Collisions Of Vector Solitons Research Articles

Nucleation of transition waves via collisions of elastic vector solitons

In this work, we show that collisions of one type of nonlinear wave can lead to generation of a different kind of nonlinear wave. Specifically, we demonstrate the formation of topological solitons (or transition waves) via collisions of elastic vector solitons, another type of nonlinear wave, in a multistable mechanical system with coupling between translational and rotational degrees of freedom. We experimentally observe the nucleation of a phase transformation arising from colliding waves, and we numerically investigate head-on and overtaking collisions of solitary waves with vectorial properties (i.e., elastic vector solitons). Unlike KdV-type solitons, which maintain their shape despite collisions, our system shows that collisions of two vector solitons can cause nucleation of a new phase via annihilation of the vector solitons, triggering the propagation of transition waves. The propagation of these depends both on the amount of energy carried by the vector solitons and on their respective rotational directions. The observation of the initiation of transition waves with collisions of vector solitons in multistable mechanical systems is an unexplored area of fundamental nonlinear wave interactions and could also prove useful in applications involving reconfigurable structures.

Some anomalous exact solutions for the four-component coupled nonlinear Schrödinger equations on complex wave backgrounds

The coupled nonlinear Schrödinger (CNLS) equations are the important models for the study of the multicomponent Bose-Einstein condensates (BECs). In this paper, we study the four-components CNLS equations via Darboux transformation and obtain the N-soliton solutions with zero seed and non-zero seed solutions(q_i=0 or q_i=e^{-2it}). The 1-soliton solution and 2-soliton solution are calculated on complex wave backgrounds, the dark-bright-bright-bright soliton solutions and dark-dark-bright-bright soliton solutions are constructed. We can obtain a new class of dark-bright-bright-bright soliton solutions, which admit one-valley dark soliton in component q_1 and triple-hump bright solitons in the other three components. The collision properties between dark-dark-bright-bright solitons are considered, and the vector solitons are expected to be much more abundant than those of previously reported vector soliton collisions.

Real-time collision dynamics of vector solitons in a fiber laser

Particle-like structures of solitons, as a result of the balance between dispersion and nonlinearity, enable remarkable elastic and inelastic soliton collisions in many fields. Despite the experimental observation of temporal vector-soliton collisions in birefringent fibers, collision dynamics of vector solitons in fiber lasers have not been revealed before, to the best of our knowledge. Here, the real-time spectral evolutions of vector solitons during collisions in a dual-comb fiber laser, which generates vector solitons with slightly different repetition rates, are captured by a time-stretch dispersive Fourier transform technique. We record the whole process of vector-soliton collisions, including the formation of weak pulses induced by cross-polarization coupling, opposite central wavelength shifts of both vector solitons, distinct intensity redistribution and dissipative energy, and gradual recovery to initial states. Furthermore, extreme collisions with strong four-wave mixing sidebands are observed by virtue of coherent coupling between the orthogonal polarization components of vector solitons. Numerical simulations match well with the experimental observations. The experimental and numerical evidences of vector-soliton collision dynamics could give insight into the understanding of nonlinear dynamics in fiber lasers and other physical systems, as well as the improvement of laser performance for application in dual-comb spectroscopy.

Conservation Laws and Mixed-Type Vector Solitons for the 3-Coupled Variable-Coefficient Nonlinear Schrödinger Equations in Inhomogeneous Multicomponent Optical Fibre

Abstract In this article, the propagation and collision of vector solitons are investigated from the 3-coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of the Lax pair, infinitely-many conservation laws are obtained. Under an integrability constraint among the variable coefficients for the group velocity dispersion (GVD), nonlinearity and fibre gain/loss, and two mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation. Influence of the variable coefficients for the GVD and nonlinearity on the vector soliton amplitudes and velocities is analysed. Through the asymptotic and graphic analysis, bound states and elastic and inelastic collisions between the vector two solitons are investigated: Not only the elastic but also inelastic collision between the 2-bright-1-dark vector two solitons can occur, whereas the collision between the 1-bright-2-dark vector two solitons is always elastic; for the bound states, the GVD and nonlinearity affect their types; with the GVD and nonlinearity being the constants, collision period decreases as the GVD increases but is independent of the nonlinearity.

