Stable vortex solitons in the two-dimensional Ginzburg-Landau equation

In the framework of the complex cubic-quintic Ginzburg-Landau equation, we perform a systematic analysis of two-dimensional axisymmetric doughnut-shaped localized pulses with the inner phase field in the form of a rotating spiral. We put forward a qualitative argument which suggests that, on the contrary to the known fundamental azimuthal instability of spinning doughnut-shaped solitons in the cubic-quintic NLS equation, their GL counterparts may be stable. This is confirmed by massive direct simulations, and, in a more rigorous way, by calculating the growth rate of the dominant perturbation eigenmode. It is shown that very robust spiral solitons with (at least) the values of the vorticity S=0, 1, and 2 can be easily generated from a large variety of initial pulses having the same values of intrinsic vorticity S. In a large domain of the parameter space, it is found that all the stable solitons coexist, each one being a strong attractor inside its own class of localized two-dimensional pulses distinguished by their vorticity. In a smaller region of the parameter space, stable solitons with S=1 and 2 coexist, while the one with S=0 is absent. Stable breathers, i.e., both nonspiraling and spiraling solitons demonstrating persistent quasiperiodic internal vibrations, are found too.

Discrete vortex solitons

Localized states in the discrete two-dimensional (2D) nonlinear Schrödinger equation is found: vortex solitons with an integer vorticity S. While Hamiltonian lattices do not conserve angular momentum or the topological invariant related to it, we demonstrate that the soliton's vorticity may be conserved as a dynamical invariant. Linear stability analysis and direct simulations concur in showing that fundamental vortex solitons, with S=1, are stable if the intersite coupling C is smaller than some critical value C((1))(cr). At C>C((1))(cr), an instability sets in through a quartet of complex eigenvalues appearing in the linearized equations. Direct simulations reveal that an unstable vortex soliton with S=1 first splits into two usual solitons with S=0 (in accordance with the prediction of the linear analysis), but then an instability-induced spontaneous symmetry breaking takes place: one of the secondary solitons with S=0 decays into radiation, while the other one survives. We demonstrate that the usual (S=0) 2D solitons in the model become unstable, at C>C((0))(cr) approximately 2.46C((1))(cr), in a different way, via a pair of imaginary eigenvalues omega which bifurcate into instability through omega=0. Except for the lower-energy S=1 solitons that are centered on a site, we also construct ones which are centered between lattice sites which, however, have higher energy than the former. Vortex solitons with S=2 are found too, but they are always unstable. Solitons with S=1 and S=0 can form stable bound states.

Bragg-grating solitons in a semilinear dual-core system

We investigate the existence and stability of gap solitons in a double-core optical fiber, where one core has the Kerr nonlinearity and the other one is linear, with the Bragg grating (BG) written on the nonlinear core, while the linear one may or may not have a BG. The model considerably extends the previously studied families of BG solitons. For zero-velocity solitons, we find exact solutions in a limiting case when the group-velocity terms are absent in the equation for the linear core. In the general case, solitons are found numerically. Stability borders for the solitons are found in terms of an internal parameter of the soliton family. Depending on the frequency omega, the solitons may remain stable for large values of the group velocity in the linear core. Stable moving solitons are also found. They are produced by interaction of initially separated solitons, which shows a considerable spontaneous symmetry breaking in the case when the solitons attract each other.

Anisotropic solitons in dipolar bose-einstein condensates

Starting with a Gaussian variational ansatz, we predict anisotropic bright solitons in quasi-2D Bose-Einstein condensates consisting of atoms with dipole moments polarized perpendicular to the confinement direction. Unlike isotropic solitons predicted for the moments aligned with the confinement axis [Phys. Rev. Lett. 95, 200404 (2005)10.1103/PhysRevLett.95.200404], no sign reversal of the dipole-dipole interaction is necessary to support the solitons. Direct 3D simulations confirm their stability.

Spatiotemporally localized solitons in resonantly absorbing bragg reflectors

We predict the existence of multidimensional solitons that are localized in both space and time ("light bullets") in two- and three-dimensional self-induced-transparency media embedded in a Bragg grating. These fully stable light bullets suggest new possibilities of signal transmission control and self-trapping of light.

Stability of multiple pulses in discrete systems

The stability of multiple-pulse solutions to the discrete nonlinear Schrödinger equation is considered. A bound state of widely separated single pulses is rigorously shown to be unstable, unless the phase shift Delta phi between adjacent pulses satisfies Delta phi=pi. This instability is accounted for by positive real eigenvalues in the linearized system. The analysis leading to the instability result does not, however, determine the linear stability of those multiple pulses for which Delta phi=pi between adjacent pulses. A direct variational approach for a two-pulse predicts that it is linearly stable if Delta phi=pi, and if the separation between the individual pulses satisfies a certain condition. The variational approach can easily be generalized to study the stability of N pulses for any N>or=3. The analysis is supplemented with a detailed numerical stability analysis.

Embedded solitons in a three-wave system

We report a rich spectrum of isolated solitons residing inside (embedded into) the continuous radiation spectrum in a simple model of three-wave spatial interaction in a second-harmonic-generating planar optical waveguide equipped with a quasi-one-dimensional Bragg grating. An infinite sequence of fundamental embedded solitons is found, each one differing by the number of internal oscillations. Branches of these zero-walkoff spatial solitons give rise, through bifurcations, to several secondary branches of walking solitons. The structure of the bifurcating branches suggests a multistable configuration of spatial optical solitons, which may find straightforward applications for all-optical switching.

Soliton collisions in the discrete nonlinear Schrödinger equation

We report analytical and numerical results for on-site and intersite collisions between solitons in the discrete nonlinear Schrödinger model. A semianalytical variational approximation correctly predicts gross features of the collision, viz., merger or bounce. We systematically examine the dependence of the collision outcome on initial velocity and amplitude of the solitons, as well as on the phase shift between them, and location of the collision point relative to the lattice; in some cases, the dependences are very intricate. In particular, merger of the solitons into a single one, and bounce after multiple collisions are found. Situations with a complicated system of alternating transmission and merger windows are identified too. The merger is often followed by symmetry breaking (SB), when the single soliton moves to the left or to the right, which implies momentum nonconservation. Two different types of the SB are identified, deterministic and spontaneous. The former one is accounted for by the location of the collision point relative to the lattice, and/or the phase shift between the solitons; the momentum generated during the collision due to the phase shift is calculated in an analytical approximation, its dependence on the solitons' velocities comparing well with numerical results. The spontaneous SB is explained by the modulational instability of a quasiflat plateau temporarily formed in the course of the collision.

Stable spinning optical solitons in three dimensions

We introduce spatiotemporal spinning solitons (vortex tori) of the three-dimensional nonlinear Schrödinger equation with focusing cubic and defocusing quintic nonlinearities. The first ever found completely stable spatiotemporal vortex solitons are demonstrated. A general conclusion is that stable spinning solitons are possible as a result of competition between focusing and defocusing nonlinearities.