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.

Solitary waves in systems with separated Bragg grating and nonlinearity

The existence and stability of solitons in a dual-core optical waveguide, in which one core has Kerr nonlinearity while the other one is linear with a Bragg grating written on it, are investigated. The system's spectrum for the frequency omega of linear waves always contains a gap. If the group velocity c in the linear core is zero, it also has two other, upper and lower (in terms of omega) gaps. If c not equal to 0, the upper and lower gaps do not exist in the rigorous sense, as each overlaps with one branch of the continuous spectrum. When c=0, a family of zero-velocity soliton solutions, filling all the three gaps, is found analytically. Their stability is tested numerically, leading to a conclusion that only solitons in an upper section of the upper gap are stable. For c not equal to 0, soliton solutions are sought for numerically. Stationary solutions are only found in the upper gap, in the form of unusual solitons, which exist as a continuous family in the former upper gap, despite its overlapping with one branch of the continuous spectrum. A region in the parameter plane (c,omega) is identified where these solitons are stable; it is again an upper section of the upper gap. Stable moving solitons are found too. A feasible explanation for the (virtual) existence of these solitons, based on an analytical estimate of their radiative-decay rate (if the decay takes place), is presented.

Ring solitons on vortices

Interaction of a ring dark or antidark soliton (RDS and RADS, respectively) with a vortex is considered in the defocusing nonlinear Schrödinger equation with cubic (for RDS) or saturable (for RADS) nonlinearities. By means of direct simulations, it is found that the interaction gives rise to either an almost isotropic or a spiral-like pattern. A transition between them occurs at a critical value of the RDS or RADS amplitude, the spiral pattern appearing if the amplitude exceeds the critical value. An initial ring soliton created on top of the vortex splits into a pair of rings moving inward and outward. In the subcritical case, the inbound ring reverses its polarity, bouncing from the vortex core, without conspicuous effect on the core. In the transcritical case, the bounced ring soliton suffers a spiral deformation, while the vortex changes its position and structure and also loses its axial symmetry. Through a variational-type approach to the system's Hamiltonian, we additionally find that the vortex-RDS and vortex-RADS interactions are, respectively, attractive and repulsive. Simulations with the vortex placed eccentrically with respect to the RDS or RADS reveal the generation of strongly localized multispot dark and/or antidark coherent structures. The occurrence of spiral-like patterns in many numerical experiments prompted an attempt to generate a spiral dark soliton, but the latter is found to suffer a core instability that converts it into a rotating dipole emitting waves in the outward direction.

Stabilized Kuramoto-Sivashinsky system

A model consisting of a mixed Kuramoto-Sivashinsky-Korteweg-de Vries equation, linearly coupled to an extra linear dissipative equation, is proposed. The model applies to a description of surface waves on multilayered liquid films. The extra equation makes it possible to stabilize the zero solution in the model, thus opening the way to the existence of stable solitary pulses. By means of perturbation theory, treating the dissipation and the instability-generating gain in the model (but not the linear coupling between the two waves) as small perturbations, and making use of the balance equation for the net momentum, we demonstrate that the perturbations may select two steady-state solitons from their continuous family existing in the absence of the dissipation and gain. In this case, the selected pulse with the larger value of the amplitude is expected to be stable, provided that the zero solution is stable. The prediction is completely confirmed by direct simulations. If the integration domain is not very large, some pulses are stable even when the zero background is unstable. An explanation for the latter finding is proposed. Furthermore, stable bound states of two and three pulses are found numerically.

Nonlinear parametric instability in double-well lattices

A possibility of a nonlinear resonant instability of uniform oscillations in dynamical lattices with harmonic intersite coupling and onsite nonlinearity is predicted. Numerical simulations of a lattice with a double-well onsite anharmonic potential confirm the existence of the nonlinear instability with an anomalous value of the corresponding power index, approximately 1.57, which is intermediate between the values 1 and 2 characterizing the linear and nonlinear (quadratic) instabilities. The anomalous power index may be a result of a competition between the resonant quadratic instability and nonresonant linear instabilities. The observed instability triggers transition of the lattice into a chaotic dynamical state.

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.

