What am I interested in?

Other examples include the KdV equation (for which the IST theory was originally developed), the non-linear Schrodinger equation (NLS), certain versions of the modified NLS, the short pulse equation, and the Toda lattice. My advisor and I were able to prove a nice spectral confinement result for the sine-Gordon equation using a Krein signature argument (see below).
I am also interested in stability theory in PDE's and ODE's. It is important to understand the stability of certain classes of solutions to such equations since you will only expect to observe the stable solutions in physical applications. For example, consider either the focusing or de-focusing cubic non-linear Schrodinger equation (NLS) given by

This equation has a spatially periodic standing wave solution which can be expressed implicitly in terms of certain elliptic integrals. One can then study the stability of such a solution by studying the linearization of the NLS around this special solution and (upon taking the Fourier transform in time) studying the resulting spectral problem. Such techniques have been used extensively in the literature and is the focus of much of my current study.

Publications

6. On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation
   27 pages. Submitted.
   Download Arxiv preprint.

5. An Index Theorem for the Stability of Periodic Traveling Waves of KdV Type
   With Jared C. Bronski and Todd Kapitula.
   31 pages.
   Download Arxiv preprint.

4. The Transverse Instability of Periodic Waves in Zakharov-Kuznetsov Type Equations
   17 pages. Submitted.
   Download Arxiv preprint.

3. Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation
   24 pages. Submitted.
   Download Arxiv preprint.

2. The Modulational Instability for a Generalized KdV Equation
   With Jared C. Bronski
   32 pages. Submitted.
   Download Arxiv preprint.

1. Krein Signatures for the Faddeev-Tahktajan Eigenvalue Problem
   With Jared C. Bronski
   Communications in Mathematical Physics, 288 no. 3: 821-846, 2009.
   Download Arxiv preprint.

Working Papers

1. Modulational Instabilities of Quasi-Periodic Solutions to the Nonlinear Schrodinger Equation

Ph. D. Thesis

• On the Stability of Periodic Solutions of Nonlinear Dispersive Equations (.pdf).

Talks and Presentations

• 2009 AMS Spring Central Sectional Meeting, March 27-29, 2009, University of Illinois at Urbana-Champaign.
  "Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation" (.pdf).
  
  
• The XIXth International Workshop on Operator Theory and its Applications, July 22-26 2008, College of William and Mary.
  "Krein Signatures for Eigenvalue Problems Associated to Integrable Systems" (.pdf).
  
  
• 32nd SIAM Southeastern-Atlantic Section Conference (SIAM-SEAS 2008) March 14-15 2008, University of Central Florida, Orlando.
  
  
• UIUC Analysis Seminar, March 6, 2008.
  
  
• UIUC Graduate Student Seminar in Analysis, September 25, 2007.
  
  
• UIUC Graduate Student Seminar in Analysis, March 3, 2007.

Undergraduate Publications:

3. Non-Destructive Testing of Thermal Resistances for a Single Inclusion in a 2-D Domain
   With N. Christian
   31 pages. Rose-Hulman Institute of Technology Undergraduate Math Journal, Vol. 6, Issue 1, 2005.
   Download .pdf
2. Non-Destructive Testing of Thermal Resistances for a Single Inclusion in a 2-D Domain
   With N. Christian
   9 pages. Ball State Undergraduate Mathematics Exchange, Fall 2004, Vol. 2, No. 2.
   Note: This is a shortened version of the full technical report written in partial fulfillment of the requirements of the 2004 REU at Rose Hulman Institute of Technology.
   Download .pdf
1. Quantum Mechanics in Quantum Computing
   8 pages. Ball State Undergraduate Mathematics Exchange, Fall 2003, Vol. 1, No. 1.
   Download .pdf

Research Experiences