It is well known that a symmetric soliton in coupled nonlinear Schrödinger (NLS) equations with the cubic nonlinearity loses its stability with the increase of the energy, featuring a transition into an asymmetric soliton via a subcritical bifurcation. A similar phenomenon was found in a...

Consider a uniform train of surface waves with a two-dimensional, bi-periodic surface pattern, propagating on deep water. One approximate model of the evolution of these waves is a pair of coupled nonlinear Schrödinger equations, which neglects any dissipation of the waves. We show that in this...

We consider a dynamical model of a superfluid Fermi gas in the Bardeen–Cooper–Schrieffer regime trapped in a periodic optical lattice (OL) potential. The model is based on an equation for complex order parameter ϕ of the superfluid, which is derived from the relevant energy density and...

In this report we present numerical simulations of nonlinear pulse propagation in air to elucidate the physical mechanism underlying the experimentally observed long distance propagation of filaments. Simulations of the nonlinear Schrödinger equation for the electromagnetic field coupled to the...

We explore spatially extended dynamical states in the discrete nonlinear Schrödinger lattice in two- and three-dimensions, starting from the anti-continuum limit. We first consider the “core” of the relevant states (either a two-dimensional “tile” or a three-dimensional “stone”),...

The possibility of physically relevant singular solutions of the nonlinear Schrödinger equation with sustained dissipation into the singularity is considered through numerical study of a nonlinear dissipative regularisation and its small dissipation limit. A new form of such dissipative...

We review a recent mean-field statistical model of self-organization in a generic class of focusing, nonintegrable nonlinear Schrödinger (NLS) equations. Such equations provide natural prototypes for nonlinear dispersive wave turbulence. The main conclusion of the theory is that the...

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate...

We numerically study the nucleation of gray solitons and vortex–antivortex pairs created by a moving impurity in, respectively, 1D and 2D Bose–Einstein condensates (BECs) confined by a parabolic potential. The simulations emulate the motion of a localized laser-beam spot through the trapped...