Identical Solitons Research Articles

Trapping Bragg solitons by a pair of defects

We study collisions of moving solitons in a fiber Bragg grating with a structure composed of two local defects of the grating, attractive or repulsive. Results are summarized in the form of diagrams showing the share of the trapped energy as a function of the soliton's velocity and defects' strength. The moving soliton can be trapped by a cavity bounded by repulsive defects; a well-defined region of the most efficient trapping is identified. The trapped soliton performs persistent oscillations in the cavity, with the frequency in the GHz range. For attractive defects, essential differences are found from the earlier studied case of the collision of a soliton with a single defect: in this case, too, there appears a well-defined region of the most efficient trapping, and the largest velocity, up to which the soliton can be captured, increases. The findings may be significant for experiments aimed at the creation of "standing-light" pulses in the fiber gratings and for related applications. Collisions between identical solitons moving across the two-defect structure are also studied. On the attractive set, soliton-soliton collisions may give rise to symmetric capture of the solitons by both defects or merger into a single pulse trapped at one defect.

Two-soliton collisions in a near-integrable lattice system.

We examine collisions between identical solitons in a weakly perturbed Ablowitz-Ladik (AL) model, augmented by either onsite cubic nonlinearity (which corresponds to the Salerno model, and may be realized as an array of strongly overlapping nonlinear optical waveguides) or a quintic perturbation, or both. Complex dependences of the outcomes of the collisions on the initial phase difference between the solitons and location of the collision point are observed. Large changes of amplitudes and velocities of the colliding solitons are generated by weak perturbations, showing that the elasticity of soliton collisions in the AL model is fragile (for instance, the Salerno's perturbation with the relative strength of 0.08 can give rise to a change of the solitons' amplitudes by a factor exceeding 2). Exact and approximate conservation laws in the perturbed system are examined, with a conclusion that the small perturbations very weakly affect the norm and energy conservation, but completely destroy the conservation of the lattice momentum, which is explained by the absence of the translational symmetry in generic nonintegrable lattice models. Data collected for a very large number of collisions correlate with this conclusion. Asymmetry of the collisions (which is explained by the dependence on the location of the central point of the collision relative to the lattice, and on the phase difference between the solitons) is investigated too, showing that the nonintegrability-induced effects grow almost linearly with the perturbation strength. Different perturbations (cubic and quintic ones) produce virtually identical collision-induced effects, which makes it possible to compensate them, thus finding a special perturbed system with almost elastic soliton collisions.

Solitons in the noisy Burgers equation.

We investigate numerically the coupled diffusion-advective type field equations originating from the canonical phase space approach to the noisy Burgers equation or the equivalent Kardar-Parisi-Zhang equation in one spatial dimension. The equations support stable right hand and left hand solitons and in the low viscosity limit a long-lived soliton pair excitation. We find that two identical pair excitations scatter transparently subject to a size-dependent phase shift and that identical solitons scatter on a static soliton transparently without a phase shift. The soliton pair excitation and the scattering configurations are interpreted in terms of growing step and nucleation events in the interface growth profile. Finally, we show that growing steps perform an anomalous random walk with dynamic exponent z=3/2.

Spatial solitons in a medium composed of self-focusing and self-defocusing layers

We introduce a model combining Kerr nonlinearity with a periodically changing sign (“nonlinearity management”) and a Bragg grating (BG). The main result, obtained by means of systematic simulations, is presented in the form of a stability diagram on the parameter plane of the model; the diagram turns out to be a universal one, as it practically does not depend on the soliton's power. Moreover, simulations of the nonlinear Schrödinger (NLS) model subjected to the same “nonlinearity management” demonstrate that the same diagram determines the stability of the NLS solitons, unless they are very narrow. The stability region of very narrow NLS solitons is much smaller, and soliton splitting is readily observed in that case. The universal diagram shows that a minimum nonzero average value of Kerr coefficient is necessary for the existence of stable solitons. Interactions between identical solitons with an initial phase difference between them are simulated too in the BG model, resulting in generation of stable moving solitons. A strong spontaneous symmetry breaking is observed in the case when in-phase solitons pass through each other due to attraction between them.

