# Research

## Differential Equations

### Wednesday, November 18, 2009

Title: Two enlarged loop algebras and their associated integrable hierarchies
Speaker: Yufeng Zhang, Mathematical School
Liaoning Normal University
Dalian, China
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

Taking a specific loop algebra, we obtain a modified Kaup-Newell hierarchy of soliton equations. Two enlarged loop algebras of the considered loop algebra are proposed and used to establish iso-spectral problems with multi-component potentials. Thus, two different types of multi-component hierarchies of soliton equations are generated, and their Hamiltonian structures are constructed by using the variational identity. The multi-component hierarchies are integrable couplings of the modified Kaup-Newell hierarchy.

### Wednesday, November 4, 2009

Title: Orthogonality of the MacDonald's Functions
Speaker: Sherwin Kouchekian
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

In this talk we show an orthogonality relation for MacDonald's functions with identical arguments but unequal complex lower indices. The orthogonality is derived heuristically and rigorously using polynomial approximation. Surprisingly this relation which comes about naturally with respect to certain PDE (Laplace's equation in spheroidal coordinate system) has not been observed before. We also give some background and motivation for the derived formula.

### Wednesday, October 28, 2009

Title: The KP hierarchy and Sato’s theory
Speaker: Wen-Xiu Ma
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

The algebra of pseudo-differential operators is used to generate the KP hierarchy within the framework of zero curvature equations and Sato's equations. The dressing operator and its wave function are the tool for constructing solutions to the KP equations and the relation of the resulting solutions with the tau-function will be presented. The extended KP hierarchy, viewed as the compatibility equations of the extended Lax equations with the KP hierarchy, will also be discussed.

### Wednesday, October 21, 2009

Title: Exact Solutions of a Nonlinear Diffusion-Convection Equation
Speaker: V. Vanaja
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

Symbolic methods are used to obtain travelling-wave solutions of the equation $$\partial u/\partial t=\left(\partial\left(u^n\right)\right)/\partial x+\left(\partial\left(u^m\right)\right)/\partial x$$ where $$n$$ and $$m$$ are integers, and $$n\ge m > 1$$. This equation models the flow of water under gravity through a homogeneous and isotropic porous medium. For certain values of $$n$$ and $$m$$ satisfying the given condition, analytical solutions of the equation as a polynomial in $$\tan(x)$$ or $$\tanh(x)$$, with integral or fractional powers are obtained, and the plots for real solutions are given. From the exact solutions, we determine whether the moisture content is positive and the seepage velocity is continuous.

### Wednesday, October 14, 2009

Title: The Carnot-Caratheodory distance, the eikonal equation and the infinite Laplacian, Part II
Speaker: Tom Bieske
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

It is known that in Euclidean $$R^n$$, the distance from the origin is smooth away from the origin and satisfies both the eikonal and infinite Laplacian equations. Generalizing to sub-Riemannian spaces, it is natural to ask if the distance from the origin still satisfies these equations. We will give a geometric condition that determines the points at which the distance satisfies these equations in an appropriate weak sense.

### Wednesday, October 7, 2009

Title: The projection method
Speaker: Florentina Tone, University of West Florida
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

In this talk I will present the results derived by M. Marion and R. Temam on the stability and error estimates of the projection method applied to the Navier-Stokes equations.

### Wednesday, September 30, 2009

Title: The Carnot-Caratheodory distance, the eikonal equation and the infinite Laplacian
Speaker: Tom Bieske
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

It is known that in Euclidean $$R^n$$, the distance from the origin is smooth away from the origin and satisfies both the eikonal and infinite Laplacian equations. Generalizing to sub-Riemannian spaces, it is natural to ask if the distance from the origin still satisfies these equations. We will give a geometric condition that determines the points at which the distance satisfies these equations in an appropriate weak sense.

### Wednesday, September 23, 2009

Title: Bi-Hamiltonian evolution equations
Speaker: Wen-Xiu Ma
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

We will discuss Hamiltonian structures and bi-Hamiltonian structures of evolution equations. A bi-Hamiltonian formulation often implies the existence of infinitely many symmetries and conservation laws, and thus, sheds light on integrability of evolution equations. Its relation with hereditary operators will be analyzed and illustrative examples of bi-Hamiltonian equations will be given.

### Wednesday, September 16, 2009

Title: Sub-equation method for finding exact solutions of nonlinear partial differential equations
Speaker: Huaitang Chen, Department of Mathematics
Linyi Normal University, China
Time: 4:30pm‐5:30pm
Place: PHY 109

#### Abstract

In the past several decades, both mathematicians and physicists have made many attempts to investigating travelling wave solutions of nonlinear partial differential equations (NLPDEs), which are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, etc. With the development of soliton theory, many effective methods have been presented.

The $$\tanh$$ method is considered to be one of the most straightforward and effective algorithm to obtain solitary wave solutions for a large NLPDEs. Since Fan presented the extended tanh method, whose key idea is to use the solutions of a Riccati equation as subequation to replace the $$\tanh$$ function in the $$\tanh$$ method, a series of rational subequation expansion method was presented, such as the Jacobi elliptic function rational expansion method, the Riccati equation rational expansion method, and the elliptic equation rational expansion method, in which the ansatzes was expressed as rational form. On the basis of the sub-equation methods, a generalized method was established by a more general form and used to construct some exact solutions of both linear and nonlinear PDEs by Profs. Ma and Lee.