SIAM-SEAS 2005 Graduate Student Papers Session GSDE
Differential Equations

Friday 2:15-4:15

Session Chair: Dr. Li Zhang, The Citadel

1. 2:15-2:30 Matthew Scott Lasater, North Carolina State University
   Simulating Resonant Tunneling Diodes with the Wigner-Poisson Equations

   Resonant tunneling diodes (RTDs) are quantum size semiconductor devices, which both theory and numerical simulation predict can sustain terahertz frequency current oscillations. The electron transport in these devices are modeled by the Wigner-Poisson equations: a nonlinear PDE which describes the time-evolution of the electrons coupled with Poisson's equation to incorporate the potential effects of the electrons. Ongoing research with an RTD involves removing it from a circuit and searching for a voltage drop across the RTD that creates these high frequency current oscillations within the device. To accomplish this, we connected our simulator to LOCA (Library of Continuation Algorithms), a software library developed at Sandia National Laboratories. These algorithms enable us to trace-out the steady-state solutions to the PDE as the voltage drop across the device is varied. An eigenvalue analysis performed by LOCA allows us to predict the development of current oscillations from just steady-state calculations. Numerical results will be presented.

2. 2:35-2:50 Robert Buckingham, Duke University
   The Long-Time Asymptotics of the Nonlinear Schrodinger Equation Shock Problem

   The long-time asymptotics of two colliding plane waves governed by the focusing nonlinear Schrodinger equation are analyzed via the inverse scattering method. The leading order terms for the three resulting regions (the initial state, a shock region, and after the shock has passed) are computed with error estimate using the steepest-descent method for Riemann-Hilbert problems. The non-decaying initial data requires a new adaptation of this method. It is found that at large times, the effect of the collision is felt in the initial state well beyond the shock front.

3. 2:55-3:10 Alan Thomas, Clemson University
   Investigating some special integrals involving associated Legendre functions

   In this presentation, we will look at several integral forms involving associated Legendre functions that come up in mathematical physics paying particular attention to one integral form that has recently arisen in modeling radiative transport. We will derive an analytic solution for the general form of this integral. Then in a few cases, we will compare analytic solutions with numerical approximations.

4. 3:15-3:30 Nick Benes, College of Charleston
   Decompositions of the KdV 2-soliton Solution

   The 1-soliton solution to the Kortweg-de Vries (KdV) equation is a bell-shaped travelling wave that moves in one dimension at a constant velocity proportional to its height. The equation also has 2-soliton solutions that look like the sum of two 1-solitons at most times. In general, the two solitons are of different heights and so they will "collide" at some point in time and then continue on, apparently unscathed. This collision has generally been interpreted in one of two ways: either as the taller soliton passing through the shorter with a phase shift during the interaction, or as a particle-like interaction with each soliton changing its height with another particle passing by them and transferring the changing mass. We consider these interpretations by presenting a survey of various methods of decomposing the 2-soliton solution into a sum of two or more functions.

5. 3:35-3:50 Silvia Madrid Jaramillo, University of Arizona, with Stéphane Lafortune and Joceline Lega
   Numerical Evaluation of the Evans function

   The Evans function may be used to study the stability of coherent structures and has applications to a wide range of problems. Instabilities are detected by locating zeros of the Evans function in the right part of the complex plane. Since in general it is difficult to calculate the Evans function explicitly, it is often useful to evaluate it numerically. I will present a method for the numerical evaluation of the Evans function. I have applied this method to the spectral stability of traveling pulse solutions for a system of two coupled nonlinear Klein-Gordon equations. This system of equations models the near-threshold dynamics of an elastic rod subject to a constant twist. I will show that the numerical results agree with an analytic criterion involving the speed of propagation and the frequency of the pulses, which is a necessary and sufficient condition for their spectral stability (S. Lafortune & J. Lega, to appear in SIAM J. Math. Analysis).

6. 3:55-4:10 Michael A. Saum, University of Tennessee Knoxville, with Ohannes Karakashian
   Implementation of an Adaptive Algorithm for a Discontinuous Galerkin Method Applied to Second Order Elliptic Problem

   Preliminary results of current research will be presented describing implementation of an adaptive algorithm used with the Discontinuous Galerkin Symmetric Interior Penalty method as applied to second order elliptic problems.