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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.

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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.

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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.

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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.

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Symbolic computation of Lax pairs of systems of partial difference equations

Willy Hereman

Colorado School of Mines

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.