Perturbation-induced dynamics of dark solitons

Negative radiation pressure in Bose-Einstein condensates

In two-component nonlinear Schrödinger equations, the force exerted by incident monochromatic plane waves on an embedded dark soliton and on dark-bright-type solitons is investigated, both perturbatively and by numerical simulations. When the incoming wave is nonvanishing only in the orthogonal component to that of the embedded dark soliton, its acceleration is in the opposite direction to that of the incoming wave. This somewhat surprising phenomenon can be attributed to the well-known negative effective mass of the dark soliton. When a dark-bright soliton, whose effective mass is also negative, is hit by an incoming wave nonvanishing in the component corresponding to the dark soliton, the direction of its acceleration coincides with that of the incoming wave. This implies that the net force acting on it is in the opposite direction to that of the incoming wave. This rather counterintuitive effect is a yet another manifestation of negative radiation pressure exerted by the incident wave, observed in other systems. When a dark-bright soliton interacts with an incoming wave in the component of the bright soliton, it accelerates in the opposite direction; hence the force is pushing it now. We expect that these remarkable effects, in particular the negative radiation pressure, can be experimentally verified in Bose-Einstein condensates.

Motion of dark solitons in a non-uniform flow of Bose-Einstein condensate

We study motion of dark solitons in a non-uniform one-dimensional flow of a Bose-Einstein condensate. Our approach is based on Hamiltonian mechanics applied to the particle-like behavior of dark solitons in a slightly non-uniform and slowly changing surrounding. In one-dimensional geometry, the condensate's wave function undergoes the jump-like behavior across the soliton, and this leads to generation of the counterflow in the background condensate. For a correct description of soliton's dynamics, the contributions of this counterflow to the momentum and energy of the soliton are taken into account. The resulting Hamilton equations are reduced to the Newton-like equation for the soliton's path, and this Newton equation is solved in several typical situations. The analytical results are confirmed by numerical calculations.

Soliton, rational, and periodic solutions for the infinite hierarchy of defocusing nonlinear Schrödinger equations

Analysis of short-pulse propagation in positive dispersion media, e.g., in optical fibers and in shallow water, requires assorted high-order derivative terms. We present an infinite-order "dark" hierarchy of equations, starting from the basic defocusing nonlinear Schrödinger equation. We present generalized soliton solutions, plane-wave solutions, and periodic solutions of all orders. We find that "even"-order equations in the set affect phase and "stretching factors" in the solutions, while "odd"-order equations affect the velocities. Hence odd-order equation solutions can be real functions, while even-order equation solutions are complex. There are various applications in optics and water waves.

Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects gives rise to rich dynamics: this extends from Poincaré-Bendixson-type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections or to space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (even in the absence of other higher-order effects) for the existence of periodic, quasiperiodic, and chaotic spatiotemporal structures. Suitable low-dimensional phase-space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

Solitons riding on solitons and the quantum Newton's cradle

The reduced dynamics for dark and bright soliton chains in the one-dimensional nonlinear Schrödinger equation is used to study the behavior of collective compression waves corresponding to Toda lattice solitons. We coin the term hypersoliton to describe such solitary waves riding on a chain of solitons. It is observed that in the case of dark soliton chains, the formulated reduction dynamics provides an accurate an robust evolution of traveling hypersolitons. As an application to Bose-Einstein condensates trapped in a standard harmonic potential, we study the case of a finite dark soliton chain confined at the center of the trap. When the central chain is hit by a dark soliton, the energy is transferred through the chain as a hypersoliton that, in turn, ejects a dark soliton on the other end of the chain that, as it returns from its excursion up the trap, hits the central chain repeating the process. This periodic evolution is an analog of the classical Newton's cradle.

Interactions of bright and dark solitons with localized PT-symmetric potentials

We study collisions of moving nonlinear-Schrödinger solitons with a PT-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solitons are found by means of numerical simulations, and in the approximate semi-analytical form. An exact solution for the dark soliton pinned by the complex PT-symmetric dipole is produced too.

Dark solitons in the presence of higher-order effects

Dark soliton propagation is studied in the presence of higher-order effects, including third-order dispersion, self-steepening, linear/nonlinear gain/loss, and Raman scattering. It is found that for certain values of the parameters a stable evolution can exist for both the soliton and the relative continuous-wave background. Using a newly developed perturbation theory we show that the perturbing effects give rise to a shelf that accompanies the soliton in its propagation. Although, the stable solitons are not affected by the shelf it remains an integral part of the dynamics otherwise not considered so far in studies of higher-order nonlinear Schrödinger models.

