Switching via wave interaction in topological photonic lattices.

A honeycomb Floquet lattice with helically rotating waveguides and an interface separating two counter-propagating subdomains is analyzed. Two topologically protected localized waves propagate unidirectionally along the interface. Switching can occur when these interface modes reach the edge of the lattice and the light splits into waves traveling in two opposite directions.

Fractional Integrable Nonlinear Soliton Equations.

Nonlinear integrable equations serve as a foundation for nonlinear dynamics, and fractional equations are well known in anomalous diffusion. We connect these two fields by presenting the discovery of a new class of integrable fractional nonlinear evolution equations describing dispersive transport in fractional media. These equations can be constructed from nonlinear integrable equations using a widely generalizable mathematical process utilizing completeness relations, dispersion relations, and inverse scattering transform techniques.

Transverse Instability of Rogue Waves.

Rogue waves are abnormally large waves which appear unexpectedly and have attracted considerable attention, particularly in recent years. The one space, one time (1+1) nonlinear Schrödinger equation is often used to model rogue waves; it is an envelope description of plane waves and admits the so-called Pergerine and Kuznetov-Ma soliton solutions. However, in deep water waves and certain electromagnetic systems where there are two significant transverse dimensions, the 2+1 hyperbolic nonlinear Schrödinger equation is the appropriate wave envelope description.

Peierls-Nabarro barrier effect in nonlinear Floquet topological insulators.

The Peierls-Nabarro barrier is a discrete effect that frequently occurs in discrete nonlinear systems. A signature of the barrier is the slowing and eventual stopping of discrete solitary waves. This work examines intense electromagnetic waves propagating through a periodic honeycomb lattice of helically driven waveguides, which serves as a paradigmatic Floquet topological insulator.

Whitham equations and phase shifts for the Korteweg-de Vries equation.

The semi-classical Korteweg-de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations.

Whitham modulation theory for the Kadomtsev- Petviashvili equation.

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

Rogue waves in nonlocal media.

The generation of rogue waves is investigated in a class of nonlocal nonlinear Schrödinger (NLS) equations. In this system, modulation instability is suppressed as the effect of nonlocality increases. Despite this fact, there is a parameter regime where the number and amplitude of the rogue events increase as compared to the standard NLS equation, which is a limit of the system when nonlocality vanishes.

Strong transmission and reflection of edge modes in bounded photonic graphene.

The propagation of linear and nonlinear edge modes in bounded photonic honeycomb lattices formed by an array of rapidly varying helical waveguides is studied. These edge modes are found to exhibit strong transmission (reflection) around sharp corners when the dispersion relation is topologically nontrivial (trivial). An asymptotic theory is developed that establishes the presence (absence) of typical edge states, including, in particular, armchair and zigzag edge states in the topologically nontrivial (trivial) case.

Integrable discrete PT symmetric model.

An exactly solvable discrete PT invariant nonlinear Schrödinger-like model is introduced. It is an integrable Hamiltonian system that exhibits a nontrivial nonlinear PT symmetry. A discrete one-soliton solution is constructed using a left-right Riemann-Hilbert formulation.

Dispersive shock wave interactions and asymptotics.

Dispersive shock waves (DSWs) are physically important phenomena that occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg-de Vries (KdV) equation is the universal model for systems with weak dispersion and weak, quadratic nonlinearity. Here we show that the long-time-asymptotic solution of the KdV equation for general, steplike data is a single-phase DSW; this DSW is the "largest" possible DSW based on the boundary data.

Integrable nonlocal nonlinear Schrödinger equation.

A new integrable nonlocal nonlinear Schrödinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws and is PT symmetric. The inverse scattering transform and scattering data with suitable symmetries are discussed.

Nonlinear shallow ocean-wave soliton interactions on flat beaches.

Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights.

Nonlinear diffraction in photonic graphene.

The nonlinear (NL) diffraction of wave packets in honeycomb lattices near Dirac points is studied. Strong nonlinearity can significantly deform the diffraction patterns from conical to triangular structure. This is described by a mean field discrete NL Dirac system and in the continuous limit by a higher-order NL Dirac system, which, in turn, is consistent with the trigonal warping of the dispersion relation.

Dark solitons in mode-locked lasers.

