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Mathematics & Statistics

Differential Equations
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Friday, April 28, 2017

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.

Friday, April 21, 2017

Title
Speaker

Time
Place

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.

Title
Speaker


Time
Place

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.

Friday, April 14, 2017

Title
Speaker
Time
Place

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.

Friday, April 7, 2017

Title
Speaker

Time
Place

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.

Friday, March 31, 2017

Title
Speaker
Time
Place

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.

Friday, March 24, 2017

Title
Speaker
Time
Place

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.

Friday, March 17, 2017

SPRING BREAK — no seminar this week.

Friday, March 10, 2017

No seminar this week.

Friday, March 3, 2017

Title
Speaker
Time
Place

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.

Friday, February 24, 2017

Title
Speaker
Time
Place

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.

Friday, February 17, 2017

Title
Speaker


Time
Place

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.

Friday, February 10, 2017

Title
Speaker


Time
Place

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.

Friday, February 3, 2017

Title
Speaker
Time
Place

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.

Friday, January 27, 2017

Title
Speaker
Time
Place

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.

Friday, January 20, 2017

Title
Speaker
Time
Place

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.