Dark breathers in granular crystals

We present a study of the existence, stability, and bifurcation structure of families of dark breathers in a one-dimensional uniform chain of spherical beads under static load. A defocusing nonlinear Schrödinger equation (NLS) is derived for frequencies that are close to the edge of the phonon band and is used to construct targeted initial conditions for numerical computations. Salient features of the system include the existence of large amplitude solutions that emerge from the small amplitude solutions described by the NLS equation, and the presence of a nonlinear instability that, to the best of the authors' knowledge, has not been observed in classical Fermi-Pasta-Ulam lattices. Finally, it is also demonstrated that these dark breathers can be detected in a physically realistic experimental settings by merely actuating the ends of an initially at rest chain of beads and inducing destructive interference between their signals.

Low-frequency wave modulations in an electronegative dusty plasma in the presence of charge variations

The effects of dust charge variations on low-frequency wave modulations in an electronegative dusty plasma are investigated. The dynamics of the modulated wave is governed by a nonlinear Schrödinger equation with a dissipative term. The dissipation arises due to the nonsteady (nonadiabatic) dust charge variations. Theoretical and numerical investigations predict the formation of dissipative bright (envelope) and dark solitons. The nonsteady charge-variation-induced dissipation reduces the modulational instability growth rate and introduces a characteristic time scale to observe bright solitons. Results are discussed in the context of electronegative dusty plasma experiments.

Reorientational versus Kerr dark and gray solitary waves using modulation theory

We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and gray optical spatial solitary waves for both the defocusing nonlinear Schrödinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used as a test bed for the modulation theory applied to the nematicon equations, which have no exact solitary wave solution. We find that the evolution of dark and gray NLS solitons, as well as nematicons, is entirely driven by the emission of diffractive radiation, in contrast to the evolution of bright NLS solitons and bright nematicons. Moreover, the steady nematicon profile is nonmonotonic due to the long-range nonlocality associated with the perturbation of the optic axis. Excellent agreement is obtained with numerical solutions of both the defocusing NLS and nematicon equations. The comparisons for the nematicon solutions raise a number of subtle issues relating to the definition and measurement of the width of a dark or gray nematicon.

Perturbations of dark solitons

A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations.

Description of the suppression of the soliton self-frequency shift by bandwidth-limited amplification

A perturbation study of the suppression of the soliton self-frequency shift by the bandwidth-limited optical amplification is proposed. The stability of the equilibrium point for the soliton amplitude and velocity identified by the adiabatic approximation of the soliton perturbation theory (SPT) is analyzed by a numerical solution of a linearized system in the neighborhood of the equilibrium point. The obtained analytical expressions for the eigenvalues of the linearized system allow the determination of the values of pulse and material parameters for which the equilibrium point is stable. A perturbation approach that leads to the research of the equation of strongly nonlinear Duffing-Van der Pol oscillator is suggested. The last equation is explored by two different methods. First, the recently obtained results for this equation by the hyperbolic perturbation method are used. Next, the hyperbolic Lindstedt-Poincare perturbation method is applied to the exploration of this equation. The equilibrium velocity of the perturbed stationary solution was calculated as a critical value of the control parameter in both methods. It turned out that the coupling of the equilibrium velocity and the amplitude of the perturbed stationary solution in both methods is similar to the relation between the soliton amplitude and velocity derived by the adiabatic approximation of SPT. The change in the form of the perturbed stationary solution has also been identified by means of the hyperbolic Lindstedt-Poincare perturbation method.

Dark soliton dynamics and interactions in continuous-wave-induced lattices

The dynamics of dark spatial soliton beams and their interaction under the presence of a continuous wave (CW), which dynamically induces a photonic lattice, are investigated. It is shown that appropriate selection of the characteristic parameters of the CW result in controllable steering of a single soliton as well as controllable interaction between two solitons. Depending on the CW parameters, the soliton angle of propagation can be changed drastically, while two-soliton interaction can be either enhanced or reduced, suggesting a reconfigurable soliton control mechanism. Our analytical approach, based on the variational perturbation method, provides a dynamical system for the dark soliton evolution parameters. Analytical results are shown in good agreement with direct numerical simulations.

Self-bending of dark and gray photorefractive solitons

We investigate the effects of diffusion on the evolution of steady-state dark and gray spatial solitons in biased photorefractive media. Numerical integration of the nonlinear propagation equation shows that the soliton beams experience a modification of their initial trajectory, as well as a variation of their minimum intensity. This process is further studied using perturbation analysis, which predicts that the center of the optical beam moves along a parabolic trajectory and, moreover, that its minimum intensity varies linearly with the propagation distance, either increasing or decreasing depending on the sign of the initial transverse velocity. Relevant examples are provided.

