My PhD has been with my advisor Bernard Deconinck at the University of Washington. My current work has been on three projects all related to water waves. The first project involves a fully analytical analysis of stability spectrum for the focusing NLS equation. The second project involves the KdV-NLS system of equations, a model which incorporates the interaction of long and short waves. The third project involves a fully nonlinear long-wave model with constant background vorticity.
Stability spectrum for focusing NLS
For the focusing NLS equation I examined the stability spectrum for stationary elliptic-type solutions. I split the solution parameter space into four qualitatively different regions. Specifically, shown in the figure is the stability spectrum in the triple figure eight region. Additionally, I found stability for solutions with respect to perturbations of integer multiples of the period, as well as give a procedure for apporximating the greatest real part of the spectrum. This work was submitted to Physica D and can be seen here
KdV-NLS system of equations
For the KdV-NLS system, I have been collaborating with Nghiem Nguyen
at Utah State University. I found a five-parameter set of Elliptic solutions. I was able to use the methods of Robert Conte and Micheline Musette to verify that these are the only elliptic solutions to the system. I then examined the spectral stability of these solutions using the Fourier-Floquet-Hill method. Shown in the figure is a plot of the spectrum for the operator for various coupling coefficients α. Related work was published in Journal of Physics A and can be seen here
Long-wave model with constant vorticity
For the constant vorticity problem, I have been expanding on the work of Ali and Kalisch by determining an explicit parametric form solution to their equation for surface waves. In a similar vein, I have an explicit formula for the pressure at the channel bed. Previously these results were found numerically, but I can now verify them analytically. In the figure I show the pressure contours for a solution, with the red line the contour of zero pressure, above this line the pressure is below atmospheric pressure. This work was submitted for publication and can be seen here.
For my undergraduate honors thesis I worked with Alvin Bayliss and Vladimir Volpert at Northwestern University. My research explored a multiple species population system model with nonlocal coupling.
Competing populations with nonlocal interactions
I extended a previously developed population model based on the well-known diffusive logistic model with nonlocal interaction, to a system involving competing species. The model involved a system of nonlinear integro-differential equations, with the nonlocal interaction characterized by convolution integrals of the population densities against specified kernel functions. In the figure we see isolated islands of population for a specific value of nonlocal coupling δ. My work was published in Physica D and can be seen here