Faculty Bibliography

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 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!

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.

In this paper we describe laser light interacting with nematic liquid crystals. The paper begins with a summary of recent experimental results of E. Braun, L. Faucheux, and A. Libchaber in which the liquid crystal sample is studied in three geometries - film, pipe, and droplet. Then, after a very brief glimpse at the history of liquid crystals, a theoretical model of the interacting system is described. In a one transverse dimensional idealization, we investigate the pipe and film configurations. In these cases the model reduces to a coupled system of nonlinear pde's - an elliptic sine-Gordon equation for the director field coupled to a Schroedinger equation for the electromagnetic field. Properties and qualitative behavior of this coupled system are described, both numerically and theoretically. As an illustrative example of boundary layer analysis of such coupled light-nematic systems, we describe calculations in the film geometry in some detail. Results of this analysis include: (i) an extension of the Frederiks bifurcation analysis to electric fields with spatial variation; (ii) the determination of the transverse scale at which self-focusing saturates in this nematic; (iii) the derivation of a nonlocal nonlinear Schroedinger equation which governs the inner structure of the laser beam. We conclude the paper with a summary of similar boundary layer calculations for light-nematic systems in other geometries