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Mathematics & Statistics

Differential Equations
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Tuesday, February 19, 2019

Title
Speaker
Time
Place

Riemann-Hilbert problems for two-component coupled mKdV systems
Fudong Wang
3:00pm-4:00pm
SOC 286

Abstract

We will discuss how to generate two-component coupled mKdV systems from the \(3\times 3\) matrix AKNS spectral problem. Based on the inverse scattering method, we will first derive the corresponding Riemann-Hilbert problems to recover the potentials and then long-time asymptotics of the combined mKdV systems by the steepest descent method.

Tuesday, February 12, 2019

Title
Speaker

Time
Place

Dynamical analysis of a laminated composite piezoelectric rectangular plate
Ni Song
North University of China
3:00pm-4:00pm
SOC 286

Abstract

The subharmonic Melnikov method is improved to investigate subharmonic orbits of a laminated composite piezoelectric rectangular plate in the case of a 1:2:4 internal resonance. Numerical simulation also shows the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.

Tuesday, February 5, 2019

Title
Speaker
Time
Place

Inverse scattering via Riemann-Hilbert problems
Wen-Xiu Ma
3:00pm-4:00pm
SOC 286

Abstract

We will talk about the inverse scattering transform through Riemann-Hilbert problems to integrable equations. Matrix Riemann-Hilbert problems on the real axis are the basis, and soliton solutions are generated while an identity jump matrix is taken. An illustrative example will be presented for a coupled system of modified Korteweg-de Vries equations.

Tuesday, January 22, 2019

Title
Speaker
Time
Place

The Hamiltonian formalism of soliton PDEs
Wen-Xiu Ma
3:00pm-4:00pm
SOC 286

Abstract

The Hamiltonian formalism will be discussed for soliton partial differential equations — a kind of partial differential equations generated from zero curvature equations. The basic tools are the trace identity over semisimple Lie algebras and the variational identity over non-semisimple Lie algebras. Illustrative examples associated with the AKNS spectral problem will be presented.

Tuesday, January 15, 2019

Title
Speaker
Time
Place

Asymptotic solutions of the nonlinear Schrödinger equation based on conservation laws
Fudong Wang
3:00pm-4:00pm
SOC 286

Abstract

We will discuss an asymptotic solution of the nonlinear Schrödinger equation in the solitonless region, which has the decay \(t^{-1/2}\) in time. The asymptotic solution contains two arbitrary functions in the amplitude and phase, respectively. The amplitude function can be uniquely determined by conservation laws, but the phase function is undermined. The method, introduced by Segur and Ablowitz, determines the leading two terms in each asymptotical expansion and can avoid using the Marchenko integral equations.