Faculty Bibliography

We describe an experimental and theoretical study of a low-power, continuous-wave laser beam interacting with a nematic liquid crystal. Experimental observations show strong self-focusing, the onset of beam undulation and filamentation. A coupled-field model is presented and reduced through a separation of scales perturbation argument. This analysis shows that the light-nematic interaction is very different from conventional self-focusing in Kerr (nonlinear Schrodinger) media, and it provides a new optical setting for the study of the physics of complex nonlinear patterns

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

The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations of N-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation. rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to any N-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation order N. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study

In this paper we rigorously show the existence and smoothness in epsilon of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameter epsilon. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions

Certain conservative discretizations of the NLS can produce irregular behavior. We consider the diagonal discretization as a conservative perturbation of the integrable discretization and study the homoclinic crossings in its nonlinear spectrum. We find that irregularity sets in when two homoclinic structures are present and, in this case, many and continual homoclinic crossings occur throughout the irregular time series. We indicate a Melnikov analysis to study the consequences of this homoclinic behavior

Some mathematical facts about Beltrami fields, which are time-independent solutions of the three-dimensional incompressible Euler equations with nontrivial helicity, are assembled. The linearization about an arbitrary, fixed Beltrami field is studied in a Hamiltonian framework. A factorization of this linearization is introduced and used to characterize null spaces for Beltrami fields with ergodic streamlines. An interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed