Faculty Bibliography

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

Numerical and asymptotic results are presented for a coupled PDE system that models recent experiments of the self-focussing of laser light in a nematic liquid crystal. This study complements previous asymptotic analyses which, in a narrow-beam limit, describe two essential features observed in nematic self-focussing - the undulation and filamentation of a laser beam. These numerical computations represent a direct emulation of the experimental configuration within the context of a paraxial PDE model. In addition to providing numerical corroboration to these earlier asymptotic analyses, these results suggest that initial focussing caustics play a critical role in the formation of beam filament pairs

The solution of the KdV equation with single-minimum initial data has a zero-dispersion limit characterized by Lax and Levermore as the solution of an infinite-dimensional constrained quadratic minimization problem. An adaptive numerical method for computing the weak limit from this characterization is constructed and validated. The method is then used to study the weak limit. Initial simple experiments confirm theoretical predictions, while experiments with more complicated data display multiphase behavior considerably beyond the scope of current theoretical analyses. The method computes accurate weak limits with multiphase structures sufficiently complex to provide useful test cases for the calibration of numerical averaging algorithms.

The theory of the focusing NLS equation under periodic boundary conditions, together with the Floquet spectral theory of its associated Zakharov-Shabat liner operator L, is developed in sufficient detail for later use in studies of perturbations of the NLS equation. ''Counting lemmas'' for the non-selfadjoint operator L, are established which control its spectrum and show that all of its eccentricities are finite in number and must reside within a finite disc D in the complex eigenvalue plane. The radius of the disc D is controlled by the H-1 norm of the potential q. For this integrable NLS Hamiltonian system, unstable tori are identified, and Backlund transformations are then used to construct global representations of their stable and unstable manifolds - ''whiskered tori'' for the NLS pde. The Floquet discriminant DELTA(lambda;q) used to introduce a natural sequence of NLS constants of motion, [F(j)(q) = DELTA(lambda = lambda(j)c(q); q), where lambda(j)c denotes the j(th) critical point of the Floquet discriminant DELTA(lambda)]. A Taylor series expansion of the constants F(j)(q), with explicit representations of the first and second variations, is then used to study neighborhoods of the whiskered tori. In particular, critical tori with hyperbolic structure are identified through the first and second variations of F(j)(q), which themselves are expressed in terms of quadratic products of eigenfunctions of L. The second variation permits identification, within the disc D, of important bifurcations m the spectral configurations of the operator L. The constant F(j)(q), as the height of the Floquet discriminant over the critical point lambda(j)c, admits a natural interpretation as a Morse function for NLS isospectral level sets. This Morse interpretation is studied in some detail. It is valid globally for the infinite tail, {F(j)(q)}\j\>N, which is associated with critical points outside the disc D. Within this disc, the interpretation is only valid locally, with the same obstruction to its globa!