Traveling unidirectional localized edge states in optical honeycomb lattices are analytically constructed. They are found in honeycomb arrays of helical waveguides designed to induce a periodic pseudo-magnetic field varying in the direction of propagation. Conditions on whether a given pseudofield supports a traveling edge mode are discussed; a special case of the pseudo-fields studied agrees with recent experiments. Interesting classes of dispersion relations are obtained. Envelopes of nonlinear edge modes are described by the classical one-dimensional nonlinear Schrödinger equation along the edge. Novel nonlinear states termed edge solitons are predicted analytically and found numerically.

A long wave multi-dimensional approximation of shallow-water waves is the bi-directional BenneyLuke (BL) equation. It yields the well-known Kadomtsev-Petviashvili (KP) equation in a quasi one-directional limit. A direct perturbation method is developed; it uses underlying conservation laws to determine the slow evolution of parameters of two space-dimensional, non-decaying solutions to the BL equation. These non-decaying solutions are perturbations of recently studied web solutions of the KP equation. New numerical simulations, based on windowing methods which are effective for non-decaying data, are presented. These simulations support the analytical results and elucidate the relationship between the KP and the BL equations and are also used to obtain amplitude information regarding particular web solutions. Additional dissipative perturbations to the BL equation are also studied.

The validity of using the tight-binding approximation for the nonlinear Schrödinger equations with a two-dimensional optical lattice is considered. This work provides a rigorous foundation for a technique based on "orbital" functions that is central to solid-state physics and nonlinear optics. Simple and honeycomb lattices are addressed, and it is therefore shown that the use of tight-binding approximations is justified in complicated situations.

We investigate the effect of constant-vorticity background shear on the properties of wavetrains in deep water. Using the methodology of Fokas (A Unified Approach to Boundary Value Problems, 2008, SIAM), we derive a higher-order nonlinear Schrödinger equation in the presence of shear and surface tension. We show that the presence of shear induces a strong coupling between the carrier wave and the mean-surface displacement. The effects of the background shear on the modulational instability of plane waves is also studied, where it is shown that shear can suppress instability, although not for all carrier wavelengths in the presence of surface tension. These results expand upon the findings of Thomas et al. (Phys. Fluids, vol. 24 (12), 2012, 127102). Using a modification of the generalized Lagrangian mean theory in Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646) and approximate formulas for the velocity field in the fluid column, explicit, asymptotic approximations for the Lagrangian and Stokes drift velocities are obtained for plane-wave and Jacobi elliptic function solutions of the nonlinear Schrödinger equation. Numerical approximations to particle trajectories for these solutions are found and the Lagrangian and Stokes drift velocities corresponding to these numerical solutions corroborate the theoretical results. We show that background currents have significant effects on the mean transport properties of waves. In particular, certain combinations of background shear and carrier wave frequency lead to the disappearance of mean-surface mass transport. These results provide a possible explanation for the measurements reported in Smith (J. Phys. Oceanogr., vol. 36, 2006, pp. 1381–1402). Our results also provide further evidence of the viability of the modification of the Stokes drift velocity beyond the standard monochromatic approximation, such as recently proposed in Breivik et al. (J. Phys. Oceanogr., vol. 44, 2014, pp. 2433–2445) in order to obtain a closer match to a range of complex ocean wave spectra.

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BioCoder is a C++ library developed at Microsoft Research, India, for the unambiguous specification of biochemical assays. This article describes language extensions to BioCoder along with a compiler and runtime system that translate and execute assays specified using BioCoder on a software simulator. The simulator mimics the behavior of laboratories-on-a-chip (LoCs) based on a droplet actuation technology called electrowetting on dielectric (EWoD). To date, prior compilers targeting similar EWoD devices are limited to assays specified as directed acyclic graphs (DAGs) and cannot handle arbitrary control flow or feedback from the LoC. The framework presented herein addresses these challenges through dynamic interpretation, thereby enlarging the space of assays that can be compiled onto EWoD devices.

Two-dimensional localized edge modes in optical honeycomb lattices are found and analyzed analytically and computationally. Weak nonlinearity and transverse modulation are found to introduce self-phase modulation in the phase and create internal nonlinear interactions as the electromagnetic field propagates through the lattice. Even with relatively strong nonlinearity localization and persistence of modes along the edge is found.

Abstract. Hill's method is a means to numerically approximate spectra of linear differential operators with periodic coefficients. In this paper, we address different issues related to the convergence of Hill's method. We show the method does not produce any spurious approximations, and that for selfadjoint operators, the method converges in a restricted sense. Furthermore, assuming convergence of an eigenvalue, we prove convergence of the associated eigenfunction approximation in the L 2 -norm. These results are not restricted to selfadjoint operators. Finally, for certain selfadjoint operators, we prove that the rate of convergence of Hill's method to the least eigenvalue is faster than any polynomial power.