Stephanos Venakides

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8].

Approximate traveling waves in linear reaction-hyperbolic equations

Linear reaction-hyperbolic equations of a general type arising in certain physiological problems do not have traveling wave solutions, but numerical computations have shown that they possess approximate traveling waves. Using singular perturbation theory, it is shown that as the rates of the chemical reactions approach

RESONANCE AND BOUND STATES IN PHOTONIC CRYSTAL SLABS

\infty

Existence and modulation of traveling waves in particles chains

, solutions approach traveling waves. The speed of the limiting wave and the first term in the asymptotic expansion are computed in terms of the underlying chemical mechanisms.

The eigenvalue problem for the focusing nonlinear Schrödinger equations: new solvable cases

Using boundary-integral projections for time-harmonic electromagnetic (EM) fields, and their numerical implementation, we analyze EM resonance in slabs of two-phase dielectric photonic crystal materials. We characterize resonant frequencies by a complex Floquet--Bloch dispersion relation

A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

\omega = W(\beta)

Drosophila morphogenesis: tissue force laws and the modeling of dorsal closure

defined by the existence of a nontrivial nullspace of a pair of boundary-integral projections parameterized by the wave number

Boundary-Integral Calculations of Two-Dimensional Electromagnetic Scattering in Infinite Photonic Crystal Slabs: Channel Defects and Resonances

\beta

Periodic generation and propagation of traveling fronts in DC voltage biased semiconductor superlattices

and the time-frequency

Soliton turbulence as a thermodynamic limit of stochastic soliton lattices

\omega