# Research

## Differential Equations

### Friday, April 24, 2015

Title: Linear and bilinear differential equations
Speaker: Rachael Doughertya
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

We will explore properties of types of bilinear differential equations, including the application of bilinear differential operators and linear subspaces of solutions. Using these properties we will compare bilinear equations with linear equations in an attempt to gain a better understanding of the relationship between these two types of differential equations.

### Friday, April 17, 2015

Title: Bäcklund transformations and bilinear forms of the perturbation systems
Speaker: Morgan Mcanally
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

We will present a class of bilinear forms and Bäcklund transformations for the perturbation systems generated from perturbations. To guarantee hereditariness from the original system to their perturbation systems, a notion of stability of bilinear structures will be introduced. We will use the KdV to model how these findings may be used.

### Friday, March 27, 2015

Title: Bargman symmetry constraint and binary nonlinearization of soliton hierarchies
Speaker: Solomon Manukure
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

A symmetry constraint for a soliton hierarchy associated with the real special linear algebra $$\mathrm{sl}(2,R)$$ is derived, and the Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite-dimensional Hamiltonian systems, integrable in Liouville’s sense.

### Friday, March 20, 2015

Title: Lump solutions to nonlinear differential equations
Speaker: Wen-Xiu Ma
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

We shall talk about lump solutions to nonlinear differential equations. Through Hirota bilinear forms, positive quadratic functions are used to present lump solutions. Examples of lump solutions to integrable equations will be shown, together with Maple computational algorithms.

### Friday, March 13, 2015

Title: Overview of explicit quasi-periodic solutions of the Kaup–Newell hierarchy, II
Speaker: Xiang Gu
Time: 1:00pm‐2:00pm
Place: NES 108

### Friday, February 27, 2015

Title: Overview of explicit quasi-periodic solutions of the Kaup–Newell hierarchy
Speaker: Xiang Gu
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

I’ll talk about an algebraic curve of an arbitrary arithmetic genus introduced by Geng et al. using the spectral matrix of the Kaup–Newell hierarchy, based on which meromorphic functions on the resulting algebraic curve are constructed and their properties are analyzed. All the flows associated with the Kaup–Newell hierarchy are then straightened out under the Abel–Jacobi coordinates, and the explicit quasi-periodic solutions are generated for the whole Kaup–Newell hierarchy including the coupled derivative nonlinear Schrödinger equations.

### Friday, February 13, 2015

Title: Nonlinear continuous integrable couplings of the AKNS hierarchy and their Hamiltonian structures
Speaker: Yujuan Zhang, College of Mathematics and Statistics
Lanzhou University, PR China
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

I will talk about a procedure for constructing nonlinear continuous integrable couplings of the AKNS hierarchy and their Hamiltonian structures. The basic tools are semi-direct sums of Lie algebras and the associated variational identities under bilinear forms. Staring from a special non-semisimple Lie algebra, I will present an example of this procedure.

### Friday, February 6, 2015

Title: Bilinear Bäcklund transformations for variable-coefficient soliton equations, II
Speaker: Xing Lyu, Beijing Jiao Tong University
PR China
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

I will talk about Bäcklund transformations (BTs), bilinear operators and exchange formulas for soliton equations. Taking the variable-coefficient two-dimensional KdV equation as an example, I will particularly discuss the relationship between its bilinear BT and its Lax pair.

### Friday, January 30, 2015

Title: Tracking Control for Neuromuscular Electrical Stimulation
Speaker: Michael Malisoff, Roy P. Daniels Professor of Mathematics
Louisiana State University
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

We present a new tracking controller for neuromuscular electrical stimulation, which is an emerging technology that artificially stimulates skeletal muscles to help restore functionality to human limbs. We use a musculoskeletal model for a human using a leg extension machine. The novelty of our work is that we prove that the tracking error globally asymptotically and locally exponentially converges to zero for any positive input delay for a general class of possible reference trajectories that must be tracked, coupled with our ability to satisfy a state constraint imposed by the physical system. The state constraint in the dynamics is that for a seated subject, the human knee cannot be bent more than plus or minus 90 degrees from the straight down position. Also, our controller only requires sampled measurements of the states instead of continuous measurements and allows perturbed sampling schedules, which can be important for practical applications where sampled measurements of the states are available, but where continuous measurements of the states is not possible. Our work is based on a new method for constructing predictor maps for a large class of nonlinear time-varying systems, which is of independent interest. Prediction is a key method for delay compensation that uses dynamic control to compensate for arbitrarily long input delays.

### Friday, January 23, 2015

Title: Soliton solutions, Bäcklund transformation and Lax pair for a coupled Burgers system via Bell polynomials
Speaker: Ömer Ünsal, University of Eskişehir Osmangazi
Turkey
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

Based on the Bell polynomial theory of integrability, we will discuss Hirota's bilinear form and soliton solutions of a coupled Burgers system. We will also present Bäcklund transformations both in binary Bell polynomial form and bilinear form. The obtained Bäcklund transformation is used to derive a Lax pair.

### Friday, January 16, 2015

Title: Bilinear Bäcklund transformations for variable-coefficient soliton equations
Speaker: Xing Lyu, Beijing Jiao Tong University
PR China
Time: 1:00pm‐2:00pm
Place: NES 108

#### Abstract

I will talk about Bäcklund transformations (BTs), bilinear operators and exchange formulas for soliton equations. Taking the variable-coefficient two-dimensional KdV equation as an example, I will particularly discuss the relationship between its bilinear BT and its Lax pair.

### Friday, January 9, 2015

Title: Selected topics in integrable couplings
Speaker: Yuan Zhou
Time: 2:30pm‐3:30pm
Place: CMC 116

Selected topics in integrable couplings
Yuan Zhou
2:30pm-3:30pm
CMC 116

#### Abstract

I will talk about the concept of integrable couplings, non-semisimple Lie algebras, matrix loop algebras, and the general procedure for generating soliton hierarchies from Lax pairs. I will also discuss examples of integrable couplings by using irreducible representations of $$\mathrm{sl}(2,R)$$.