# Research

## Differential Equations

### Tuesday, December 3, 2013

Title: Tri-linear equations and their resonant solutions by Bell polynomials
Speaker: Yuan Zhou
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

We shall talk about a class of tri-linear differential operators described by triple Bell polynomials. The superposition principle will be applied to the construction of resonant solutions of exponential waves for the tri-linear equations. A few illustrative examples will be presented.

### Tuesday, November 26, 2013

Title: Bi-integrable couplings of Dirac equations
Speaker: Dr. Wenying Zhang, Shanghai University
Shanghai, PR China
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

A non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy.

### Tuesday, November 19, 2013

Title: Solitary waves and solitons to nonlinear evolutionary equations
Speaker: Dr. Shuimeng Yu, Jiangnan University
Wuxi, PR China
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

First, definitions and characteristics of solitary waves and solitons will be introduced. Second, some analytical solution methods will be discussed. Finally, an application example of the solution methods will be given.

### Tuesday, November 12, 2013

Title: Lie symmetry method and its application to PDEs
Speaker: Dr. Shoufeng Shen, Zhejiang University of Technology
Hangzhou, PR China
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

We will talk about the Lie symmetry method and its application to PDEs, and pay particular attention to Kac-Moody-Virasoro symmetries for high dimensional integrable systems.

### Tuesday, November 5, 2013

Title: A spectral problem from $$\mathrm{so}(3,R)$$ and its associated soliton hierarchy
Speaker: Morgan Mcanally
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

I will talk about a hierarchy of soliton equations from the zero curvature equations associated with the real Lie algebra $$\mathrm{so}(3,R)$$. In particular, I will show you using Maple how to generate the hierarchy of soliton equations and that the resulting infinitely many vector fields commute.

### Tuesday, October 29, 2013

Title: Darboux transformations for a twisted derivation and its applications to several classical integrable systems, Part II
Speaker: Dr. Chunxia Li, Capital Normal University
Beijing, PR China
Time: 1:00pm‐2:00pm
Place: EDU 257

### Tuesday, October 22, 2013

Title: Image restoration based on PDEs
Speaker: Dr. Hongyi Liu, Nanjing University of Science and Technology
Nanjing, PR China
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

We would like to discuss image restoration based on PDE models. The topics of the talk include digital image processing, mathematics model of image restoration, and recent developments of image restoration based on PDEs.

### Tuesday, October 15, 2013

Title: Darboux transformations for a twisted derivation and its applications to several classical integrable systems
Speaker: Dr. Chunxia Li, Capital Normal University
Beijing, PR China
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

In my talk, I will first introduce a twisted derivation and then construct its general Darboux transformations. Based on the general Darboux transformation, we're able to derive Darboux transformations for the KP equation, $$(2+1)$$-dimensional Toda lattice equation, fully discrete KP equation, super KdV equation and $$q$$-deformed Toda lattice equation. By iteration of their Darboux transformations, we are able to recover determinant solutions to the above-mentioned soliton equations and obtain quasideterminant solutions to their noncommutative analogues.

### Tuesday, October 8, 2013

Title: Applications of Bell polynomials and the linear superposition principle to generalized bilinear equations, Part II
Speaker: Xiang Gu
Time: 1:00pm‐2:00pm
Place: EDU 257

### Tuesday, October 1, 2013

Title: Applications of Bell polynomials and the linear superposition principle to generalized bilinear equations
Speaker: Xiang Gu
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

Following Emmanuel A. Appiah's talk on Ma's generalized bilinear equations, I would like to discuss applications of Bell polynomials and the linear superposition principle to solution subspaces of the generalized bilinear equations. Illustrative examples will also be presented for group discussion.

### Tuesday, September 24, 2013

Title: Generalized Bilinear Differential Equations, Part II
Speaker: Emmanuel Appiah
Time: 1:00pm‐2:00pm
Place: EDU 257

### Tuesday, September 17, 2013

Title: Generalized Bilinear Differential Equations
Speaker: Emmanuel Appiah
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

We will introduce a kind of generalized bilinear differential operator and explore instances where the linear superposition principle applies to the corresponding bilinear differential equations.

### Tuesday, September 10, 2013

Title: A soliton hierarchy associated with $$\mathrm{so}(3,R)$$, Part II
Speaker: Solomon Manukure
Time: 1:00pm‐2:00pm
Place: EDU 257

We will continue to talk about zero curvature equations and construct a hierarchy of integrable equations from the Lie algebra $$\mathrm{so}(3,R)$$.

### Tuesday, September 3, 2013

Title: A soliton hierarchy associated with $$\mathrm{so}(3,R)$$
Speaker: Solomon Manukure
Time: 1:00pm‐2:00pm
Place: EDU 257

#### Abstract

We will talk about zero curvature equations and construct a hierarchy of integrable equations from the Lie algebra $$\mathrm{so}(3,R)$$.