Oscillations of dark solitons in trapped Bose-Einstein condensates

We consider a one-dimensional defocusing Gross-Pitaevskii equation with a parabolic potential. Dark solitons oscillate near a center of the potential trap and their amplitude decays due to radiative losses (sound emission). We develop a systematic asymptotic multiscale expansion method in the limit when the potential trap is flat. The first-order approximation predicts a uniform frequency of oscillations for the dark soliton of arbitrary amplitude. The second-order approximation predicts the nonlinear growth rate of the oscillation amplitude, which results in decay of the dark soliton. The results are compared with previous publications and numerical computations.

Spatial solitons in chiral media

We study theoretically the nonlinear propagation of a narrow optical wave packet through a cholesteric liquid crystal. We derive the equations governing the weakly nonlinear dynamics of an optical field by taking into account the coupling with the liquid crystal. We constructed the solution as the superposition of four narrow wave packets centered around the linear eigenmodes of the helical structure whose corresponding envelopes A are slowly varying functions of their arguments. We found a system of four coupled equations to describe the resulting vector wave packet which has some integration constants and that under special conditions reduces to the nonlinear Schrödinger equation with space-dependent coefficients. We solved this equation both, using a variational approach and performing numerical calculations. We calculated analytically the soliton spatial scales, the transported power, the nonlinear refraction index, and its wavelength dependence, showing that this has its maxima at the edges of the reflection band. We also exhibit the existence of some other exact but non-self-focused solutions.

Perturbation theory for dark solitons: inverse scattering transform approach and radiative effects

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.

Controlled generation of dark solitons with phase imprinting

The generation of dark solitons in Bose-Einstein condensates with phase imprinting is studied by mapping it into the classic problem of a damped driven pendulum. We provide a simple but powerful scheme, designing the phase imprint for various desired outcomes of soliton generation. For a given phase step, we derive a formula for the number of dark solitons traveling in each direction, and examine the physics behind the generation of counterpropagating dark solitons.

Traveling solitons in the parametrically driven nonlinear Schrödinger equation

We show that the (undamped) parametrically driven nonlinear Schrödinger equation has wide classes of traveling soliton solutions, some of which are stable. For small driving strengths stable nonpropagating and moving solitons co-exist while strongly forced solitons can only be stable when moving sufficiently fast.

Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose-Einstein condensate

The cubic nonlinear Schrödinger equation is the quasi-one-dimensional limit of the mean-field theory which models dilute gas Bose-Einstein condensates. Stationary solutions of this equation can be characterized as soliton trains. It is demonstrated that for repulsive nonlinearity a soliton train is stable to initial stochastic perturbation, while for attractive nonlinearity its behavior depends on the spacing between individual solitons in the train. Toroidal and harmonic confinement, both of experimental interest for Bose-Einstein condensates, are considered.

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.

Stability of localized structures in the Swift-Hohenberg equation

We show that nonmonotonic (oscillatory) decay of the boundaries of phase domains is crucial for the stability of localized structures in systems described by Swift-Hohenberg equation. The less damped (more oscillatory) are the boundaries, the larger are the existence ranges of the localized structures. For very weakly damped spatial oscillations, higher-order localized structures are possible.

Drift instability of dark solitons in saturable media

It is shown that dark solitons display an inherent instability for a strong saturation of the nonlinear refractive index provided that the background intensity exceeds a certain critical value. Such an instability manifests itself as a drift of a black soliton that transform into a gray soliton and radiation.

Instabilities of dark solitons

Optical dark solitons described by the generalized nonlinear Schrödinger equation are discussed, and the criterion of soliton instability is presented. This analytical criterion is confirmed numerically for an exactly solvable model of nonlinear saturation.

Transmission control of dark solitons by means of nonlinear gain

Transmission control of dark solitons against various perturbations is possible by use of amplifiers with nonlinear gain. Stable propagation of gray and/or black solitons is achieved even in the presence of the Raman effect and mutual interactions between neighboring dark solitons.

Dark solitons in dispersive quadratic media

We analyze dark solitons supported by second-order parametric interactions between the fundamental and second harmonics in the presence of dispersion and group-velocity mismatch. We reveal various types of solitary waves, including a family of two-wave dark solitons propagating on a modulationally stable background, dark solitons with nonmonotonic tails, and bound states of dark solitons.