Quadratic solitons: existence versus excitation

Whole families of stable optical solitons are known to exist in media with quadratic nonlinearity, and evidence of such existence has been demonstrated experimentally during the last years in several materials. However, soliton existence does not always guarantee soliton excitation. On the contrary, suitable conditions must be employed in each particular setting for solitons to be efficiently, or even actually, generated. We overview the recent developments along this line, and illustrate their paramount practical importance by discussing in detail the differences encountered with identical soliton families when they are to be excited in up-conversion or down-conversion processes. Limitations of both, the ideal theoretical predictions and the accessible experimental data are highlighted.

Soliton complex dynamics in strongly dispersive medium

The concept of soliton complex in a nonlinear dispersive medium is proposed. It is shown that strongly interacting identical topological solitons in the medium can form bound soliton complexes which move without radiation. This phenomenon is considered to be universal and applicable to various physical systems. The soliton complex and its “excited” states are described analytically and numerically as solutions of nonlinear dispersive equations with the fourth and higher order spatial or mixed derivatives. The dispersive sine-Gordon (dSG), double and triple sine-Gordon, and piecewise linear models are studied in detail. Mechanisms and conditions of the formation of soliton complexes, and peculiarities of their stationary dynamics are investigated. A phenomenological approach to the description of the complexes and the classification of all the possible complex states are proposed. Some examples of physical systems, where the phenomenon can be experimentally observed, are briefly discussed.

A potential of incoherent attraction between multidimensional solitons

We obtain analytical expressions for an effective potential of interaction between two- and three-dimensional (2D and 3D) solitons (including the case of 2D vortex solitons) belonging to two different modes which are incoherently coupled by cross-phase modulation. The derivation is based on calculation of the interaction term in the full Hamiltonian of the system. An essential peculiarity is that, in the 3D case, as well as in the case of 2D solitons with unequal masses, the main contribution to the interaction potential originates from a vicinity of one or both solitons, similarly to what was recently found in the 2D and 3D single-mode systems, while in the case of identical 2D solitons, the dominating area covers all the space between the solitons. Unlike the single-mode systems, stable bound states of mutually orbiting solitons are possible in the bimodal system.

Nature of the parametrically excited bound soliton state

The internal dynamics of the parametrically excited bound state of double solitons is explored, in particular, the periodic ``collision'' behavior of the identical solitons. The results reveal the collapse-recreation mechanism of the endless ``collisions,'' in which momentum of each soliton always flows toward the symmetric center of the state. It is also found that damping dissipation plays an important role in maintaining the regular dynamics of ``collisions.'' In weakly dissipative media, the parametric excitation of some other internal oscillation mode of high frequency can have the soliton-soliton interaction irregular, or can even break the spatial symmetry.

Complex Toda lattice and its application to the theory of interacting optical solitons

A class of linear and nonlinear dynamical problems that arise when studying the modulation of trains of nearly identical soliton pulses of the nonlinear Schrödinger equation is introduced. In the simplest case the dynamics of the nonlinear Schrödinger equation can be reduced to an equation that is a complex extension of the integrable Toda lattice equation, so that the latter asymptotically models the former in the case of large intersoliton separations.

Synchronous Spatiotemporal Oscillation Between Two Parametrically Excited Bound States

The bound state is one of the elemental localized excitations in a parametrically driven system. In the molecule-like structure, the two identical solitons can interact with each other as a classic "oscillator". Here the dynamics of two interacting bound states is explored by both experiment and computation. The experiment reveals several important dynamical features, in particular, the polarity ordering rule that determines the stability of a compound structure, and the synchronization in the coupling pairs of opposite polarity which leads to a symmetric and harmonic spatiotemporal oscillation. All these interesting phenomena are numerically simulated with the parametrically driven, damped nonlinear Schrödinger equation, and their nonlinear dynamics is further concluded as a stability diagram in parameter space.

Intrinsic localized spin wave modes in easy-axis antiferromagnetic chains

Two types of intrinsic localized spin wave modes are found in perfect antiferromagnetic chains of classical spins with on-site easy-axis anisotropy: the amplitude is either double or single peaked. In the small spin deviation limit, both types become identical envelope solitons. The degree of localization increases as either the maximum spin deviation or the ratio of the anisotropy constant to the exchange coupling constant increase. However, only the single peaked intrinsic localized mode is stable with regard to a noise perturbation.

