Faculty Bibliography

We establish the semiclassical limit of the one-dimensional defocusing cubic nonlinear Schrodinger (NLS) equation. Complete integrability is exploited to obtain a global characterization of the weak limits of the entire NLS hierarchy of conserved densities as the field evolves from reflectionless initial data under all the associated commuting flows. Consequently, this also establishes the zero-dispersion limit of the modified Korteweg-de Vries equation that resides in that hierarchy. We have adapted and clarified the strategy introduced by Lax and Levermore to study the zero-dispersion limit of the Korteweg-de Vries equation, expanding it to treat entire integrable hierarchies and strengthening the limits obtained. A crucial role is played by the convexity of the underlying log-determinant with respect to the times associated with the commuting flows. (C) 1999 John Wiley & Sons, Inc

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

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]

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 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

In recent experiments investigating the nonlinear interaction between light and nematic liquid crystals, Braun et al. (1993) observed complex optical beam structures that were generated by the strong self-focussing of laser light. For a simplified partial differential equation (PDE) model which captures the essential coupling between optical refraction and nematic deformation, we demonstrate two of the experimentally observed features-undulation and filamentation. For the mathematical analysis, we develop a novel asymptotic representation for this strongly coupled nonlinear system which exploits the natural separation of scales at which these optical structures are created by the self-focussing process. This approach uses geometrical optics, paraxial optics, and scale-separation to identify tractable outer and inner problems. The outer problem describes the undulation of the beam, and is given by a free-boundary problem for the distortion of the nematic crystal. The inner problem describes the filamentation of the beam, and is given by a nonlocal-nonlinear Schrodinger (NLS) equation for evolution of the light wave. For the outer problem, we demonstrate analytically the existence of small amplitude undulations of the beam. Large amplitude undulations are studied numerically. For the inner problem, waveguide modes are constructed. Simulations of the nonlocal NLS show that the interaction of these modes generates filamentary beam structures. Thus the PDE model, when reduced asymptotically into two decoupled systems at two distinct spatial scales, produces a theoretical corroboration of the unusual nonlinear optical behavior of undulation, as well as a nonlinear mechanism for filamentation

Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrodinger Equation (NLS) have been shown to be responsible for the generation of numerical instabilities and chaos. In this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the NLS. Using heuristic Mel'nikov methods, the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system. The structure of the persistent homoclinic orbits that are predicted by the Mel'nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos. The Mel'nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments. In a neighborhood of the persistent homoclinic orbits, the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study