Dark soliton formation in mode-locked lasers is investigated by means of a power-energy saturation model that incorporates gain and filtering saturated with energy, and loss saturated with power. It is found that general initial conditions evolve (mode-lock) into dark solitons under appropriate requirements also met in experimental observations. The resulting pulses are essentially dark solitons of the unperturbed nonlinear Schrödinger equation.

Collision-induced timing shifts in dispersion-managed soliton systems.

The frequency and timing shifts associated with dispersion-managed solitons in a wavelength-division multiplexed system are computed by the numerically efficient Poisson sum technique. Analytical formulas are attainable by use of this approach with a Gaussian approximation for the soliton. The results are favorably compared with known results for the frequency shift.

Intrachannel pulse interactions in dispersion-managed transmission systems: energy transfer.

Energy transfer between adjacent pulses as a result of nonlinear intrachannel interactions in a transmission system with strong periodic dispersion management is analyzed. With small average dispersion, the energy change between nonzero bits grows linearly with respect to distance in the presence of loss and lumped amplification, whereas in a lossless system the energy change is found to be significantly suppressed. The analytical results compare favorably with the results of direct numerical simulation of the full system.

Solitons in two-dimensional lattices possessing defects, dislocations, and quasicrystal structures.

Localized nonlinear modes, or solitons, are obtained for the two-dimensional nonlinear Schrödinger equation with various external potentials that possess large variations from periodicity, i.e., vacancy defects, edge dislocations, and quasicrystal structure.

Noise-induced linewidth in frequency combs.

Frequency combs generated by trains of pulses emitted from mode-locked lasers are analyzed when the center time and phase of the pulses undergo noise-induced random walk, which broadens the comb lines. Asymptotic analysis and computation reveal that, when the standard deviation of the center-time jitter of the nth pulse scales as n(p/2) where p is a jitter exponent, the linewidth of the kth comb line scales as k(2/p). The linear-dispersionless (p=1) and pure-soliton (p=3) dynamics in lasers are derived as special cases of this time-frequency duality relation.

Asymptotic analysis of collision-induced timing shifts in return-to-zero quasi-linear systems with predispersion and postdispersion compensation.

An asymptotic method for calculating the collision-induced frequency and timing shifts for quasi-linear pulses in return-to-zero, wavelength-division multiplexed systems with predispersion and postdispersion compensation is developed. Predictions of the asymptotic theory agree well with quadrature and direct numerical simulations. Using this theory, computational savings of many orders of magnitude can be realized over direct numerical simulations.

Spectral renormalization method for computing self-localized solutions to nonlinear systems.

A new numerical scheme for computing self-localized states--or solitons--of nonlinear waveguides is proposed. The idea behind the method is to transform the underlying equation governing the soliton, such as a nonlinear Schrödinger-type equation, into Fourier space and determine a nonlinear nonlocal integral equation coupled to an algebraic equation. The coupling prevents the numerical scheme from diverging.

Wave dynamics in optically modulated waveguide arrays.

A model describing wave propagation in optically modulated waveguide arrays is proposed. In the weakly guided regime, a two-dimensional semidiscrete nonlinear Schrödinger equation with the addition of a bulk diffraction term and an external "optical trap" is derived from first principles, i.e.

Reduction of collision-induced timing shifts in dispersion-managed quasi-linear systems with periodic-group-delay dispersion compensation.

Periodic-group-delay (PGD) dispersion-compensation modules were recently proposed as mechanisms to alleviate collision-induced timing shifts in dispersion-managed (DM) systems. Frequency and timing shifts in quasi-linear DM systems with PGDs were obtained, and it is shown that significant reductions are achieved when even a small fraction of the total dispersion is compensated for by PGDs.

Carrier-envelope phase slip of ultrashort dispersion-managed solitons.

The carrier-envelope phase slip of an ultrashort pulse circulating in a mode-locked Ti:sapphire laser is analyzed. The laser cavity is modeled by a dispersion- and nonlinearity-managed nonlinear Schrödinger equation. The combined contributions to the phase slip induced by nonlinear phase and nonlinear dispersion are found to approach zero for strong dispersion maps.

Dispersion management for randomly varying optical fibers.

An asymptotic theory for optical pulse propagation in a dispersion-managed (DM) fiber with random dispersion is presented. The validity of the theory is verified with direct numerical simulation. The equations that describe the slow evolution of initial pulses have special solutions that, for fibers with moderate noise in the dispersion profile, perform much better than ideal DM solitons optimized for the unperturbed fiber.