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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>');)

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.