Finite-time collapse of N classical fields described by coupled nonlinear Schrödinger equations.

We prove the finite-time collapse of a system of N classical fields, which are described by N coupled nonlinear Schrödinger equations. We derive the conditions under which all of the fields experiences this finite-time collapse. Finally, for two-dimensional systems, we derive constraints on the number of particles associated with each field that are necessary to prevent collapse.

Conical refraction and nonlinearity.

We study conical refraction in crystals where both diffraction and nonlinearity are present. We develop a new set of evolution equations. We find that nonlinearity induces a modulational instability when it is defocussing as well as focussing. We also examine the evolution of incident beams which contain analytic singularities, and in particular optical vortices, which do not feel the effect of conical refraction.

Polygonal planforms and phyllotaxis on plants.

We demonstrate how phyllotaxis (the arrangement of leaves on plants) and the ribbed, hexagonal, or parallelogram planforms on plants can be understood as the energy-minimizing buckling pattern of a compressed sheet (the plant's tunica) on an elastic foundation. The key idea is that the elastic energy is minimized by configurations consisting of special triads of periodic deformations. We study the conditions that lead to continuous or discontinuous transitions between patterns, state testable predictions, and suggest experiments to test the theory.

Coherent structures and entropy in constrained, modulationally unstable, nonintegrable systems.

Many studies have shown that nonintegrable systems with modulational instabilities constrained by more than one conservation law exhibit universal long time behavior involving large coherent structures in a sea of small fluctuations. We show how this behavior can be explained in detail by simple thermodynamic arguments.

Laser-induced breakdown versus self-focusing for focused picosecond pulses in water.

We present numerical studies of nonlinear propagation for picosecond pulses focused in water. Depending on the pulse duration and focusing conditions, for some input powers self-focusing may precede laser-induced breakdown and vice versa. We derive a criterion that predicts the relative roles of laser-induced breakdown and self-focusing.

Short-pulse conical emission and spectral broadening in normally dispersive media.

The nonlinear Schrödinger equation predicts conical emission that is due to spatiotemporal propagation of short pulses in normally dispersive, cubically nonlinear media. This effect is a direct consequence of a four-wave interaction.

Self-focusing threshold in normally dispersive media.

The threshold at which self-focusing initially dominates the dynamics of short-pulse propagation in normally dispersive bulk media, causing an explosive increase in peak intensity, is estimated analytically and verified numerically. Intensity-dependent propagation effects such as spectral broadening also occur explosively at this threshold.

Reflection and transmission of self-focused channels at nonlinear dielectric interfaces.

We show with an extensive numerical study that the global reflection and transmission properties of a finite-width optical self-focused channel incident at an oblique angle to an interface separating two self-focusing nonlinear media can be categorized into three distinct regimes of behavior as the incidence angle is varied through the angle for total internal reflection. The largest regime in parameter space is the nonlinear regime, where a channel undergoes either total internal reflection or transmission, in marked contrast to the well-known linear Snell's law behavior. The beam asymptotics in this latter region are quantitatively explained by a recent equivalent-particle theory.