# Research

## Differential Equations

### Friday, April 29, 2011

Title: Wronskian solutions of the two-dimensional Toda molecule equation, Part II
Speaker: Junyi Tu
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Bi-directional Wronskian solutions and double Wronskian solutions are constructed for the two-dimensional Toda molecule equation. Two determinant identities including the Jacobi identity are the base of construction.

### Friday, April 22, 2011

Title: Wronskian solutions and Grammian solutions to the two-dimensional Toda lattice equation, Part II
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

The Plücker relation and Pfaffian identities are applied to the two-dimensional Toda lattice equation. The resulting Wronskian and Pfaffian formulations generate two kinds of determinant solutions to the two-dimensional Toda lattice equation.

### Tuesday, April 19, 2011

Title: $$W$$-algebras in CKP and BKP hierarchies
Speaker: Jingsong He, Ningbo University
Zhejiang, PR China
Time: 4:00pm‐5:00pm
Place: PHY 108

#### Abstract

This talk aims to construct additional symmetries associated with the constrained CKP and BKP hierarchies, and further to give the action of the symmetry flows on the eigenfunction and adjoint eigenfunction. We further show that their acting on the space of the wave operator, $$\partial_k^*$$ forms two centerless algebras $$W^{cC}_{1+\infty}$$ and $$W^{cB}_{1+\infty}$$-subalgebra of centerless $$W_{1+\infty}$$, respectively.

### Friday, April 15, 2011

Title: Wronskian solutions and Grammian solutions to the two-dimensional Toda lattice equation
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

The Plücker relation and Pfaffian identities are applied to the two-dimensional Toda lattice equation. The resulting Wronskian and Pfaffian formulations generate two kinds of determinant solutions to the two-dimensional Toda lattice equation.

### Friday, April 8, 2011

Title: Wronskian solutions of the two-dimensional Toda molecule equation
Speaker: Junyi Tu
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Bi-directional Wronskian solutions and double Wronskian solutions are constructed for the two-dimensional Toda molecule equation. Two determinant identities including the Jacobi identity are the base of construction.

### Friday, April 1, 2011

Title: Multi-wave solutions for nonlinear equations by the multiple exp-function method
Speaker: Wen-Xiu Ma
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

A multiple exp-function method is proposed for constructing multi-wave solutions of nonlinear partial differential equations. The method is oriented towards ease of use and capability of computer algebra systems, and provides a direct and systematical solution procedure. Using Maple, an application of the approach is made to the $$3+1$$ dimensional potential-Yu-Toda-Sasa-Fukuyama equation, and so, exact $$1$$-wave, $$2$$-wave and $$3$$-wave solutions are generated, along with $$1$$-soliton, $$2$$-soliton and $$3$$-soliton type solutions.

### Friday, March 25, 2011

Title: Wronski-type and Gramm-type Pfaffian solutions to the coupled KP equation, Part II
Speaker: Alrazi Abdeljabbar
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Pfaffian techniques are used to present Pfaffian structure of the coupled KP equation. The resulting Pfaffian formulation yields two kinds of Pfaffian solutions to the coupled KP equation. Such a solution process leads to Pfaffianization of the KP equation.

### Friday, March 11, 2011

Title: Wronski-type and Gramm-type Pfaffian solutions to the coupled KP equation
Speaker: Alrazi Abdeljabbar
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Pfaffian techniques are used to present Pfaffian structure of the coupled KP equation. The resulting Pfaffian formulation yields two kinds of Pfaffian solutions to the coupled KP equation. Such a solution process leads to Pfaffianization of the KP equation.

### Friday, March 4, 2011

Title: Pfaffian structure of the BKP equation
Speaker: Yaning Tang, Northwest Polytechnic University
P.R. China
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Pfaffian techniques are used to present Pfaffian structure of the BKP equation. The resulting Pfaffian formulation generates a class of Pfaffian solutions to the BKP equation.

### Friday, February 25, 2011

Title: Pfaffian structure of soliton equations, Part II
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Using Pfaffian techniques, we discuss the common structure of soliton equations and show how fundamental soliton equations, such as the KP, BKP, coupled KP, Toda lattice and Toda molecule equations, resolve themselves into Pfaffian identities.

### Friday, February 18, 2011

Title: Pfaffian structure of soliton equations
Speaker: Alrazi Abdeljabbar
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

Using Pfaffian techniques, we discuss the common structure of soliton equations and show how fundamental soliton equations, such as the KP, BKP, coupled KP, Toda lattice and Toda molecule equations, resolve themselves into Pfaffian identities.

### Friday, February 11, 2011

Title: Pfaffian identities in soliton theory, Part II
Speaker: Alrazi Abdeljabbar
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

We discuss a few fundamental Pfaffian identities in soliton theory. These include expansion formulae, addition formulae and derivative formulae for Pfaffians. The computation using Pfaffians is much simpler than using determinants in generating soliton solutions to Hirota bilinear equations.

### Friday, February 4, 2011

Title: Pfaffian identities in soliton theory
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

We discuss a few fundamental Pfaffian identities in soliton theory. These include expansion formulae, addition formulae and derivative formulae for Pfaffians. The computation using Pfaffians is much simpler than using determinants in generating soliton solutions to Hirota bilinear equations.

### Friday, January 28, 2011

Title: Determinant and pfaffian solutions of Hirota bilinear equations, Part III
Speaker: Yaning Tang, Northwest Polytechnic University
P.R. China
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

We discuss some facts about determinants and Pfaffians, and explore determinant and Pfaffian solutions to Hirota bilinear equations. Laplace expansions of determinants and Jacobi identities for determinants will be used to present PlÃ¼cker relations, perfect square formulae, addition formulae and derivative formulae for determinants and Pfaffians.

### Friday, January 21, 2011

Title: Determinant and pfaffian solutions of Hirota bilinear equations, Part II
Speaker: Yaning Tang, Northwest Polytechnic University
P.R. China
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

We discuss some facts about determinants and Pfaffians, and explore determinant and Pfaffian solutions to Hirota bilinear equations. Laplace expansions of determinants and Jacobi identities for determinants will be used to present Plücker relations, perfect square formulae, addition formulae and derivative formulae for determinants and Pfaffians.

### Friday, January 14, 2011

Title: Determinant and pfaffian solutions of Hirota bilinear equations
Speaker: Yaning Tang, Northwest Polytechnic University
P.R. China
Time: 9:30am‐10:30am
Place: NES 104

#### Abstract

We discuss some facts about determinants and Pfaffians, and explore determinant and Pfaffian solutions to Hirota bilinear equations. Laplace expansions of determinants and Jacobi identities for determinants will be used to present Plücker relations, perfect square formulae, addition formulae and derivative formulae for determinants and Pfaffians.