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Mathematics & Statistics

(Leader:

**Title**

**Speaker**

**Time**

**Place**

Linear and bilinear differential equations

Rachael Dougherty

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Bäcklund transformations and bilinear forms of the perturbation systems

Morgan Mcanally

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Bargman symmetry constraint and binary nonlinearization of soliton hierarchies

Solomon Manukure

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Lump solutions to nonlinear differential equations

Wen-Xiu Ma

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Overview of explicit quasi-periodic solutions of the Kaup–Newell hierarchy, II

Xiang Gu

1:00pm-2:00pm

NES 108

**Title**

**Speaker**

**Time**

**Place**

Overview of explicit quasi-periodic solutions of the Kaup–Newell hierarchy

Xiang Gu

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Nonlinear continuous integrable couplings of the AKNS hierarchy and their Hamiltonian structures

Yujuan Zhang

College of Mathematics and Statistics

Lanzhou University

PR China

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Bilinear Bäcklund transformations for variable-coefficient soliton equations, II

Xing Lyu

Beijing Jiao Tong University

PR China

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Tracking Control for Neuromuscular Electrical Stimulation

Michael Malisoff

Roy P. Daniels Professor of Mathematics

Louisiana State University

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Soliton solutions, Bäcklund transformation and Lax pair for a coupled Burgers system via Bell polynomials

Ömer Ünsal

University of Eskişehir Osmangazi

Turkey

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

Bilinear Bäcklund transformations for variable-coefficient soliton equations

Xing Lyu

Beijing Jiao Tong University

PR China

1:00pm-2:00pm

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.

**Title**

**Speaker**

**Time**

**Place**

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 \(\mathrm{sl}(2,R)\).