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Mathematics & Statistics

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**Title**

**Speaker**

**Time**

**Place**

Inverse scattering for a nonlocal reverse-time six-component AKNS system of fourth-order and its exact solutions

Alle Adjiri

12:30pm–1:45pm

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.

**Title**

**Speaker**

**Time**

**Place**

Binary Darboux transformation for nonlocal integrable equations

Wen-Xiu Ma

12:30pm–1:45pm

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.

**Title**

**Speaker**

**Time**

**Place**

From the inverse scattering transform to the Darboux transformation

Yehui Huang

North China Electric Power University

China

12:30pm–1:45pm

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.

**Title**

**Speaker**

**Time**

**Place**

Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero background

Yong Zhang

Shandong University of Science and Technology

China

12:30pm–1:45pm

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.

**Title**

**Speaker**

**Time**

**Place**

\(L^2\)-Sobolev space correspondence from jump matrices to potentials in the ZS-AKNS spectral problem, Part II

Fudong Wang

12:30pm–1:45pm

CMC 108

**Title**

**Speaker**

**Time**

**Place**

\(L^2\)-Sobolev space correspondence from jump matrices to potentials in the ZS-AKNS spectral problem

Yehui Huang

North China Electric Power University

China

12:30pm–1:45pm

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.

**Title**

**Speaker**

**Time**

**Place**

\(L^2\)-Sobolev space bijectivity in the inverse scattering theory for the ZS-AKNS system, Part II

Fudong Wang

12:30pm–1:45pm

CMC 108

**Title**

**Speaker**

**Time**

**Place**

\(L^2\)-Sobolev space bijectivity in the inverse scattering theory for the ZS-AKNS system

Fudong Wang

12:30pm–1:45pm

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 \(L^2\).