41 research outputs found
Solitons supported by localized nonlinearities in periodic media
Nonlinear periodic systems, such as photonic crystals and Bose-Einstein
condensates (BECs) loaded into optical lattices, are often described by the
nonlinear Schr\"odinger/Gross-Pitaevskii equation with a sinusoidal potential.
Here, we consider a model based on such a periodic potential, with the
nonlinearity (attractive or repulsive) concentrated either at a single point or
at a symmetric set of two points, which are represented, respectively, by a
single {\delta}-function or a combination of two {\delta}-functions. This model
gives rise to ordinary solitons or gap solitons (GSs), which reside,
respectively, in the semi-infinite or finite gaps of the system's linear
spectrum, being pinned to the {\delta}-functions. Physical realizations of
these systems are possible in optics and BEC, using diverse variants of the
nonlinearity management. First, we demonstrate that the single
{\delta}-function multiplying the nonlinear term supports families of stable
regular solitons in the self-attractive case, while a family of solitons
supported by the attractive {\delta}-function in the absence of the periodic
potential is completely unstable. We also show that the {\delta}-function can
support stable GSs in the first finite gap in both the self-attractive and
repulsive models. The stability analysis for the GSs in the second finite gap
is reported too, for both signs of the nonlinearity. Alongside the numerical
analysis, analytical approximations are developed for the solitons in the
semi-infinite and first two finite gaps, with the single {\delta}-function
positioned at a minimum or maximum of the periodic potential. In the model with
the symmetric set of two {\delta}-functions, we study the effect of the
spontaneous symmetry breaking of the pinned solitons. Two configurations are
considered, with the {\delta}-functions set symmetrically with respect to the
minimum or maximum of the potential
A nonpolynomial Schroedinger equation for resonantly absorbing gratings
We derive a nonlinear Schroedinger equation with a radical term, in the form
of the square root of (1-|V|^2), as an asymptotic model of the optical medium
built as a periodic set of thin layers of two-level atoms, resonantly
interacting with the electromagnetic field and inducing the Bragg reflection. A
family of bright solitons is found, which splits into stable and unstable
parts, exactly obeying the Vakhitov-Kolokolov criterion. The soliton with the
largest amplitude, which is |V| = 1, is found in an explicit analytical form.
It is a "quasi-peakon", with a discontinuity of the third derivative at the
center. Families of exact cnoidal waves, built as periodic chains of
quasi-peakons, are found too. The ultimate solution belonging to the family of
dark solitons, with the background level |V| = 1, is a dark compacton, also
obtained in an explicit analytical form. Those bright solitons which are
unstable destroy themselves (if perturbed) attaining the critical amplitude,
|V| = 1. The dynamics of the wave field around this critical point is studied
analytically, revealing a switch of the system into an unstable phase.
Collisions between bright solitons are investigated too. The collisions between
fast solitons are quasi-elastic, while slowly moving ones merge into breathers,
which may persist or perish (in the latter case, also by attaining |V| = 1).Comment: Physical Review A, in pres
Scattering of slow-light gap solitons with charges in a two-level medium
The Maxwell-Bloch system describes a quantum two-level medium interacting
with a classical electromagnetic field by mediation of the the population
density. This population density variation is a purely quantum effect which is
actually at the very origin of nonlinearity. The resulting nonlinear coupling
possesses particularly interesting consequences at the resonance (when the
frequency of the excitation is close to the transition frequency of the
two-level medium) as e.g. slow-light gap solitons that result from the
nonlinear instability of the evanescent wave at the boundary. As nonlinearity
couples the different polarizations of the electromagnetic field, the
slow-light gap soliton is shown to experience effective scattering whith
charges in the medium, allowing it for instance to be trapped or reflected.
This scattering process is understood qualitatively as being governed by a
nonlinear Schroedinger model in an external potential related to the charges
(the electrostatic permanent background component of the field).Comment: RevTex, 14 pages with 5 figures, to appear in J. Phys. A: Math. Theo
Translated from Pis'ma v Zhurnal Éksperimental'no oe i
Investigations of the dynamics of nonlinear wave processes involving solitary nonlinear waves, or solitons, are continuously attracted great interest in various areas of natural sciences and engineering In this work, the problem concerning the excitation of the internal mode in the standing Bragg soliton of the self-induced transparency with perturbed envelops of the direct and inverse Bloch waves is solved. It is shown that two internal modes close in shape can be simultaneously excited at low and zero frequencies. As a result of beatings of these modes, a periodic energy exchange arises between the internal-mode fields and the resonant subsystem of two-level atoms in the Bragg soliton, which results in the appearance of oscillations in the inversion of excited atoms in the Bragg soliton. The solution is generalized to the case of a slowly moving soliton. Such a soliton is already perturbed due not only to the profile deformation, but also to inversion oscilla- A new solution of two-wave Maxwell-Bloch equations has been obtained analytically and numerically. It describes the propagation of an oscillating nonlinear optical solitary wave, or optical zoomeron, in a one-dimensional periodic resonant Bragg structure. It has been shown that the appearance of large oscillations in the velocity and total amplitude of Bloch modes of the pulse is caused by beating of internal modes of the perturbed Bragg soliton