391 research outputs found
Unstaggered-staggered solitons in two-component discrete nonlinear Schr\"{o}dinger lattices
We present stable bright solitons built of coupled unstaggered and staggered
components in a symmetric system of two discrete nonlinear Schr\"{o}dinger
(DNLS) equations with the attractive self-phase-modulation (SPM) nonlinearity,
coupled by the repulsive cross-phase-modulation (XPM) interaction. These mixed
modes are of a "symbiotic" type, as each component in isolation may only carry
ordinary unstaggered solitons. The results are obtained in an analytical form,
using the variational and Thomas-Fermi approximations (VA and TFA), and the
generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the
stability. The analytical predictions are verified against numerical results.
Almost all the symbiotic solitons are predicted by the VA quite accurately, and
are stable. Close to a boundary of the existence region of the solitons (which
may feature several connected branches), there are broad solitons which are not
well approximated by the VA, and are unstable
A model of a dual-core matter-wave soliton laser
We propose a system which can generate a periodic array of solitary-wave
pulses from a finite reservoir of coherent Bose-Einstein condensate (BEC). The
system is built as a set of two parallel quasi-one-dimensional traps (the
reservoir proper and a pulse-generating cavity), which are linearly coupled by
the tunneling of atoms. The scattering length is tuned to be negative and small
in the absolute value in the cavity, and still smaller but positive in the
reservoir. Additionally, a parabolic potential profile is created around the
center of the cavity. Both edges of the reservoir and one edge of the cavity
are impenetrable. Solitons are released through the other cavity's edge, which
is semi-transparent. Two different regimes of the intrinsic operation of the
laser are identified: circulations of a narrow wave-function pulse in the
cavity, and oscillations of a broad standing pulse. The latter regime is
stable, readily providing for the generation of an array containing up to
10,000 permanent-shape pulses. The circulation regime provides for no more than
40 cycles, and then it transforms into the oscillation mode. The dependence of
the dynamical regime on parameters of the system is investigated in detail.Comment: Journal of Physics B, in pres
Helical vs. fundamental solitons in optical fibers
We consider solitons in a nonlinear optical fiber with a single polarization
in a region of parameters where it carries exactly two distinct modes, the
fundamental one and the first-order helical mode. From the viewpoint of
applications to dense-WDM communication systems, this opens way to double the
number of channels carried by the fiber. Aside from that, experimental
observation of helical (spinning) solitons and collisions between them and with
fundamental solitons are issues of fundamental interest. We introduce a system
of coupled nonlinear Schroedinger equations for fundamental and helical modes,
which have nonstandard values of the cross-phase-modulation coupling constants,
and investigate, analytically and numerically, results of "complete" and
"incomplete" collisions between solitons carried by the two modes. We conclude
that the collision-induced crosstalk is partly attenuated in comparison with
the usual WDM system, which sometimes may be crucially important, preventing
merger of the colliding solitons into a breather. The interaction between the
two modes is found to be additionally strongly suppressed in comparison with
that in the WDM system in the case when a dispersion-shifted or
dispersion-compensated fiber is used.Comment: a plain latex file with the text and two ps files with figures.
Physica Scripta, in pres
Optical Bistability in Nonlinear Optical Coupler with Negative Index Channel
We discuss a novel kind of nonlinear coupler with one channel filled with a
negative index material (NIM). The opposite directionality of the phase
velocity and the energy flow in the NIM channel facilitates an effective
feedback mechanism that leads to optical bistability and gap soliton formation
Renormalization Group Theory for a Perturbed KdV Equation
We show that renormalization group(RG) theory can be used to give an analytic
description of the evolution of a perturbed KdV equation. The equations
describing the deformation of its shape as the effect of perturbation are RG
equations. The RG approach may be simpler than inverse scattering theory(IST)
and another approaches, because it dose not rely on any knowledge of IST and it
is very concise and easy to understand. To the best of our knowledge, this is
the first time that RG has been used in this way for the perturbed soliton
dynamics.Comment: 4 pages, no figure, revte
On the (Non)-Integrability of KdV Hierarchy with Self-consistent Sources
Non-holonomic deformations of integrable equations of the KdV hierarchy are
studied by using the expansions over the so-called "squared solutions" (squared
eigenfunctions). Such deformations are equivalent to perturbed models with
external (self-consistent) sources. In this regard, the KdV6 equation is viewed
as a special perturbation of KdV equation. Applying expansions over the
symplectic basis of squared eigenfunctions, the integrability properties of the
KdV hierarchy with generic self-consistent sources are analyzed. This allows
one to formulate a set of conditions on the perturbation terms that preserve
the integrability. The perturbation corrections to the scattering data and to
the corresponding action-angle variables are studied. The analysis shows that
although many nontrivial solutions of KdV equations with generic
self-consistent sources can be obtained by the Inverse Scattering Transform
(IST), there are solutions that, in principle, can not be obtained via IST.
Examples are considered showing the complete integrability of KdV6 with
perturbations that preserve the eigenvalues time-independent. In another type
of examples the soliton solutions of the perturbed equations are presented
where the perturbed eigenvalue depends explicitly on time. Such equations,
however in general, are not completely integrable.Comment: 16 pages, no figures, LaTe
Squared Eigenfunctions for the Sasa-Satsuma Equation
Squared eigenfunctions are quadratic combinations of Jost functions and
adjoint Jost functions which satisfy the linearized equation of an integrable
equation. In this article, squared eigenfunctions are derived for the
Sasa-Satsuma equation whose spectral operator is a system, while
its linearized operator is a system. It is shown that these squared
eigenfunctions are sums of two terms, where each term is a product of a Jost
function and an adjoint Jost function. The procedure of this derivation
consists of two steps: first is to calculate the variations of the potentials
via variations of the scattering data by the Riemann-Hilbert method. The second
one is to calculate the variations of the scattering data via the variations of
the potentials through elementary calculations. While this procedure has been
used before on other integrable equations, it is shown here, for the first
time, that for a general integrable equation, the functions appearing in these
variation relations are precisely the squared eigenfunctions and adjoint
squared eigenfunctions satisfying respectively the linearized equation and the
adjoint linearized equation of the integrable system. This proof clarifies this
procedure and provides a unified explanation for previous results of squared
eigenfunctions on individual integrable equations. This procedure uses
primarily the spectral operator of the Lax pair. Thus two equations in the same
integrable hierarchy will share the same squared eigenfunctions (except for a
time-dependent factor). In the Appendix, the squared eigenfunctions are
presented for the Manakov equations whose spectral operator is closely related
to that of the Sasa-Satsuma equation.Comment: 18 page
Affine T-varieties of complexity one and locally nilpotent derivations
Let X=spec A be a normal affine variety over an algebraically closed field k
of characteristic 0 endowed with an effective action of a torus T of dimension
n. Let also D be a homogeneous locally nilpotent derivation on the normal
affine Z^n-graded domain A, so that D generates a k_+-action on X that is
normalized by the T-action. We provide a complete classification of pairs (X,D)
in two cases: for toric varieties (n=\dim X) and in the case where n=\dim X-1.
This generalizes previously known results for surfaces due to Flenner and
Zaidenberg. As an application we compute the homogeneous Makar-Limanov
invariant of such varieties. In particular we exhibit a family of non-rational
varieties with trivial Makar-Limanov invariant.Comment: 31 pages. Minor changes in the structure. Fixed some typo
A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator
\We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
for suitably restricted,
complementary classes of initial data
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