# Research

## Differential Equations

### Tuesday, April 30, 2019

Title: Darboux transformations on integrable systems
Speaker: Morgan McAnally, The University of Tampa
Time: 3:00pm‐4:00pm
Place: SOC 286

#### Abstract

In this seminar, we will discuss the basic idea of a Darboux transformation. Then we see how it was applied to the KdV. This led to an application to a whole hierarchy of partial differential equations, the AKNS system. We will discuss how the AKNS system is constructed as well as a generalized D-Kaup-Newell system. Lastly, we apply Darboux transformations to get explicit solutions.

### Tuesday, April 23, 2019

Title: Inverse scattering on the line and related Gelfand-Levitan-Marchenko integral equation
Time: 3:00pm‐4:00pm
Place: SOC 286

#### Abstract

In this seminar, I am going to talk about how to generalize the Storm-Liouville problem to a direct and inverse scattering problem on the line, through the AKNS-ZS scheme using a $$2\times2$$ eigenvalue problem, in order to solve the KdV and NLS equations.

### Tuesday, April 9, 2019

Title: Existence of global weak solutions to a system of barotropic compressible Navier-Stokes flows with degenerate viscosity coefficients
Speaker: Liqin Zhang, Xiamen Institute of Technology
China
Time: 4:00pm‐5:00pm
Place: SOC 286

#### Abstract

In this seminar, I will discuss about the global existence of weak solutions of a system of barotropic compressible Navier-Stokes equations possessing degenerate viscosities in two or three dimensional periodic domains. The result also answered an open problem on fluid mechanics proposed by P. L. Lions in 1998.

### Tuesday, April 2, 2019

Title: Introduction to Quantum Mechanics — II
Speaker: Xiang Gu
Time: 4:00pm‐5:00pm
Place: SOC 286

#### Abstract

As a follow-up to last week's seminar, I will first briefly introduce the rigged Hilbert space in Quantum Mechanics. Then by reviewing the simple but very classical model of harmonic oscillators, it is expected that we could see how a classical mechanic system is “quantized” when switched to the world of quantum mechanics.

### Tuesday, March 26, 2019

Title: Introduction to Quantum Mechanics — I
Speaker: Xiang Gu
Time: 4:00pm‐5:00pm
Place: SOC 286

#### Abstract

Starting from the wave-particle-duality, we shall briefly discuss why the description of quantum mechanics differs so much from classical mechanics. Next, we will review the fundamental postulates of quantum mechanics (postulates in physics are just like axioms in mathematics) in both languages of physics and mathematical analysis (functional interpretations in Hilbert space).

### Tuesday, March 19, 2019

Title: Long-time asymptotics via nonlinear steepest decent
Speaker: Wen-Xiu Ma
Time: 3:00pm‐4:00pm
Place: SOC 286

#### Abstract

We will talk about long-time asymptotics for nonlinear equations integrable by the inverse scattering transform. The basis is a kind of oscillatory Riemann-Hilbert problems generated from matrix spectral problems, and deforming the associated Riemann-Hilbert problems via the nonlinear steepest descent method lead to the required asymptotics for integrable equations. Applications to vector modified Korteweg-de Vries equations will be discussed.

### Tuesday, March 12, 2019

No seminar this week due to Spring Break.

### Tuesday, March 5, 2019

Title: Darboux transformations and analytic solutions for a type of NLS equations with higher order nonlinear terms
Fenghua Qi
Speaker: Fenghua Qi, Beijing Wuzi University
Time: 3:00pm‐4:00pm
Place: SOC 286

#### Abstract

In some cases, higher-order nonlinear terms are required to be incorporated into the nonlinear Schröedinger equation to describe certain physical phenomena. We will discuss the construction of Darboux transformations for a type of nonlinear Schrödinger equations with higher order nonlinear terms. Based on the obtained Darboux transformations, various analytic solutions will be generated.

### Tuesday, February 26, 2019

Title: Riemann-Hilbert problems for two-component coupled mKdV systems, Part II
Speaker: Fudong Wang
Time: 3:00pm‐4:00pm
Place: SOC 286

### Tuesday, February 19, 2019

Title: Riemann-Hilbert problems for two-component coupled mKdV systems
Speaker: Fudong Wang
Time: 3:00pm‐4:00pm
Place: 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: Dynamical analysis of a laminated composite piezoelectric rectangular plate
Speaker: Ni Song, North University of China
Time: 3:00pm‐4:00pm
Place: 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: Inverse scattering via Riemann-Hilbert problems
Speaker: Wen-Xiu Ma
Time: 3:00pm‐4:00pm
Place: 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: The Hamiltonian formalism of soliton PDEs
Speaker: Wen-Xiu Ma
Time: 3:00pm‐4:00pm
Place: 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: Asymptotic solutions of the nonlinear Schrödinger equation based on conservation laws
Speaker: Fudong Wang
Time: 3:00pm‐4:00pm
Place: 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.