-Hilbert problems with arbitrary-order poles and potentially severe spectral singularities in a beautifully simple and unified way.

### Tuesday, February 11, 2020

Title: $$L^2$$-Sobolev space correspondence from jump matrices to potentials in the ZS-AKNS spectral problem, Part II
Speaker: Fudong Wang
Time: 12:30pm‐1:45pm
Place: CMC 108

### Tuesday, February 4, 2020

Title: $$L^2$$-Sobolev space correspondence from jump matrices to potentials in the ZS-AKNS spectral problem
Speaker: Yehui Huang, North China Electric Power University
Time: 12:30pm‐1:45pm
Place: CMC 108

#### Abstract

Following the forward scattering theory, this week we will discuss about the inverse scattering theory for the ZS-AKNS spectral problem. More specifically, we will talk about the correspondence from jump matrices to potentials, and related decay and regularity properties.

### Tuesday, January 28, 2020

Title: $$L^2$$-Sobolev space bijectivity in the inverse scattering theory for the ZS-AKNS system, Part II
Speaker: Fudong Wang
Time: 12:30pm‐1:45pm
Place: CMC 108

### Tuesday, January 21, 2020

Title: $$L^2$$-Sobolev space bijectivity in the inverse scattering theory for the ZS-AKNS system
Speaker: Fudong Wang
Time: 12:30pm‐1:45pm
Place: CMC 108

#### Abstract

In this seminar talk, we will discuss the $$L^2$$-Sobolev space bijectivity of the scattering and inverse scattering transforms associated with the ZS-AKNS system, presented by Xin Zhou in 1998. The theory is a natural counterpart to the $$L^2$$-Fourier theory. The main result is to establish the relations of integral decay and regularity orders between the potential and the reflection coefficients. The proof is primarily based on a few of Riemann-Hilbert factorization problems. The basic $$L^1$$-theory of the inverse scattering transforms for general first-order spectral problems, by Beals and Coifman in 1984, offers a main framework for the analysis of the inverse scattering transforms in $$L^2$$.