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# Differential Equations (Leader: Prof. Wen-Xiu Ma <mawx (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'mawx' + '&#64;' + 'usf.edu' + '">Prof. Wen-Xiu Ma</a>');)

## Wednesday, April 16, 2014

Title
Speaker

Time
Place

A $$3\times 3$$ matrix spectral problem based on the Lie algebra $$\mathrm{so}(3,R)$$
Hongcai Ma
Donghua University
Shanghai, China
4:00pm-5:00pm
NES 103

Abstract

We present a $$3\times 3$$ spectral problem, based on the three-dimensional real special orthogonal Lie algebra $$\mathrm{so}(3,R)$$, and construct a hierarchy of commuting bi-Hamiltonian soliton equations by zero curvature equations associated with the spectral problem. An illustrative example of soliton equations is computed, together with its associated bi-Hamiltonian structure.

## Wednesday, April 9, 2014

Title
Speaker
Time
Place

Introduction to complexiton Wronskian solutions to the KdV equations
Yuan Zhou
4:00pm-5:00pm
NES 103

Abstract

Complexiton solutions are a very important class of exact solutions of integrable equations. We will first give an introduction into Wronskian solutions to the KdV equations and some generalizations, and then introduce complexiton solutions of the KdV equations based on Wronskian determinants. The Wronskian technique is powerful in finding complexiton solutions and we will illustrate the technique by some examples.

## Wednesday, April 2, 2014

Title
Speaker
Time
Place

Introduction to group analysis and invariant solutions of integral-differential equations
Xiang Gu
4:00pm-5:00pm
NES 103

Abstract

We shall give an introduction into applications of group analysis to integro-differential equations (IDEs). A few methods for constructing symmetries and finding invariant solutions of IDEs, which form generalizations of the classical scheme to the construction of determining equations of an admitted Lie group, will be presented, together with applications to simple model equations.

## Wednesday, March 26, 2014

Title
Speaker

Time
Place

Generalized WKI soliton hierarchies associated with $$\mathrm{sl}(2,R)$$ and $$\mathrm{so}(3,R)$$
Shuimeng Yu
Jiangnan University
China
4:00pm-5:00pm
NES 103

Abstract

The deduction of generalized WKI spectral problems and modification terms will be discussed. The generalized WKI soliton hierarchies associated with $$\mathrm{sl}(2,R)$$ and $$\mathrm{so}(3,R)$$ will be then presented.

## Friday, March 21, 2014

Title
Speaker

Time
Place

Symbolic computation of Lax pairs of systems of partial difference equations
Willy Hereman
11:00am-12:00pm
CMC 204

Abstract

A three-step method due to Bobenko & Suris and Nijhoff to derive Lax pairs for scalar partial difference equations is extended to systems which are defined on a quadrilateral and consistent around the cube.

Lax pairs will be presented for several systems including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schrodinger and Boussinesq-type lattice systems. Previously unknown Lax pairs will be presented for systems of partial difference equations recently derived by Hietarinta. The method is algorithmic and is being implemented in Mathematica.

## Wednesday, March 5, 2014

Title
Speaker

Time
Place

A generalized Kaup-Newell hierarchy and its bi-integrable couplings
Dr. Yuqin Yao
China Agricultural University
Beijing, China
4:00pm-5:00pm
NES 103

Abstract

A new type of generalization of the Kaup-Newell spectral problem is proposed and the corresponding generalized Kaup-Newell hierarchy of soliton equations is worked out. Its Hamiltonian structures are furnished by using the trace identity. Based on a class of non-semisimple matrix loop algebra, bi-integrable couplings of the generalized hierarchy are constructed.

## Wednesday, February 26, 2014

Title
Speaker
Time
Place

λ-symmetries, μ-symmetries and integrable couplings
Wen-Xiu Ma
4:00pm-5:00pm
NES 103

Abstract

λ-symmetries for ODEs and μ-symmetries for PDEs will be recognized as generalized symmetries of particular integrable couplings, and a scheme to generate a new type of λ-symmetries and μ-symmetries will be presented by using bi-integrable couplings. A few open questions on integrable couplings will be discussed as well.

## Wednesday, February 19, 2014

Title
Speaker
Time
Place

A counterpart of the BPT Hierarchy
Emmanuel Appiah
4:00pm-5:00pm
NES 103

Abstract

Based on the Tu scheme, a Liouville integrable hierarchy (associated with $$\mathrm{so}(3,R)$$ ) of soliton equations is generated, which possesses Hamiltonian structures.

## Wednesday, February 12, 2014

Title
Speaker

Time
Place

Hidden symmetries and applications to 2D and 3D wave equations
Chunxia Li
Capital Normal University
China
4:00pm-5:00pm
NES 103

Abstract

I will first explain what hidden symmetries are and talk about theorems underlying. Then by taking the linear 2D and 3D wave equations as examples, I will explain the general process to calculate hidden symmetries.

## Wednesday, February 5, 2014

Title
Speaker

Time
Place

The λ-symmetry method for difference equations
Shoufeng Shen
Zhejiang University of Technology
4:00pm-5:00pm
NES 103

Abstract

We shall discuss the λ-symmetry method for difference equations. Applications will be given to a few lower order difference equations.

## Wednesday, January 29, 2014

Title
Speaker
Time
Place

λ-symmetries, differential invariants and reductions of differential equations
Wen-Xiu Ma
4:00pm-5:00pm
NES 103

Abstract

We shall discuss λ-symmetries, related differential invariants and applications to reductions of differential equations. Examples illustrating possible ways of applying λ-symmetries will be given and generalizations of λ-symmetries will be made from telescopic vector fields.

## Wednesday, January 22, 2014

Title
Speaker
Time
Place

A generalization of a Kaup-Newell type spectral problem
Solomon Manukure
4:00pm-5:00pm
NES 103

Abstract

A Kaup-Newell type spectral problem associated with $$\mathrm{so}(3,R)$$ is introduced and a soliton hierarchy associated with this spectral problem is found. Further, a generalization of this spectral problem is given and its generalized soliton hierarchy is also found.

## Wednesday, January 15, 2014

Title
Speaker

Time
Place

Integrable couplings and associated matrix Lie algebras
Wenying Zhang
Shanghai University
Shanghai, PR China
4:00pm-5:00pm
NES 103

Abstract

We will continue to discuss how to generate integrable couplings from zero curvature equations associated with matrix Lie algebras. The key of the study is a class of matrix Lie algebras consisting of block matrices. The corresponding Hamiltonian structures of the resulting soliton equations will be furnished by the variational identity defined over non-semisimple Lie algebras.