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Mathematics & Statistics

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**Title**

**Speaker**

**Time**

**Place**

Principal chiral field equations and their solitons

Sili Zuo

Northwest University

Xian, China

1:00pm-2:00pm

CHE 302

**Abstract**

The principal chiral field is one of the classical models of the field theory having geometrical meaning. We shall talk about how the principal chiral field equation can be derived from the Lagrangian density on the special unitary group. The principal chiral field equation also admits an integral Lax pair, and so the theory of the principal chiral field is quiet similar to the theory of \(N\)-waves. We shall briefly outline soliton solutions of the principal chiral field equation by the Riemann-Hilbert method.

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Matrosov's Approach

Michael Malisoff

Louisiana State University

2:00pm-3:00pm

CHE 302

**Abstract**

The construction of strict Lyapunov functions is important for proving stability and robustness properties for nonlinear control systems. The Matrosov approach involves combining known nonstrict Lyapunov functions for nonlinear systems with one or more so-called auxiliary functions, to build strict Lyapunov functions. This seminar will present a method for building the required auxiliary functions, based on iterated Lie derivatives, and will include an application to a Lotka-Volterra dynamics. It will include the relevant definitions to make it understandable to graduate students and others who are familiar with the basic theory of nonlinear ordinary differential equations.

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**Speaker**

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Systems integrable by the Riemann problem technique

Hai-Qiang Zhang

University of Shanghai for Science and Technology

PR China

1:00pm-2:00pm

CHE 302

**Abstract**

The gauge transformation for an over-defined system of differential equations is introduced. New solutions of the system are constructed by the gauge transformation and the Riemann-Hilbert problem. Moreover, we’ll talk about how to dress the bare solution with the Riemann-Hilbert problem.

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**Speaker**

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Solitons and their universal interactions

Jing Tian

1:00pm-2:00pm

CHE 302

**Abstract**

We will study a procedure for solving an integrable system of the N-wave problem. By studying a Riemann-Hilbert problem related to the scattering data of an associated spectral problem, we will compute diverse soliton solutions and explore their universal interaction phenomena. Cases of the breakdown of a superposition soliton and the merge of elementary solitions will be discussed.

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**Speaker**

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Integrable reductions of the \(N\)-wave problem

Xuelin Yong

North China Electric Power University

1:00pm-2:00pm

CHE 302

**Abstract**

A few special integrable reductions that are consistent with the \(N\)-wave equations are discussed. The physically relevant and important local \(N\)-wave equations which describe interaction of multiple waves of different types are presented. Moreover, new nonlocal symmetry reductions which yield systems of multi-particle interaction equations are also derived.

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**Speaker**

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Riemann-Hilbert problems in solving inverse scattering problems for 1st-order matrix spectral systems

Xiang Gu

1:00pm-2:00pm

CHE 302

**Abstract**

We shall briefly review how the Riemann-Hilbert problem is used in solving the inverse scattering problem for a first-order matrix spectral system. The potential matrix will be retrieved from the asymptotic behavior of the unique solution to the normalized Riemann-Hilbert problem.

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Riemann-Hilbert problems with zeros

Fudong Wang

1:00pm-2:00pm

CHE 302

**Abstract**

I will talk about how to solve Riemann-Hilbert Problems with zeros. The canonical normalization problems will be considered. The projection operators will be basic tools in presenting unique solutions.

SPRING BREAK — no seminar this week.

No seminar this week.

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**Speaker**

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Unitary invariant integrable systems with multiple components

Ahmed Ahmed

1:00pm-2:00pm

CHE 302

**Abstract**

A general construction of bi-Hamiltonian integrable systems from inelastic curve flows in symmetric spaces will be talked about, which allows us to derive multi-component integrable systems of mKdV-type, NLS-type and sine-Gordon type. Three examples are associated with Hermitian spaces: \(\mathrm{SU}(n+1)/\mathrm{U}(n)\), \(\mathrm{SO}(n+2)/SO(n)\times\mathrm{SO}(2)\) and \(\mathrm{SO}(2n)/\mathrm{U}(n)\), and the resulting integrable systems in these spaces exhibit unitary invariance.

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**Speaker**

**Time**

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Applications of a refined invariant subspace method to two systems of evolution equations

Sumayah Batwa

1:00pm-2:00pm

CHE 302

**Abstract**

The invariant subspace method is one of the existing approaches to exact solutions of PDEs. In this talk, the invariant subspace method is refined and further applied to two-component nonlinear systems. Exact solutions with generalized separated variables are presented for the two considered systems.

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**Speaker**

**Time**

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Binary Darboux transformation for the coupled Sasa-Satsuma equations

Hai-Qiang Zhang

University of Shanghai for Science and Technology

Shanghai, China

1:00pm-2:00pm

CHE 302

**Abstract**

A binary Darboux transformation is constructed for the coupled Sasa-Satsuma equations, and its \(N\)-times iterative version is expressed in terms of quasideterminants. The resulting transformation allows one to generate a series of explicit exact solutions from either vanishing or non-vanishing backgrounds. The breather, single- and double-hump bright vector solitons, and anti-dark solitons are presented from the once-iterated transformation.

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**Speaker**

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Rarefaction problem of the focusing nonlinear Schrödinger equation

Deng-Shan Wang

Beijing Information Science and Technology University

Beijing, China

1:00pm-2:00pm

CHE 302

**Abstract**

In this talk, the long-time asymptotic behaviors of two separable plane waves of the focusing nonlinear Schrödinger equation are analyzed via the Riemann-Hilbert formulation. It is found that there are two asymptotic regions in space-time: the plane-wave region and elliptic wave region with a one-phase wave. The leading-order terms for the two regions are computed with error estimates using the steepest-descent method for Riemann-Hilbert problems.

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A Riemann-Hilbert problem for solving a generalized Sasa–Satsuma equation

Yuan Zhou

1:00pm-2:00pm

CHE 302

**Abstract**

A generalized Sasa–Satsuma equation, associated with a \(3\times 3\) matrix spectral problem, will be solved by the Riemann–Hilbert approach. The spectral analysis of its Lax pair will be first discussed, and \(N\)-soliton solutions will be then obtained by solving a particular Riemann–Hilbert problem with vanishing scattering coefficients.

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**Speaker**

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From matrix loop algebras to integrable Hamiltonian equations

Wen-Xiu Ma

1:00pm-2:00pm

CHE 302

**Abstract**

We address the problem of generating integrable Hamiltonian equations via matrix loop algebras. Lax pairs and semi-direct sum decompositions are basic tools in the generating formulation. Hamiltonian structures are furnished by either the trace identity or the variational identity, thereby yielding infinitely many conservation laws.

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**Speaker**

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An application of the Riemann-Hilbert approach to the Harry Dym equation

Xiang Gu

1:00pm-2:00pm

CHE 302

**Abstract**

A brief introduction is given on the Riemann-Hilbert approach. A recent work will be discussed, which applies the Riemann-Hilbert approach to solve the Harry Dym equation on the real line. Major focus will be on how to construct the solution of a decay initial value problem via the corresponding \(2\times 2\) matrix Riemann-Hilbert problem on the complex plane.