# Research

## Differential Equations

### Thursday, April 26, 2018

Title: The Riemann-Hilbert method for integrable equations
Speaker: Wen-Xiu Ma
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

We will talk about the Riemann-Hilbert method for solving integrable equations. The starting point is a pair of matrix spectral problems, called a Lax pair. Soliton solutions can be generated from Riemann-Hilbert problems on the real axis with an identity jump matrix. An illustrative example, associated with an arbitrary order matrix spatial spectral problem, will be presented.

### Thursday, April 19, 2018

Title: Lump and interaction solutions to an extended $$(2+1)$$-dimensional Boussinesq equation
Speaker: Hui Wang, Shanghai Maritime University
Shanghai, China
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

I will briefly talk about lump and interaction solutions to nonlinear partial differential equations. The example I will focus on is an extended $$(2+1)$$-dimensional Boussinesq equation, whose lump and interaction solutions will be explored.

### Thursday, April 12, 2018

Title: Derive longtime asymptotics of the MKdV equation by Painlevé II
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

We will discuss longtime asymptotics for the MKdV equation by the isomonodromy deformation of the second Painlevé equation.

### Thursday, April 5, 2018

Title: On some 2D orthogonal $$q$$-polynomials
Speaker: Ruiming Zhang, Northwest A&F University
PR China
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

This talk is based on a joint work with Professor Mourad E. H. Ismail recently published on AMS Transactions, in which we have generalized several families of 2D orthogonal polynomials used in probability theory and physics (a.k.a. Zernike polynomials). In this talk, I introduce two $$q$$-analogues of the 2D-Hermite polynomials that are polynomials in two complex variables. For both families, I present explicit formulas, raising and lowering operator relations, generating functions, and Rodrigues formulas. These $$q$$-orthogonal polynomials also have unexpected connections to the theory of integer partitions.

### Thursday, March 29, 2018

Title: The factorization problem on a fixed cross
Speaker: Yuan Zhou
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

The factorization problem on a fixed cross is solved explicitly in terms of parabolic cylinder functions. The solution is then used to present the longtime asymptotics of the mKdV equation.

### Thursday, March 22, 2018

Title: Evaluating integrals on two crosses in oscillatory Riemann-Hilbert problems by the steepest descent method
Speaker: Xiang Gu
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

With the help of the Cauchy operators, we will discuss how to separate contributions from two crosses in a contour, and explore long-time asymptotic behavior of the corresponding integrals.

### Thursday, March 15, 2018

Spring Break — no seminar this week.

### Thursday, March 8, 2018

Title: Steepest descent method for oscillatory Riemann-Hilbert problems with truncated contours II
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: NES 104

### Thursday, March 1, 2018

Title: Steepest descent method for oscillatory Riemann-Hilbert problems with truncated contours
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

We will discuss how to use the Cauchy operator and apply its properties, say, $$L^2$$ boundedness, to convert argumented Riemann-Hilbert problems to truncated Riemann-Hilbert problems with controlled error terms.

### Thursday, February 22, 2018

Title: Studies on exact solutions of partial differential and differential-difference equations
Speaker: Yongli Sun, Beijing University of Chemical Technology
PR China
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

In this seminar, I will briefly introduce my recent research. One is about exact solutions of some partial differential equations, based on the transformed rational function method. The other is about soliton solutions to the Blaszak-Marciniak lattice II, by the Darboux transformation.

### Thursday, February 15, 2018

Title: A matrix spectral problem based on $$\mathrm{so}(3,R)$$ and its associated soliton hierarchy
Speaker: Sumayah Batwa
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

In this talk, I will discuss a spectral problem associated with the real Lie algebra $$\mathrm{so}(3,R)$$ and generate a hierarchy of soliton equations from zero curvature equations linked with the spectral problem. Moreover, I will give an illustrative example of soliton equations with a bi-Hamiltonian structure.

### Thursday, February 8, 2018

Title: Contour deformation in oscillatory Riemann-Hilbert problems associated with the mKdV equation III
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: NES 104

### Thursday, February 1, 2018

Title: Nonlocal symmetries with pseudopotentials
Speaker: Xiazhi Hao, East China Normal University
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

In this talk, I will discuss nonlocal symmetries by including pseudopotentials, which are equivalent to Lie point symmetries of a prolonged system. Moreover, I will show that through the transformations resulted from nonlocal symmetries, exact solutions to the original system can be generated.

### Thursday, January 25, 2018

Title: Contour deformation in oscillatory Riemann-Hilbert problems associated with the mKdV equation II
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: NES 104

### Thursday, January 18, 2018

Title: Contour deformation in oscillatory Riemann-Hilbert problems associated with the mKdV equation
Speaker: Fudong Wang
Time: 2:00pm‐3:00pm
Place: NES 104

#### Abstract

I will discuss about how to extend oscillatory Riemann-Hilbert problems to an augmented contour. The focus will be on how to apply the steepest descent method introduced by Percy Deift and Xin Zhou to analyze long time asymptotics for the mKdV equation.