Mixed-type vector solitons of theN-coupled mixed derivative nonlinear Schrödinger equations from optical fibers

We investigate the mixed-type vector solitons of the $N$-coupled mixed derivative nonlinear Schr\odinger ($N$-CMDNLS) equations from optical fibers. Mixed-type ($m$-bright-$n$-antidark or $m$-bright-$n$-dark) vector-soliton solutions are derived via the Hirota method. Such vector solitons are found to be independent of the sign of cubic nonlinearity in the $N$-CMDNLS equations, which is different from the case of the coupled nonlinear Schr\odinger equations. The parameter conditions for the shape-preserving and shape-changing collisions of the mixed-type vector solitons are also given by an asymptotic analysis. As an example, the mixed-type vector-soliton collisions in the 3-CMDNLS equations are graphically illustrated. Those results may be useful to the ultrashort-pulse propagation in nonlinear fibers.

Energy-exchange collisions of vector solitons in the N-coupled mixed derivative nonlinear Schrödinger equations from the birefringent optical fibers

The pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers is governed by the coupled mixed derivative nonlinear Schrödinger equations. The energy-exchange interactions between colliding bright vector optical solitons in the N-coupled nonlinear system are investigated, with complete or partial energy switching among the multiple components of each bright vector optical soliton. The analytical bright soliton solutions are presented with Hirota's bilinear method. Such energy-exchange dynamics between two vector solitons among the N components are addressed in detail by performing the asymptotic analysis for the bright two-soliton solution. Based on the energy-exchange properties, possible applications are proposed for the design of logical gates, fiber directional couplers, and quantum information processors.

Bright N-Soliton Solutions to the Vector Hirota Equation from Nonlinear Optics with Symbolic Computation

Under investigation in this paper is the vector Hirota (VH) equation which governs the simultaneous propagation of multiple interacting femtosecond pulses in a certain type of coupled optical waveguides. By the Nth iterated Darboux transformation starting from the zero potential, the VH equation is found to admit the bright N-soliton solutions in terms of the multi-component Wronskian. Asymptotic formulae of the bright N-soliton solutions are derived for any given set of spectral parameters, which allows us to directly analyze the collision dynamics of VH solitons. Via symbolic computation, some collision properties possessed by the two- and three-soliton solutions are revealed from four aspects: the asymptotic patterns of the colliding solitons, parametric conditions for the amplitude-preserving collisions, phase shifts induced by the vector-soliton collisions, and soliton state changes described by the generalized linear fractional transformations.

Quantum-like information processing using vector solitons

The concept of simulating the quantum logic via collisions of vector solitons (Janutka 2006 J. Phys. A: Math. Gen. 39 12505, 2007 J. Phys. A: Math. Theor. 40 10813) is developed in the direction of designing a true quantum-information processor that is based on mesoscopic objects, solitons. In this concept, quantum-like information is encoded in the vector-soliton (polarization) parameters. An exponential increase of the logical-operating speed compared to that achievable in the earlier simulation schemes is found to be possible due to a coherent conversion of a 2n-component vector soliton that carries an n-cebit of information into an ensemble of 2n−1 two-component and 2n−2 four-component vector pulses. Two solid-state circuits (transmitting magnetic solitons or fluxons of long Josephson junctions) which enable such a pulse conversion are proposed.

Numerical simulations of collision behaviors of optical solitons in birefringent fibres

Vector-soliton collisions in birefringent nonlinear optical fibers are investigated. The multi-symplectic structure of the coupled nonlinear Schrödinger (CNLS) system was obtained directly from the variational principle. The multi-symplectic structure was discretizated by the multi-symplectic spectral method. Numerical experiment results showed that the multi-symplectic spectral method has the excellent behavior in the long-time numerical calculation. The collisions of the solitary waves are sensitive to the coupled constant β and the initial velocity V0.

