We study the cubic- (focusing-)quintic (defocusing) nonlinear Schrödinger equation in two transverse dimensions. We discuss a family of stationary traveling waves, including rarefaction pulses and vortex-antivortex pairs, in a background of critical amplitude. We show that these rarefaction pulses can be generated inside a flattop soliton when a smaller bright soliton collides with it. The fate of the evolution strongly depends on the relative phase of the solitons. Among several possibilities, we find that the dark pulse can reemerge as a bright soliton.

We present two simple designs of matter-wave beam splitters in a trapped Bose-Einstein Condensate (BEC). In our scheme, identical pairs of atomic solitons are produced by an adequate control -in time and/or space -of the scattering length. Our analysis is performed by numerical integration of the Gross-Pitaevskii equation and supported by several analytic estimates. Our results show that these devices can be implemented in the frame of current BEC experiments. The system has potential applications for the construction of a soliton interferometer.

We discuss the possibility of emitting vector solitons from a two-component elongated BEC by manipulating in time the inter-or intraspecies scattering lengths with Feshbach resonance tuning. We present different situations which do not have an analog in the single species case. In particular, we show vector soliton outcoupling by tuning the interspecies forces, how the evolution in one species is controlled by tuning the dynamics of the other, and how one can implement the so-called supersolitons. The analysis is performed by numerical simulations of the one-dimensional Gross-Pitaevskii equation. Simple analytic arguments are also presented in order to give a qualitative insight.

We analyze a system of three two-dimensional nonlinear Schrödinger equations coupled by linear terms and with the cubic-quintic (focusing-defocusing) nonlinearity. We consider two versions of the model: conservative and parity-time (PT) symmetric. These models describe triple-core nonlinear optical waveguides, with balanced gain and losses in the PT-symmetric case. We obtain families of soliton solutions and discuss their stability. The latter study is performed using a linear stability analysis and checked with direct numerical simulations of the evolutional system of equations. Stable solitons are found in the conservative and PT-symmetric cases. Interactions and collisions between the conservative and PT-symmetric solitons are briefly investigated, as well.

We present a numerical study of the cubic-quintic nonlinear Schrödinger equation in two transverse dimensions, relevant for the propagation of light in certain exotic media. A well-known feature of the model is the existence of flat-top bright solitons of fixed intensity, whose dynamics resembles the physics of a liquid. They support traveling wave solutions, consisting of rarefaction pulses and vortex-antivortex pairs. In this work, we demonstrate how the vortex-antivortex pairs can be generated in bright soliton collisions displaying destructive interference followed by a snake instability. We then discuss the collisional dynamics of the dark excitations for different initial conditions. We describe a number of distinct phenomena including vortex exchange modes, quasielastic flyby scattering, solitonlike crossing, fully inelastic collisions, and rarefaction pulse merging.