## MEMS Actuator Analysis

Brenda McLellan, Chenmei Xu, Prof. Yisong Yang, and I completed a joint research project titled
which has been submitted for publication.

In this project, we study the dynamical behavior of an undamped electrostatic MEMS actuator with one-degree of freedom subject to a Casimir force. In such a situation, the well-known mathematical difficulty associated with an inverse quadratic term due to a Coulomb force is supplemented with an inverse quartic term due to the joint application of a Casimir force. We show that the small Coulomb and Casimir force situations, described by sufficiently low values of two positive parameters, $\lambda$ and $\mu$, respectively, are characterized by one-stagnation-point periodic motions and there exists a unique critical pull-in curve in the $(\lambda,\mu)$ coordinate quadrant beyond which a finite-time touch down or collapse of the actuator takes place. We demonstrate how to locate and approximate the pull-in curve. When mechanical nonlinearity such as that due to the presence of a cubic elastic force term is considered in the equation of motion, we show that a similar three-phase oscillation-pull-in-finite-time-touchdown phenomenon occurs and that pull-in curves are actually enhanced or elevated by nonlinear elasticity.

Research in this project was motivated by the recent paper, "Dynamics of Electrostatic MEMS Actuators", of Yisong Yang, R.  Zhang, and L. Zhao.

## Modeling Electron-Phonon Interactions in Fullerenes

Watson Markson, together with Joseph Esposito and I started a project studying the family of molecules known as "fullerenes"-molecules consisting entirely of Carbon atoms lying in a grid, which is of interest in the field of nanotechnology. Applications vary through distinct fields of engineering, from electrical to optical to biomedical. Electrical engineers hope fullerene-based transistors will replace today’s silicon-based ones and possibly provide high-temperature
superconductors; cancer researchers hope to use fullerenes to aid in radiation therapy’s accuracy, and optical engineers use fullerenes to make the blackest materials ever made.

It is thought that many of the unique properties exhibited by fullerenes, that are of interest because of their wide array of applications, are caused by phonon-electron interactions on the grid of Carbon atoms. We study a modified nonlinear Schrödinger equation modeling the phonon-electron interactions. Using tools from nonlinear functional analysis and solitons in field theory we attempt to develop a mathematical existence theory for this model. We also attempt to give a numerical description of the solutions.

## Dynamics of BPS Equations Modeling Kink Solitons in a Monopole Confinement Problem

Alex Arakelian and Errol Elbasan studied kink solitons arising in the study of Auzzi, Bolognesi, and Shifman presented in a recent study by Chen, Li, and Yang of a  monopole confinement problem in $N=2$ supersymmetric QCD. From the kink energy functional, Alex and Errol obtained the corresponding Euler-Lagrange equations, which are a second order coupled nonlinear system of differential equations. Then using the well known BPS trick (named after the seminal works of Bogomol'nyi, Prasad, and Sommerfield) they re-derived a lower bound of the kink energy and reduced the 2nd order system to a first-order coupled nonlinear system of differential equations of the BPS type.