Zero wave resistance for ships moving in shallow channels at supercritical speeds

This paper deals with the wave pattern and wave resistance of a slender ship moving steadily at supercritical speed in a shallow water channel. Using, successively, linear and nonlinear shallow-water wave theory it is demonstrated that, if the hull form is adapted to speed and channel geometry according to certain rules, the ship waves can be made to form a localized pattern around the ship that moves at the same speed as the ship and at the same time the associated wave resistance can be made to vanish. In the nonlinear case, the zero-wave-resistance ship hull is derived from a KP equation solution of the oblique interaction of two identical solitons. This astonishing phenomenon may be called shallow-channel superconductivity.

Soliton interactions on dual-core fibers.

Dynamics of ultrashort pulse propagation in a dual-core fiber are governed by a set of coupled nonlinear Schr\odinger equations. We apply numerical and analytical techniques to study the nature of the inelastic interactions between identical soliton pulses propagating in close proximity on separate cores. We find that the soliton interactions can change from repulsive at small values of the coupling coefficient to attractive at large values. Representative experimental parameters are discussed.

Discrete symmetries and statistics of solitons in the presence of topological actions

Abstract We show that the intrinsic charge conjugation or parity of a soliton-antisoliton pair in the chiral soliton model equals minus one if the solitons are fermions, and plus one if they are bosons. The result is intimately connected to the sign for rotating a soliton through 2π, or for interchanging two identical solitons, and holds in any theory where a topological term contributes to the statistics of solitons.

Periodic solutions of the DABO equation as a sum of repeated solitons

It is shown that the periodic solution of the non-linear DABO equation can be written as an infinite sum of an equally spaced row of identical Lorentzian solitons. This may be considered as a generalisation of a similar result which has been found for the KdV equation and is related to the clean-interaction properties of colliding solitons under non-linear coupling.

Phenomenological quantization scheme in a nonlinear Schroumldinger equation.

A phenomenological quantization scheme is carried out for the collision process of identical solitons of a nonlinear Schr\"odinger equation. An effective potential is determined from the classical expression for the time delay. The corresponding effective $S$ matrix reproduces the exact $S$ matrix of the quantized nonlinear field as well as the ground-state binding energy for large $N$, but it is not adequate to describe the scattering process in the low-energy region.

Intermittency and Solitons in the Driven Dissipative Nonlinear Schrödinger Equation

The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.

Statistics of Yang-Mills solitons

We determine the statistics and the spin of isolated Yang-Mills monopoles in eigenstates of their electric and magnetic charge. Exchange of solitons is defined using the translation operator of a companion paper; and under exchange, state vectors representingN identical solitons change sign precisely when the angular momentum of each soliton is half-integral.

Possibility of Solitions with Charge ± e/2 in Highily Correlated 1:2 Salts of Tetracyanoqutnodimethane (TCNQ)

Abstract Solitons with fractional charge ± e/2 might be experimentally realized in the highly correlated 1:2 salts of TCNQ if the quarter filled TCNQ chains of these compounds correspond to linear Hubbard chanins dominated by the on-site Coulomb repulsion. The low-teperature current carrying excitations of the TCNQ chains would consist of thermally activated pairs of such fractionally charged solitons. The addition of a single electron to a TCNQ chain would lead to the formation of two identical solitons with half-integer charge.

Interaction of plane Langmuir solitons

The formation and interaction of Langmuir solitons is governed by the Schrödinger equation with a self-consistent potential. Generally, two kinds of interaction - quasi-static and dynamic - are considered. In the former case, the problem reduces to solving the Schrödinger equation with a cubic term, while in the latter, the self-consistent potential satisfies the wave equation. It is maintained that in the quasi-static description, solitons do not asymptotically interact with each other if uninteresting shifts of the soliton phases are disregarded. In the dynamic approach - which is physically more consistent - solitons may interact with each other and with acoustic waves. The article considers the problem of the interaction of two identical solitons moving towards each other. Semi-quantitative estimates have been used in order to determine the region of parameters in which fusion of solitons is possible. The numerical modelling results which refine the analytical estimates are discussed. The analysis shows that there is practically no region where the quasi-static approximation can be applied, since this approximation disregards the low-frequency motions which are fundamentally important for the process of Langmuir turbulence instability. The process of soliton formation from a non-symmetrical Langmuir-wave packet is investigated. A study is made of the influence of the boundary conditions, especially the periodic ones, on the formation and interaction of solitons. Applying numerical modelling, the authors show that great caution must be exercised in using the periodic boundary conditions for unbounded systems. Wave boundary conditions have been suggested.