Research Interests

My interests include integrable systems, Riemann-Hilbert problems, the analysis of boundary-value problems and collocation methods. The bulk of my PhD research has centered around numerical nonlinear steepest descent --- the implementation of the Deift and Zhou method of nonlinear steepest descent numerically. I have also worked on stability problems in laser physics, contributed to the development of the Unified Transform Method and developed an alternate characterization of the so-called finite-genus solutions of the Korteweg-de Vries equation. Short descriptions of projects can be found below.

Numerical Nonlinear Steepest Descent

The effective computation of the inverse scattering transform, orthogonal polynomials with exponential weights, the Painlevé transcendents and other nonlinear special functions is accomplished by combining the Deift and Zhou method of nonlinear steepest descent with a method for the numerical solution of Riemann-Hilbert problems. Under mild assumptions on the numerical method, this combination produces an appoximation that is uniformly and spectrally convergent over large parameter ranges. Below we plot a solution of the Korteweg-de Vries (KdV) equation while simultaneously displaying the contours of the Riemann--Hilbert problem that is being solved to compute the value of the solution q at each point.

Finite-Genus KdV

Riemann-Hilbert methods are also used to derive representations of the finite-genus solutions of the KdV equation. This presents an alternative to the now-classical theta function approach. The Riemann-Hilbert problem is derived by representing scalar-valued analytic functions on a hyperelliptic Riemann surface by vector-valued functions on a cut version of the complex plane. These vector-valued functions satisfy a Riemann--Hilbert problem which can be solved numerically. An animation of a genus two solution of the KdV equation is found here:

A genus two solution of the KdV equation

Instability of Spatial Solitons

Instabilities of spatial solitons in a (2+1)-dimensional nonlinear Schrödinger equation have been well-studied since the work of Zakharov and Rubenchik in 1979. Through accurate computations using Hill's method we closely examine the growth rate of instabilities. Hill's method allows us to not only capture the maximal growth rate in the so-called snake and oscillatory snake regions but also the growth rate for the sub-dominant oscillatory neck instability. These rates are experimentally demonstrated by the group of M. Haelterman using a 2D waveguide array. See below for the computations of these instabilities. We animate the evolution of the equilibrium solution plus the first-order correction obtained using Hill's method.

Evolution of the neck instability
Evolution of the oscillatory snake instability