University of Texas at Arlington

Title: Exact Solutions to Integrable Nonlinear Evolution Equations

Director: Tuncay Aktosun

Email:

aktosun@uta.edu

Dates of Program: June 4 - July 31, 2012

Summary:

A certain class of nonlinear partial differential equations and nonlinear partial differential-difference equations, known as integrable evolution equations, will be studied. The equations to be considered include the Korteweg-de Vries equation, the nonlinear SchrÃ¶dinger equation, the modified Korteweg-de Vries equation, the sine-Gordon equation, the Toda lattice equation, the Langmuir lattice equation, and the discrete nonlinear SchrÃ¶dinger equation. Some solution formulas will be constructed for certain solutions to such nonlinear equations by using a triplet of constant matrices and using matrix exponentials. Physical characteristics of those solutions will be analyzed in relation to the matrix triplet. In our summer research program, four minority students will obtain research experience by analyzing various integrable evolution equations in a systematic way, deriving explicit solution formulas for them, studying properties of such solutions, and visualizing the time evolution of those solutions by developing Mathematica codes. In addition to gaining basic research and computational skills, the student researchers will learn about opportunities at the graduate level and will gain skills to present and disseminate research findings.

The objectives are to provide the participants with meaningful research experience, to show them the enjoyment of doing research, to encourage them to pursue advanced degrees in mathematical sciences, and to increase research participation by minority groups. At the end of the program the participants will prepare a written report and give a presentation of their research. The program will be enhanced with some scientific, cultural, and social activities. In addition to participating in quality research, the student researchers will be developing basic research and computational skills, learn about opportunities at the graduate level, and gain skills to present and disseminate research findings.

Student Researchers Supported by MAA: