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 investigated analytically and numerically. The vortices and higherorder solitons show the transverse symmetry-breaking azimuthal instability below some critical power. We find the threshold of the vortex soliton stabilization using the linear stability analysis and direct numerical simulations. The higher-order solitons, which have a central peak and one or more surrounding rings, are also demonstrated to be stabilized in nonlocal nonlinear media. Using direct numerical simulations, we find a class of radially asymmetric, dipole-like solitons and show that, at sufficiently high power, these structures are stable.

We present a simple approach for finding N -soliton solution and the corresponding Jost solutions of the derivative nonlinear Scrödinger equation with nonvanishing boundary conditions. Soliton perturbation theory based on the inverse scattering transform method is developed. As an application of the present theory we consider the action of the diffusive-type perturbation on a single bright/dark soliton.

We have performed numerical analysis of the two-dimensional ͑2D͒ soliton solutions in Bose-Einstein condensates with nonlocal dipole-dipole interactions. For the modified 2D Gross-Pitaevski equation with nonlocal and attractive local terms, we have found numerically different types of nonlinear localized structures such as fundamental solitons, radially symmetric vortices, nonrotating multisolitons ͑dipoles and quadrupoles͒, and rotating multisolitons ͑azimuthons͒. By direct numerical simulations we show that these structures can be made stable.

We study the bound states of two-dimensional bright solitons in nonlocal nonlinear media. The general properties and stability of these multisolitary structures are investigated analytically and numerically. We have found that a steady bound state of coherent nonrotating and rotating solitary structures (azimuthons) can exist above some threshold power. A dipolar nonrotating multisoliton occurs to be stable within the finite range of the beam power. Azimuthons turn out to be stable if the beam power exceeds some threshold value. The bound states of three or four nonrotating solitons appear to be unstable.Comment: 8 pages, 6 figures, Submitted to Phys. Lett.

A perturbation theory for dark solitons of the nonlinear Schrödinger equation is developed. The theory is based on the inverse scattering transform method. Equations describing dynamics discrete (solitonic) and continuous (radiative) scattering data in the presence of perturbations are derived for N-soliton case. Adiabatic equations for soliton parameters and the perturbation-induced radiative field are obtained. The problem of the absence of a threshold for the creation of dark solitons under the action of a perturbation is discussed. A temporal one-soliton pulse with random initial perturbation and a spatial soliton with linear gain and two-photon absorption are considered as examples of application of the developed theory.