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# MY RESEARCH

*PhD*

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.

*Undergrad*

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.

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