Research

I am an applied analyst with broad research interests in integrable systems, orthogonal polynomials, and nonlinear waves. My expertise, in particular, is in the analysis of Riemann-Hilbert problems, which often appear in the asymptotic analysis of these systems.

For brief descriptions of past and current projects, please see below. 
For a complete list of my publications, please see Publications.

The nth particle has position q(n, t). Except when n = 0 and n = N, the nth particle in the finite Toda lattice interacts with its two nearest neighbors.  
Source: Teschl, G., "Almost everything you always wanted to know about the Toda equation"
























If the off-diagonal entries are positive and the diagonal entries are real, then the above matrix is a Jacobi matrix. 
Long time asymptotics of the finite Toda lattice
The finite Toda lattice is a mass-spring system consisting of N particles, each interacting only with its nearest neighbors. The model was proposed by Toda in 1967 to describe the motion of particles in a one-dimensional crystal [Toda]. Today, 50 years later, the model has taken center stage, serving as a canonical model in integrable systems and statistical physics.

In the case of non-periodic boundary conditions, the behavior of the particles in long time was first determined by Moser [1]. He proved that asymptotically the positions of the particles are linear functions of time. Moreover, he determined the asymptotic velocities of the particles in terms of the initial data for the system.

In joint work with R. Jenkins and K. McLaughlin [2], we used Riemann-Hilbert techniques to improve upon Moser's classical long time formulas. The techniques we used allow one, in principle, to compute the complete asymptotic expansions of the positions and velocities of the particles. In particular, we obtained explicit estimates on the associated error in Moser's classical formulas for the asymptotic positions and velocities. 

References:
[1] Moser, J., "Finitely many mass points on the line under the influence of an exponential potential -- an integrable system," pp. 467-497, Lecture Notes in Phys., Vol. 38, 1975. 
[2] Jenkins, R., McLaughlin, K., Pounder, K., "The inverse spectral problem for Jacobi matrices and applications to the finite Toda lattice," preprint


Nearly singular Jacobi matrices
The inverse spectral problem for Jacobi matrices (i.e., real, symmetric, tridiagonal matrices with positive off-diagonal entries) is to determine a Jacobi matrix from its spectral data -- i.e., its eigenvalues and spectral weights. The classical solution to this problem exploits the remarkable connection between Jacobi matrices and discrete orthogonal polynomials on the real line. 

In joint work with R. Jenkins and K. McLaughlin [1], we proved that if the spectral weights of a Jacobi matrix are of disparate sizes, then some of the off-diagonal entries of the Jacobi matrix are near zero. We call such matrices "nearly singular Jacobi matrices." The key tool in our analysis was the characterization of discrete orthogonal polynomials on the real line in terms of a family of Riemann-Hilbert problems.

References:
[1] Jenkins, R., McLaughlin, K., Pounder, K., "The inverse spectral problem for Jacobi matrices and applications to the finite Toda lattice," preprint.