Collisions between spatiotemporal solitons of different dimensionality in a planar waveguide

A (2+1)-dimensional nonlinear Schrödinger equation including third-order dispersion is a natural model of a waveguide, in which strong temporal dispersion is induced by a grating in order to make the existence of two-dimensional spatiotemporal solitons possible. By means of analytical and numerical methods, we demonstrate that this model may support, simultaneously, stable dark quasi-one-dimensional (stripe) solitons and two-dimensional elevation solitons ("antidark solitons") in the form of weakly localized "lumps." The spatial position of lumps can be controlled by passing stripe dark solitons through them in an arbitrary direction. To substantiate this mechanism, we analytically calculate a position shift generated by a headon collision between the stripe and lump. The obtained results are in good agreement with direct numerical simulations.

Interactions of dispersion-managed solitons in wavelength-division-multiplexed optical transmission lines

We investigate interactions between pulses in dispersion-managed multichannel wavelength-division-multiplexed soliton systems, using an improved variational approximation. The frequency shifts are found to be smallest for moderate, i.e., relatively short-scale, dispersion management. The position shifts increase monotonically with map strength.

Stable localized vortex solitons

We demonstrate that parametric interaction of a fundamental beam with its second harmonic in bulk media, in the presence of self-defocusing third-order nonlinearity, gives rise to the first ever examples of completely stable localized ring-shaped solitons with intrinsic vorticity n=1 and n=2. The stability is demonstrated both in direct simulations and by computing eigenvalues of the corresponding linearized equations. A potential application of the (2+1)-dimensional ring solitons in optics is a possibility to design a reconfigurable multichannel system guiding signal beams.

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.

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.

Shock wave dynamics in a discrete nonlinear Schrodinger equation with internal losses

Propagation of a shock wave (SW), converting an energy-carrying domain into an empty one, is studied in a discrete version of the normal-dispersion nonlinear Schrodinger equation with viscosity, which may describe, e.g., an array of optical fibers in a weakly lossy medium. It is found that the SW in the discrete model is stable, as well as in its earlier studied continuum counterpart. In a strongly discrete case, the dependence of the SWs velocity upon the amplitude of the energy-carrying background is found to obey a simple linear law, which differs by a value of the proportionality coefficient from a similar law in the continuum model. For the underdamped case, the velocity of the shock wave is found to be vanishing along with the viscosity constant. We argue that the latter feature is universal for long but finite systems, both discrete and continuum. The dependence of the SW's width on the parameters of the system is also discussed.

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.

Stabilization of dark solitons in the cubic ginzburg-landau equation

The existence and stability of exact continuous-wave and dark-soliton solutions to a system consisting of the cubic complex Ginzburg-Landau (CGL) equation linearly coupled with a linear dissipative equation is studied. We demonstrate the existence of vast regions in the system's parameter space associated with stable dark-soliton solutions, having the form of the Nozaki-Bekki envelope holes, in contrast to the case of the conventional CGL equation, where they are unstable. In the case when the dark soliton is unstable, two different types of instability are identified. The proposed stabilized model may be realized in terms of a dual-core nonlinear optical fiber, with one core active and one passive.

Three-dimensional walking spatiotemporal solitons in quadratic media

Two-parameter families of chirped stationary three-dimensional spatiotemporal solitons in dispersive quadratically nonlinear optical media featuring type-I second-harmonic generation are constructed in the presence of temporal walk-off. Basic features of these walking spatiotemporal solitons, including their dynamical stability, are investigated in the general case of unequal group-velocity dispersions at the fundamental and second-harmonic frequencies. In the cases when the solitons are unstable, the growth rate of a dominant perturbation eigenmode is found as a function of the soliton wave number shift. The findings are in full agreement with the stability predictions made on the basis of a marginal linear-stability curve. It is found that the walking three-dimensional spatiotemporal solitons are dynamically stable in most cases; hence in principle they may be experimentally generated in quadratically nonlinear media.

Spontaneous pattern formation in driven nonlinear lattices

We demonstrate the spontaneous formation of spatial patterns in a damped, ac-driven cubic Klein-Gordon lattice. These patterns are composed of arrays of intrinsic localized modes characteristic for nonlinear lattices. We analyze the modulation instability leading to this spontaneous pattern formation. Our calculation of the modulational instability is applicable in one- and two-dimensional lattices; however, in the analyses of the emerging patterns we concentrate particularly on the two-dimensional case.

