University of Texas at Arlington

Title: Exact Solutions to Nonlinear Partial Differential Equations

Director(s): Tuncay Aktosun, Minerva Cordero

Dates of Program: June 3-July 15, 2007

Summary: The research topic is exact solutions to two nonlinear partial differential equations (PDEs), known as the Korteweg-de Vries (KdV) equation u_t-6uu_x+u_xxx=0 and the nonlinear SchrÃ¶dinger (NLS) equation iu_t+u_xx+2|u|²u=0. The goal is to obtain exact solutions to these equations in a systematic way in terms of elementary functions (exponential, trigonometric, and polynomial functions of x and t). Exact solutions will be constructed in terms of triplets A, B, C, where A is a constant pÃ?p matrix, B is a constant column p-vector, and C is a constant p-row vector, where p is any positive integer. Such solutions involve matrix exponentials. In the special case p=1 such solutions include the so-called one-soliton solution. The special case where A is a diagonal matrix includes all multisoliton solutions. Besides working on the theoretical and analytical aspects, the participants will use the symbolic software Mathematica to animate and display such solutions explicitly.