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Soliton stability and collapse in the discrete nonpolynomial Schrodinger equation with dipole-dipole interactions
Abstract
The stability and collapse of fundamental unstaggered bright solitons in the discrete Schrodinger equation with the nonpolynomial on-site nonlinearity, which models a nearly one-dimensional Bose-Einstein condensate trapped in a deep optical lattice, are studied in the presence of the long-range dipole-dipole (DD) interactions. The cases of both attractive and repulsive contact and DD interaction are considered. The results are summarized in the form of stability/collapse diagrams in the parametric space of the model, which demonstrate that the the attractive DD interactions stabilize the solitons and help to prevent the collapse. Mobility of the discrete solitons is briefly considered too.
This article offers a comprehensive survey of results obtained for solitons
and complex nonlinear wave patterns supported by purely nonlinear lattices
(NLs), which represent a spatially periodic modulation of the local strength
and sign of the nonlinearity, and their combinations with linear lattices. A
majority of the results obtained, thus far, in this field and reviewed in this
article are theoretical. Nevertheless, relevant experimental settings are
surveyed too, with emphasis on perspectives for implementation of the
theoretical predictions in the experiment. Physical systems discussed in the
review belong to the realms of nonlinear optics (including artificial optical
media, such as photonic crystals, and plasmonics) and Bose-Einstein
condensation (BEC). The solitons are considered in one, two, and three
dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the
review are their existence, stability, and mobility. Although the field is
still far from completion, general conclusions can be drawn. In particular, a
novel fundamental property of 1D solitons, which does not occur in the absence
of NLs, is a finite threshold value of the soliton norm, necessary for their
existence. In multidimensional settings, the stability of solitons supported by
the spatial modulation of the nonlinearity is a truly challenging problem, for
the theoretical and experimental studies alike. In both the 1D and 2D cases,
the mechanism which creates solitons in NLs is principally different from its
counterpart in linear lattices, as the solitons are created directly, rather
than bifurcating from Bloch modes of linear lattices.
We predict solitary vortices in quasi-planar condensates of dipolar atoms, polarized parallel to the confinement direction, with the effective sign of the dipole-dipole interaction inverted by means of a rapidly rotating field. Energy minima corresponding to vortex solitons with topological charges {% \ell}=1 and 2 are predicted for moderately strong dipole-dipole interaction, using an axisymmetric Gaussian ansatz. The stability of the solitons with is confirmed by full 3D simulations, whereas their counterparts with are found to be unstable against splitting into a set of four fragments (quadrupole). Comment: 6 pages, 6 figures
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