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Mathematics & Statistics

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**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.

SPRING BREAK — no seminar this week.

No seminar this week.

**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.

**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.

**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.

**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.

**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.

**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.

**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.