Faculty Bibliography

Bistable responses of Fabry-Perot cavities and optical arrays in the presence of diffraction and diffusion are studied both analytically and numerically. The model is a pair of nonlinear Schrodinger equations coupled through a diffusion equation. The numerical computations are based on a split-step method, with three distinct characteristics. In these diffusion-dominated arrays with weak diffraction, this study demonstrates that focusing nonlinearity can improve the response characteristics significantly. The primary results of the study are that (1) for diffusion-dominated media a small amount of diffraction is sufficient to alter optical bistability significantly; (2) focusing nonlinearities enhance optical bistability in comparison with defocusing nonlinearities; (3) in diffusion-dominated media these focusing-defocusing effects are quite distinct from self-focusing behavior in Kerr media; (4) in the presence of diffraction the response of the array can be described analytically by a reduced map, whose derivation can be viewed as an extension of Firth's diffusive model to include weak diffraction; (5) this map is used to explain analytically certain qualitative features of bistability, as observed in the numerical experiments; and (6) optimal spacing predictions are made with a reduced map and verified with numerical simulations of small all-optical arrays. (C) 1999 Optical Society of America [S0740-3224(99)00107-1]

Dispersive wave turbulence is studied numerically for a class of one-dimensional nonlinear wave equations. Both deterministic and random (white noise in time) forcings are studied. Four distinct stable spectra are observed-the direct and inverse cascades of weak turbulence (WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each spectrum can describe long-time behavior, and each can be only metastable (with quite diverse lifetimes)-depending on details of nonlinearity, forcing, and dissipation. Cases of a long-live MMT transient state dcaying to a state with WT spectra, and vice-versa, are displayed. In the case of freely decaying turbulence, without forcing, both cascades of weak turbulence are observed. These WT states constitute the clearest and most striking numerical observations of WT spectra to date-over four decades of energy, and three decades of spatial, scales. Numerical experiments that study details of the composition, coexistence, and transition between spectra are then discussed, including: (i) for deterministic forcing, sharp distinctions between focusing and defocusing nonlinearities, including the role of long wavelength instabilities, localized coherent structures, and chaotic behavior; (ii) the role of energy growth in time to monitor the selection of MMT or WT spectra; (iii) a second manifestation of the MMT spectrum as it describes a self-similar evolution of the wave, without temporal averaging; (iv) coherent structures and the evolution of the direct and inverse cascades; and (v) nonlocality (in k-space) in the transferral process.

We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schrodinger equation. Devillard and Souillard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such time-harmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerical determination of the eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex plane. The distributions are distinctly different for focusing and defocusing nonlinearities. We argue further that these instabilities are connected with resonances in a Schrodinger problem, and interpret the earlier numerical simulations of Caputo, Newell, and Shelley, and of Shelley in terms of these instabilities. Finally, in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time-harmonic state observed in the simulations of Shelley

A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number. It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (\k\(-1/3)) spectrum compared with the steeper (\k\(-3/4)) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states

The existence of homoclinic orbits, for a finite-difference discretized form of a damped and driven perturbation of the focusing nonlinear Schroedinger equation under even periodic boundary conditions, is established. More specifically, for external parameters on a codimension I submanifold, the existence of homoclinic orbits is established through an argument which combines Melnikov analysis with a geometric singular perturbation theory and a purely geometric argument (called the ''second measurement'' in the paper). The geometric singular perturbation theory deals with persistence of invariant manifolds and fibration of the persistent invariant manifolds. The approximate location of the codimension 1 submanifold of parameters is calculated. (This is the material in Part I.) Then, in a neighborhood of these homoclinic orbits, the existence of ''Smale horseshoes'' and the corresponding symbolic dynamics are established in Part II [21]

The persistence of homoclinic orbits for certain perturbations of the integrable nonlinear Schrodinger equation under even periodic boundary conditions is established. More specifically, the existence of a symmetric pair of homoclinic orbits is established for the perturbed NLS equation through an argument that combines Melnikov analysis with a geometric singular perturbation theory for the PDE. (C) 1996 John Wiley & Sons, Inc