227 research outputs found

### The modified nonlinear Schroedinger equation: Facts and artefacts

It is argued that the integrable modified nonlinear Schroedinger equation
with the nonlinearity dispersion term is the true starting point to
analytically describe subpicosecond pulse dynamics in monomode fibers. Contrary
to the known assertions, solitons of this equation are free of self-steepining
and the breather formation is possible.Comment: LaTeX2e, 8 pages, 1 figure, to be published in Europ. Phys. J.

### Perturbation theory for the modified nonlinear Schr{\"o}dinger solitons

The perturbation theory based on the Riemann-Hilbert problem is developed for
the modified nonlinear Schr{\"o}dinger equation which describes the propagation
of femtosecond optical pulses in nonlinear single-mode optical fibers. A
detailed analysis of the adiabatic approximation to perturbation-induced
evolution of the soliton parameters is given. The linear perturbation and the
Raman gain are considered as examples.Comment: 22 pages, Latex, no figures. Submitted to Physica

### Nonlinear von Neumann-type equations: Darboux invariance and spectra

Generalized Euler-Arnold-von Neumann density matrix equations can be solved
by a binary Darboux transformation given here in a new form:
$\rho[1]=e^{P\ln(\mu/\nu)}\rho e^{-P\ln(\mu/\nu)}$ where $P=P^2$ is explicitly
constructed in terms of conjugated Lax pairs, and $\mu$, $\nu$ are complex. As
a result spectra of $\rho$ and $\rho[1]$ are identical. Transformations
allowing to shift and rescale spectrum of a solution are introduced, and a
class of stationary seed solutions is discussed.Comment: Phys.Lett.A - in prin

### Non-Linear Evolution Equations with Non-Analytic Dispersion Relations in 2+1 Dimensions. Bilocal Approach

A method is proposed of obtaining (2+1)-dimensional non- linear equations
with non-analytic dispersion relations. Bilocal formalism is shown to make it
possible to represent these equations in a form close to that for their
counterparts in 1+1 dimensions.Comment: 13 pages, to be published in J. Phys.

### Dynamics of subpicosecond dispersion-managed soliton in a fibre: A perturbative analysis

A model is studied which describes a propagation of a subpicosecond optical
pulse in dispersion-managed fibre links. In the limit of weak chromatic
dispersion management, the model equation is reduced to a perturbed modified
NLS equation having a nonlinearity dispersion term. By means of the
Riemann--Hilbert problem, a perturbation theory for the soliton of the modified
NLS equation is developed. It is shown in the adiabatic approximation that
there exists a unique possibility to suppress the perturbation-induced shift of
the soliton centre at the cost of proper matching of the soliton width and
nonlinearity dispersion parameter. In the next-order approximation, the
spectral density of the radiation power emitted by a soliton is calculated.Comment: 16 pages, 3 figures, to appear in J. Mod. Optic

### The Semiclassical Modified Nonlinear Schroedinger Equation I: Modulation Theory and Spectral Analysis

We study an integrable modification of the focusing nonlinear
Schroedinger equation from the point of view of semiclassical asymptotics. In
particular, (i) we establish several important consequences of the mixed-type
limiting quasilinear system including the existence of maps that embed the
limiting forms of both the focusing and defocusing nonlinear Schroedinger
equations into the framework of a single limiting system for the modified
equation, (ii) we obtain bounds for the location of discrete spectrum for the
associated spectral problem that are particularly suited to the semiclassical
limit and that generalize known results for the spectrum of the nonselfadjoint
Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family
of initial data for which we solve the associated spectral problem in terms of
special functions for all values of the semiclassical scaling parameter. We
view our results as part of a broader project to analyze the semiclassical
limit of the modified nonlinear Schroedinger equation via the noncommutative
steepest descent procedure of Deift and Zhou, and we also present a
self-contained development of a Riemann-Hilbert problem of inverse scattering
that differs from those given in the literature and that is well-adapted to
semiclassical asymptotics.Comment: 56 Pages, 21 Figure

### Two binary Darboux transformations for the KdV hierarchy with self-consistent sources

Two binary (integral type) Darboux transformations for the KdV hierarchy with
self-consistent sources are proposed. In contrast with the Darboux
transformation for the KdV hierarchy, one of the two binary Darboux
transformations provides non auto-B\"{a}cklund transformation between two n-th
KdV equations with self-consistent sources with different degrees. The formula
for the m-times repeated binary Darboux transformations are presented. This
enables us to construct the N-soliton solution for the KdV hierarchy with
self-consistent sources.Comment: 19 pages, LaTeX, no figures, to be published in Journal of
Mathematical Physic

### Full-time dynamics of modulational instability in spinor Bose-Einstein condensates

We describe the full-time dynamics of modulational instability in F=1 spinor
Bose-Einstein condensates for the case of the integrable three-component model
associated with the matrix nonlinear Schroedinger equation. We obtain an exact
homoclinic solution of this model by employing the dressing method which we
generalize to the case of the higher-rank projectors. This homoclinic solution
describes the development of modulational instability beyond the linear regime,
and we show that the modulational instability demonstrates the reversal
property when the growth of the modulation amplitude is changed by its
exponential decay.Comment: 6 pages, 2 figures, text slightly extended, a reference adde

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