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The Washington Research Foundation Fellowship

Nate Bottman, Mathematics - 2008-09 WRFF

Nate Bottman. 2008 WRFF recipient.I entered the UW four years ago through the Early Entrance Program.  All I knew about myself was that I loved math more than anything else, so fresh out of the first quarter of Steffen Rohde's brilliant honors calculus class, I started working with Bernard Deconinck.  Our goal was to determine the stability properties of some solutions of the nonlinear Schrödinger equation, which is a fundamental equation in mathematical physics, governing the behavior of modulated waves in nonlinear media.  The project wasn't without its obstacles --- over the first summer I spent, I was unknowingly replicating some work done in England during the 1970s --- but I kept at it, and the result of our labors is a general method for computing the stability spectra of stationary solutions of integrable equations.

Math has taken me all over the world, including to Moscow, where I spent four months in 2007.  There I both fell in love with Russia --- I'm a Russian major now, as well as a math major --- and began working with another of my collaborators, George Shabat, on another project in numerical analysis and elliptic integral theory.  I have been able to give a few talks on my work --- one in Atlanta, one in Vancouver, B.C., and too many to count at the UW, through the nonlinear waves research group --- and the hospitality and inclusiveness of the worldwide math community has always amazed me.  Thanks in part to the Washington Research Foundation, I don't plan on stopping working on unsolved math problems anytime soon.

In my spare time, I enjoy doing math, watching Russian films, baking bread, and drinking coffee.

Mentor: Bernard Deconinck, Applied Mathematics

Project Title: Analytically Determining the Spectra of Stationary Solutions of Integrable Equations

Abstract: Recent years have seen a lot of activity around the stability analysis of stationary
periodic solutions of integrable equations. Some of this work has been
analytical, some numerical, but all approaches would bene t from a method for
analytically determining the stability spectra of periodic stationary solutions of
integrable equations. Bernard Deconinck and I propose a method to completely
determine these stability spectra. This method relies on the squared-eigenfunction
connection between the stability spectrum and the Lax pair spectrum, so often used
in the soliton case. We present explicit determinations of the stability spectra of all
periodic stationary solutions of the defocusing nonlinear Schrödinger equation and
of the Korteweg-de Vries equation, and partial results towards analytically determining
the stability spectra of periodic stationary solutions of the focusing nonlinear
Schrdinger equation. Our method is relatively simple and has far-ranging implications
for how stability spectra of stationary solutions of integrable equations will
be computed in the future.