# Research

## Differential Equations

### Friday, December 2, 2011

Title: Refining the invariant subspace method of evolution equatio
Speaker: Wen-Xiu Ma
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

The invariant subspace method is refined, with a view to shedding light on unity and diversity of exact solutions to evolution equations. The crucial idea is to take solution subspaces of linear ordinary differential equations as invariant subspaces that evolution equations admit. A few of examples will be given to illustrate the refined approach.

### Friday, November 18, 2011

Title: Mittag-Leffler Functions and Fractional Differential Equations
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

The Mittag-Leffler method has been used in different areas of mathematics. The main aim of the method is to prove solvability of linear fractional differential equations. Moreover, Mittag-Leffler functions with two parameters play a pivotal role and appear regularly in solutions of fractional differential equations. In this talk, we would like to discuss applications of Mittag-Leffler functions in the study of fractional differential equations.

### Friday, November 4, 2011

Title: Existence of solutions to the fractional-order diffusion and wave equations
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to fractional orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus. We would like to introduce some basics of fractional calculus and discuss the existence of solutions to the fractional-order diffusion and wave equations.

### Friday, October 28, 2011

Title: Vector fields and $$1$$-forms on manifolds
Speaker: Junyi Tu
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We would like to discuss some basic structures on real smooth and complex holomorphic manifolds, including vector fields and $$1$$-forms, flows and the straightening theorem.

### Friday, October 21, 2011

Title: Loop algebras and integrable couplings
Speaker: Wen-Xiu Ma
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We will talk about integrable couplings of soliton equations. The desired integrability is exhibited through the variational identities over semi-direct sums of Lie algebras. A few illustrative examples are given to show how to generate integrable couplings from loop algebras.

### Friday, October 14, 2011

Title: Hamiltonian structures and bi-Hamiltonian structures of integrable equations
Speaker: Wen-Xiu Ma
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We explore Hamiltonian structures and bi-Hamiltonian structures of integrable equations by the zero curvature formulation. The key tool is the trace identity associated with loop algebras. We shed light on some basic aspects of the theory by an often-quoted example — the AKNS soliton hierarchy.

### Friday, October 7, 2011

h3

Title: Hirota bilinear equations and vertex operators
Speaker: Junyi Tu
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We will discuss a connection between Hirota bilinear equations and vertex operators. In particular, we will show how it works from the $$1$$-soliton solution to the $$N$$-soliton solution of the KdV equation.

### Friday, September 30, 2011

Title: A study on the complex-valued partial differential equations
Speaker: Netra Khanal, University of Tampa
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We will discuss the blow-up solutions of complex-valued Burgers and KdV equations and the regularity of series-type solutions of complex KdV-Burgers equation under some mild conditions.

### Friday, September 23, 2011

Title: The pseudo-differential operator representation for the hierarchy of KdV equations, Part II
Speaker: Mengshu Zhang
Time: 10:00am‐11:00am
Place: LIF 269

### Friday, September 16, 2011

Title: The pseudo-differential operator representation for the hierarchy of KdV equations
Speaker: Mengshu Zhang
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We will talk about the higher order KdV equations by considering their pseudodifferential operator representation, and compute infinitely many commuting symmetries through the Lax pairs of pseudodifferential operators.

### Friday, September 9, 2011

Title: The KdV equation and its symmetries, Part II
Speaker: Jinghan Meng
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We will continue to talk about symmetries of the KdV equation in terms of infinitesimal transformations by a nonlinear evolution equation. The KdV equation is one of typical integrable evolution equations. We will also show how to drive it through the compatibility conditions of two spectral problems.

### Friday, September 2, 2011

Title: The KdV equation and its symmetries
Speaker: Jinghan Meng
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

We will talk about symmetries of the KdV equation in terms of infinitesimal transformations by a nonlinear evolution equation. The KdV equation is one of typical integrable evolution equations. We will also show how to drive it through the compatibility conditions of two spectral problems.

### Friday, August 26, 2011

Title: Symbolic computation of analytic solutions for nonlinear differential equations
Speaker: Yinping Liu, East China Normal University
P.R. China
Time: 10:00am‐11:00am
Place: LIF 269

#### Abstract

In this talk, I will give a brief introduction about symbolic computations on nonlinear differential equations. The outline of my talk is as follows:

1. The Elliptic Equation method to construct different types of exact solutions for nonlinear evolution equations, and an automated derivation program.
2. An algorithm to construct auto-BTs for given nonlinear evolution equations, and a program for automated derivation of auto-BTs as well as the corresponding superposition formulas.
3. A program for automated derivation of analytic approximate solutions for nonlinear differential equations with initial or boundary conditions.
4. A frame of a database for differential equations.