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Mathematics & Statistics

(Leader: Prof. Wen-Xiu Ma)

**Title**

**Speaker**

**Time**

**Place**

The Hadamard variational formula and connection of the Laplacian growth problem to soliton theory

Erik Lundberg

1:00pm-2:00pm

NES 103

**Abstract**

We would like to gain some understanding of several recent papers revealing a connection between Laplacian growth and integrable hierarchies in soliton theory. The Laplacian growth problem is to describe the evolution of a growing air bubble surrounded by oil confined between two plates. We choose as our starting point to review the first order variation of the Green's function of a domain with respect to a perturbation of its boundary. This formula goes back to Hadamard. In the context of moving boundaries related to the Laplacian growth problem, the Hadamard variational formula leads to a set of commuting flows with respect to infinitely many “times”. This resembles the integrable hierarchies studied in soliton theory, and in fact it can be transformed to equations which exactly correspond to well-known instances of such hierarchies.

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Solutions of the Extended Kadomtsev-Petviashvili-Boussinesq Equation by the Hirota Direct Method

Asli Pekcan

Scientific and Technological Research Council of Turkey

Ankara, TURKEY

1:00pm-2:00pm

NES 103

**Abstract**

We show that the Hirota direct method can be applied to some non-integrable equations in higher dimensions. For this purpose, we consider the extended Kadomtsev-Petviashvili-Boussinesq (eKPBo) equation with \(M\) variables: $$ \left(u_{xxx}-6uu_x\right)+a_{11}u_{xx}+2\sum_{k=2}^M a_{1k}u_{xx_{i}}+\sum_{i,j=2}^M a_{ij}u_{x_{i}x_i}=0, $$ where \(a_{ij}=a_{ji}\) are constants and \(\left(x_i\right)=\left(x,t,y,z,\dotsc,x_M\right)\). We will give the results on existence of three-solitons for \(M=3\) and \(M=4\) with a detailed analysis. Then we will discuss generalization of the resulting results to any larger integer \(M > 4\).

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**Speaker**

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Integrable Peakon and Cuspon Equations

Zhijun (George) Qiao

University of Texas-Pan American

1:00pm-2:00pm

NES 103

**Abstract**

In my talk, I will introduce integrable peakon and cuspon equations and present a basic approach how to get peakon solutions. Those equations include the well-known Camassa-Holm (CH), the Degasperis-Procesi (DP), and other new peakon equations with M/W-shape peakon soutions. I take the CH case as a typical example to explain the details. My presentation is based on my previous work (The Camassa-Holm Hierarchy, \(N\)-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold, Communications in Mathematical Physics 239, 309-341). I will show that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in:

- a new Neumann-like \(N\)-dimensional system when it is restricted into a symplectic submanifold of \(R^{2N}\), which is proven to be integrable by using the Dirac-Poisson bracket and the \(r\)-matrix process; and
- a new Bargmann-like \(N\)-dimensional system when it is considered in the whole \(R^{2N}\), which is proven to be integrable by using the standard Poisson bracket and the \(r\)-matrix process.

The whole CH hierarchy (both positive and negative orders) is shown to have the parametric solutions, which obey the corresponding constraint relation. In particular, the CH equation, constrained to a symplectic submanifold in \(R^{2N}\), has the parametric solutions. Moreover, solving the parametric representation of the solution on the symplectic submanifold gives a class of a new algebro-geometric solution of the CH equation. In the end of my talk, some open problems are also addressed for discussion.

**Title**

**Speaker**

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On Some Algebraic Properties of Semi-Discrete Hyperbolic Type Equations

Asli Pekcan

Scientific and Technological Research Council of Turkey

Ankara, TURKEY

1:00pm-2:00pm

NES 103

**Abstract**

Nonlinear semi-discrete equations of the form \(\partial t/\partial x(n+1)=f(t(n),t(n+1),\partial t/\partial x(n))\) are studied. An adequate algebraic formulation of the Darboux integrability is discussed and the attempt to adopt this notion to the classification of Darboux integrable chains has been undertaken.

