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We present a brief overview of the basic concepts of the soliton stability theory and discuss some characteristic examples of the instability-induced soliton dynamics, in application to spatial optical solitons described by the NLS-type…
We present gap solitons (GSs) that can be created in free nearly two-dimensional (2D) space in dipolar spinor Bose-Einstein condensates with the spin-orbit coupling (SOC), subject to tight confinement, with size $a_{\perp }$, in the third…
In the present work we consider the subject of dark fractional solitary waves in the realm of generalized (fractional) forms of the nonlinear Schr\"odinger (NLS) equation. While earlier studies have examined such states in the realm of real…
We introduce a system with one or two amplified nonlinear sites ("hot spots", HSs) embedded into a two-dimensional linear lossy lattice. The system describes an array of evanescently coupled optical or plasmonic waveguides, with gain…
Quantized nonlinear lattice models are considered for two different classes, boson and fermionic ones. The quantum discrete nonlinear Schroedinger model (DNLS) is our main objective, but its so called modified discrete nonlinear (MDNLS)…
We study stability and dynamics of the single cylindrically symmetric solitary structures and dipolar solitonic molecules in spatially nonlocal media. The main properties of the solitons, vortex solitons, and dipolar solitons are…
We consider effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schr{\"{o}}dinger equation as a paradigm model. For fundamental solitons, we develop a variational…
Recently we have derived an effective one-dimensional nonpolynomial Schr\"odinger equation (NPSE) [Phys. Rev. A {\bf 65}, 043614 (2002)] that accurately describes atomic Bose-Einstein condensates under transverse harmonic confinement. In…
Dispersive PDEs are important both in applications (wave phenomena e.g. in hy- drodynamics, nonlinear optics, plasma physics, Bose-Einstein condensates,...) and a mathematically very challenging class of partial differential equations,…
We investigate the existence and especially the linear stability of single and multiple-charge quantized vortex states of nonlinear Schroedinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions.…
Quasi-one-dimensional solitons that may occur in an elongated Bose-Einstein condensate become unstable at high particle density. We study two basic modes of instability and the corresponding bifurcations to genuinely three-dimensional…
We study the orbital stability and instability of single-spike bound states of semiclassical nonlinear Schrodinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model…
We put forward a model for trapping stable optical vortex solitons (VSs) with high topological charges $m$. The cubic-quintic nonlinear medium with an imprinted ring-shaped modulation of the refractive index is shown to support two branches…
We demonstrate a possibility of the creation of stable optical solitons combining one continuous and one discrete coordinate, with embedded vorticity, in an array of planar waveguides with intrinsic cubic-quintic nonlinearity. The same…
Dynamics of solitons is considered in the framework of an extended nonlinear Schr\"odinger equation (NLSE), which is derived from a Zakharov-type model for wind-driven high-frequency (HF) surface waves in the ocean, coupled to damped…
The stability of two-dimensional bright vortex solitons in a media with focusing cubic and defocusing quintic nonlinearities is investigated analytically and numerically. It is proved that above some critical beam powers not only one- and…
Dynamical and self-trapping properties of two-dimensional (2D) binary mixtures of Bose-Einstein condensates (BECs) in cross-combined lattices consisting of a one-dimensional (1D) linear optical lattice (LOL) in the $x-$ direction for the…
Five-component minimum-energy bound states and mobile vector solitons of a spin-orbit-coupled quasi-one-dimensional hyperfine-spin-2 Bose-Einstein condensate are studied using the numerical solution and variational approximation of a…
Discrete nonlinear Schr\"oginger equation (DNLS) of the form, $i \frac{dC_n} {dt}$ = $C_{n+1}$ + $C_{n-1}$ - $ \chi_n [|C_{n+1}|^2 + |C_{n-1}|^2 - 2 |C_n|^2] C_n$ is used to study the formation of stationary localized states in one…
We analyze the generation and mobility of discrete solitons in Bose-Einstein condensates confined in an optical lattice under realistic experimental conditions. We discuss first the creation of 1D discrete solitons, for both attractive and…