3,922 research outputs found
Nonlinear Propagation of Light in One Dimensional Periodic Structures
We consider the nonlinear propagation of light in an optical fiber waveguide
as modeled by the anharmonic Maxwell-Lorentz equations (AMLE). The waveguide is
assumed to have an index of refraction which varies periodically along its
length. The wavelength of light is selected to be in resonance with the
periodic structure (Bragg resonance). The AMLE system considered incorporates
the effects non-instantaneous response of the medium to the electromagnetic
field (chromatic or material dispersion), the periodic structure (photonic band
dispersion) and nonlinearity. We present a detailed discussion of the role of
these effects individually and in concert. We derive the nonlinear coupled mode
equations (NLCME) which govern the envelope of the coupled backward and forward
components of the electromagnetic field. We prove the validity of the NLCME
description and give explicit estimates for the deviation of the approximation
given by NLCME from the {\it exact} dynamics, governed by AMLE. NLCME is known
to have gap soliton states. A consequence of our results is the existence of
very long-lived {\it gap soliton} states of AMLE. We present numerical
simulations which validate as well as illustrate the limits of the theory.
Finally, we verify that the assumptions of our model apply to the parameter
regimes explored in recent physical experiments in which gap solitons were
observed.Comment: To appear in The Journal of Nonlinear Science; 55 pages, 13 figure
Bright and Gap Solitons in Membrane-Type Acoustic Metamaterials
We study analytically and numerically envelope solitons (bright and gap
solitons) in a one-dimensional, nonlinear acoustic metamaterial, composed of an
air-filled waveguide periodically loaded by clamped elastic plates. Based on
the transmission line approach, we derive a nonlinear dynamical lattice model
which, in the continuum approximation, leads to a nonlinear, dispersive and
dissipative wave equation. Applying the multiple scales perturbation method, we
derive an effective lossy nonlinear Schr\"odinger equation and obtain
analytical expressions for bright and gap solitons. We also perform direct
numerical simulations to study the dissipation-induced dynamics of the bright
and gap solitons. Numerical and analytical results, relying on the analytical
approximations and perturbation theory for solions, are found to be in good
agreement
Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schr\"odinger equation
We study various properties of solutions of an extended nonlinear
Schr\"{o}dinger (ENLS) equation, which arises in the context of geometric
evolution problems -- including vortex filament dynamics -- and governs
propagation of short pulses in optical fibers and nonlinear metamaterials. For
the periodic initial-boundary value problem, we derive conservation laws
satisfied by local in time, weak (distributional) solutions, and
establish global existence of such weak solutions. The derivation is obtained
by a regularization scheme under a balance condition on the coefficients of the
linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS
model. Next, we investigate conditions for the existence of traveling wave
solutions, focusing on the case of bright and dark solitons. The balance
condition on the coefficients is found to be essential for the existence of
exact analytical soliton solutions; furthermore, we obtain conditions which
define parameter regimes for the existence of traveling solitons for various
linear dispersion strengths. Finally, we study the modulational instability of
plane waves of the ENLS equation, and identify important differences between
the ENLS case and the corresponding NLS counterpart. The analytical results are
corroborated by numerical simulations, which reveal notable differences between
the bright and the dark soliton propagation dynamics, and are in excellent
agreement with the analytical predictions of the modulation instability
analysis.Comment: 27 pages, 5 figures. To be published in Journal of Physics A:
Mathematical and Theoretica
Traveling dark-bright solitons in a reduced spin-orbit coupled system: application to Bose-Einstein condensates
In the present work, we explore the potential of spin-orbit (SO) coupled
Bose-Einstein condensates to support multi-component solitonic states in the
form of dark-bright (DB) solitons. In the case where Raman linear coupling
between components is absent, we use a multiscale expansion method to reduce
the model to the integrable Mel'nikov system. The soliton solutions of the
latter allow us to reconstruct approximate traveling DB solitons for the
reduced SO coupled system. For small values of the formal perturbation
parameter, the resulting waveforms propagate undistorted, while for large
values thereof, they shed some dispersive radiation, and subsequently distill
into a robust propagating structure. After quantifying the relevant radiation
effect, we also study the dynamics of DB solitons in a parabolic trap,
exploring how their oscillation frequency varies as a function of the bright
component mass and the Raman laser wavenumber
Dissipative Kerr solitons in optical microresonators
This chapter describes the discovery and stable generation of temporal
dissipative Kerr solitons in continuous-wave (CW) laser driven optical
microresonators. The experimental signatures as well as the temporal and
spectral characteristics of this class of bright solitons are discussed.
Moreover, analytical and numerical descriptions are presented that do not only
reproduce qualitative features but can also be used to accurately model and
predict the characteristics of experimental systems. Particular emphasis lies
on temporal dissipative Kerr solitons with regard to optical frequency comb
generation where they are of particular importance. Here, one example is
spectral broadening and self-referencing enabled by the ultra-short pulsed
nature of the solitons. Another example is dissipative Kerr soliton formation
in integrated on-chip microresonators where the emission of a dispersive wave
allows for the direct generation of unprecedentedly broadband and coherent
soliton spectra with smooth spectral envelope.Comment: To appear in "Nonlinear optical cavity dynamics", ed. Ph. Grel
Dynamics of Lattice Kinks
In this paper we consider two models of soliton dynamics (the sine Gordon and
the \phi^4 equations) on a 1-dimensional lattice. We are interested in
particular in the behavior of their kink-like solutions inside the Peierls-
Nabarro barrier and its variation as a function of the discreteness parameter.
We find explicitly the asymptotic states of the system for any value of the
discreteness parameter and the rates of decay of the initial data to these
asymptotic states. We show that genuinely periodic solutions are possible and
we identify the regimes of the discreteness parameter for which they are
expected to persist. We also prove that quasiperiodic solutions cannot exist.
Our results are verified by numerical simulations.Comment: 50 pages, 10 figures, LaTeX documen
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