Azimuthal instability of spinning spatiotemporal solitons

We find one-parameter families of three-dimensional spatiotemporal bright vortex solitons (doughnuts, or spinning light bullets), in dispersive quadratically nonlinear media. We show that they are subject to a strong instability against azimuthal perturbations, similarly to the previously studied (2+1)-dimensional bright spatial vortex solitons. The instability breaks the spinning soliton into several fragments, each being a stable nonspinning light bullet.

Spontaneous symmetry breaking and switching in planar nonlinear optical antiwaveguides

We consider guided light beams in a nonlinear planar structure described by the nonlinear Schrodinger equation with a symmetric potential hill. Such an "antiwaveguide" (AWG) structure induces a transition from symmetric to asymmetric modes via a transcritical pitchfork bifurcation, provided that the beam's power exceeds a certain critical value. It is shown analytically that the asymmetric modes always satisfy the Vakhitov-Kolokolov (necessary) stability criterion; nevertheless, the application of a general Jones' theorem shows that the AWG modes are always unstable. To realize the actual character of the instability, we perform direct numerical simulations, which reveal that a deflecting instability, which drives the asymmetric beam into the cladding without giving rise to fanning or stripping of the beam, sets in after a propagation distance of approximately 16 transverse widths of the AWG's core. The symmetry-breaking bifurcation, in combination with the deflecting instability, may be used to design an all-optical switch. The switching can easily be controlled by means of a symmetry-breaking "hot spot" that acts upon an initial symmetric beam launched with a power exceeding the bifurcation value.

Nonsteady condensation and evaporation waves

We study the motion of a phase transition (PT) front at a constant temperature between stable and metastable states in fluids with the van der Waals equation of state. We focus on a case of relatively large metastability and low viscosity, when no steadily moving PT front exists. Simulating the one-dimensional hydrodynamic equations, we find that the PT front generates acoustic shocks in forward and backward directions. Through this mechanism, the nonsteady PT front drops its velocity and eventually stops. The shock wave may shuttle between the PT front and the system's edge, rarefaction waves appearing in the shuttle process. If the viscosity is below a certain threshold, a turbulent state sets in.

Crystallization kinetics and self-induced pinning in cellular patterns

Within the framework of the Swift-Hohenberg model it is shown numerically and analytically that the front propagation between cellular and uniform states is determined by periodic nucleation events triggered by the explosive growth of the localized zero-eigenvalue mode of the corresponding linear problem. We derive an evolution equation for this mode using asymptotic analysis, and evaluate the time interval between nucleation events, and hence the front speed. In the presence of noise, we find the velocity exponent of "thermally activated" front propagation (creep) beyond the pinning threshold.

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.

Stable solitons of quadratic ginzburg-landau equations

We present a physical model based on coupled Ginzburg-Landau equations that supports stable temporal solitary-wave pulses. The system consists of two parallel-coupled cores, one having a quadratic nonlinearity, the other one being effectively linear. The former core is active, with bandwidth-limited amplification built into it, while the latter core has only losses. Parameters of the model can be easily selected so that the zero background is stable. The model has nongeneric exact analytical solutions in the form of solitary pulses ("dissipative solitons"). Direct numerical simulations, using these exact solutions as initial configurations, show that they are unstable; however, the evolution initiated by the exact unstable solitons ends up with nontrivial stable localized pulses, which are very robust attractors. Direct simulations also demonstrate that the presence of group-velocity mismatch (walkoff) between the two harmonics in the active core makes the pulses move at a constant velocity, but does not destabilize them.

Three-dimensional spinning solitons in the cubic-quintic nonlinear medium

We find one-parameter families of three-dimensional spatiotemporal bright vortex solitons (doughnuts, or spinning light bullets), in bulk dispersive cubic-quintic optically nonlinear media. The spinning solitons display a symmetry-breaking azimuthal instability, which leads to breakup of the spinning soliton into a set of fragments, each being a stable nonspinning light bullet. However, in some cases the instability is developing so slowly that the spinning light bullets may be regarded as virtually stable ones, from the standpoint of an experiment with finite-size samples.