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Six Expanding Integrable Models of the Nonlinear Schrödinger Equation

Yufeng Zhang

Mathematical School

Liaoning Normal University

P. R. China

1:00pm-2:00pm

NES 103

**Abstract**

Five Lie algebras are introduced, for which six integrable hierarchies are generated under the frame of zero curvature equations. Their reduced cases exhibit six expanding integrable models of the nonlinear Schrödinger equation, including a perturbation nonlinear expanding integrable model, in which the potential functions are all nonlinear.

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Hamiltonian Structures of Soliton Equations

Wen-Xiu Ma

1:00pm-2:00pm

NES 103

**Abstract**

We will talk about how to determine Hamiltonian structures for soliton equations associated with semi-direct sums of Lie algebras, with a focus on specific unsolved problems. The basic techniques are the variational and component-trace identities.

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Properties of Index Transform in Modeling of Nanostructures

Sherwin Kouchekian

1:00pm-2:00pm

NES 103

**Abstract**

We will discuss modeling of nanostructures by considering certain PDE boundary value problem. Also we will show the use of index transform such as Mehler-Fock and Kontrovich-Lebedev in these situations.

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Approximation of solutions of PDEs by pure exponentials

Boris Shekhtman

1:00pm-2:00pm

NES 103

**Abstract**

Let \(G\) be an \(N\)-dimensional space formed by the solutions of a homogeneous system of PDEs with constant coefficients. It is obvious that, in one variable, this space can be approximated by the linear combination of \(N\) exponential functions. It is far less obvious that the same is true in two variables. It is surprising that the result is false in three or more variables. These and several related issues are to be discussed in the talk.

**Title**

**Speaker**

**Time**

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Lie Algebraic Method of Integrable Couplings

Yufeng Zhang

Mathematical School

Liaoning Normal University

P. R. China

1:00pm-2:00pm

NES 103

**Abstract**

The talk focuses on the Lie algebraic method of generating integrable couplings. We will recall the concept of integrable couplings, and briefly show the importance of studying integrable couplings. The problem of coupling integrable couplings will be also discussed, and several examples will be given.

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Some open problems involving differential equations

Arcadii Grinshpan

1:00pm-2:00pm

NES 103

**Abstract**

The areas of discussion will include fuel injection, homeomorphisms of the complex plane, and univalent functions in the unit disk. I am going to focus on the three open problems that involve ordinary and partial differential equations. Partial results will be given.

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Construction of Soliton Solutions to Nonlinear Partial Differential Equations

Huaitang Chen

Linyi Normal University

P. R. China

1:00pm-2:00pm

NES 103

**Abstract**

Exact solutions of nonlinear partial differential equations (PDEs) play an important role in applications in the physical sciences. It is an important problem in mathematical physics how to construct soliton solutions of nonlinear PDEs. I would like to talk about the following ways: (a) to change nonlinear PDEs into linear ones by function transformations, (b) to turn PDEs into ordinary ones, (c) to change higher dimensional PDEs into lower dimensional ones, and (d) to change complicated PDEs into simpler ones. Several examples will be given to illustrate those methods.

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The Transformed Rational Function Method for Nonlinear Equations

Wen-Xiu Ma

1:00pm-2:00pm

NES 103

**Abstract**

The general idea behind direct and exact methods for nonlinear equations is to decompose partial differential equations into integrable ordinary differential equations. The so-called transformed rational function method provides a systematical and convenient handling of the solution process of nonlinear equations, unifying many existing methods such as the tanh-function method, the homogeneous balance method, the exp-function method and the mapping method. Its key point is to search for rational solutions to the resulting variable-coefficient ordinary differential equations. Applications to higher-dimensional problems show the diversity of exact solutions to nonlinear equations.