Error of quantum-logic simulation via vector-soliton collisions

In a concept of simulating the quantum logic with vector solitons by the author (Janutka 2006 J. Phys. A: Math. Gen. 39 12505), the soliton polarization is thought of as a state vector of a system of cebits (classical counterparts of qubits) switched via collisions with other solitons. The advantage of this method of information processing compared to schemes using linear optics is the possibility of the determination of the information-register state in a single measurement. Minimization of the information-processing error for different optical realizations of the logical systems is studied in the framework of a quantum analysis of soliton fluctuations. The problem is considered with relevance to general difficulties of the quantum error-correction schemes for the classical analogies of the quantum-information processing.

Observation of Temporal Vector Soliton Propagation and Collision in Birefringent Fiber

We report the experimental observation of temporal vector soliton propagation and collision in a linearly birefringent optical fiber. To the best of the authors' knowledge, this is both the first demonstration of temporal vector solitons with two mutually incoherent component fields, and of vector soliton collisions in a Kerr nonlinear medium. Collisions are characterized by an intensity redistribution between the two components, and the experimental results agree with numerical predictions of the coupled nonlinear Schrödinger equation.

Simulation of quantum logic via collisions of vector solitons

The polarization of a multi-component vector soliton (of the Manakov type) can be thought of as a state vector of a system of qubits (a register of quantum information). A change of this state on demand via colliding the register pulse with other solitons is shown to be possible with arbitrary accuracy. The parameters (polarization, pulse width, velocity) of the register-switching solitons corresponding to the computationally universal set of quantum gates are found. Physical realizations of information processing using effects of the self-focusing of optical media or of the self-induced transparency are considered.

Inelastic vector soliton collisions: a lattice-based quantum representation.

Lattice-based quantum algorithms are developed for vector soliton collisions in the completely integrable Manakov equations, a system of coupled nonlinear Schrödinger (coupled-NLS) equations that describe the propagation of pulses in a birefringent fibre of unity cross-phase modulation factor. Under appropriate conditions the exact 2-soliton vector solutions yield inelastic soliton collisions, in agreement with the theoretical predictions of Radhakrishnan et al. (1997 Phys. Rev. E56, 2213). For linearly birefringent fibres, quasi-elastic solitary-wave collisions are obtained with emission of radiation. In a coupled integrable turbulent NLS system, soliton turbulence is found with mode intensity spectrum scaling as kappa(-6).

Complexity and regularity of vector-soliton collisions.

In this paper, we extensively investigate the collision of vector solitons in the coupled nonlinear Schrödinger equations. First, we show that for collisions of orthogonally polarized and equal-amplitude vector solitons, when the cross-phase modulational coefficient beta is small, a sequence of reflection windows similar to that in the phi(4) model arises. When beta increases, a fractal structure unlike phi(4)'s gradually emerges. But when beta is greater than one, this fractal structure disappears. Analytically, we explain these collision behaviors by a variational model that qualitatively reproduces the main features of these collisions. This variational model helps to establish that these window sequences and fractal structures are caused entirely or partially by a resonance mechanism between the translational motion and width oscillations of vector solitons. Next, we investigate collision dependence on initial polarizations of vector solitons. We discovered a sequence of reflection windows that is phase induced rather than resonance induced. Analytically, we have derived a simple formula for the locations of these phase-induced windows, and this formula agrees well with the numerical data. Last, we discuss collision dependence on relative amplitudes of initial vector solitons. We show that when vector solitons have different amplitudes, the collision structure simplifies. Feasibility of experimental observation of these results is also discussed at the end of the paper.

Information transfer via cascaded collisions of vector solitons

We demonstrate experimentally the transport of information from one vector (Manakov-like) spatial soliton to another via collisions with a third, intermediate soliton.

Resonance- and phase-induced window sequences in vector-soliton collisions

In this Letter, we report two new discoveries in vector-soliton collisions of the nonintegrable coupled nonlinear Schrodinger equations. The first discovery is that, at low cross-phase-modulational coefficients, a sequence of reflection windows similar to that in the φ 4 model appears in the collision of two orthogonally polarized and equal-amplitude solitons. This window sequence is caused by a resonance between the translational motion and width oscillations of vector solitons. Analytically, we explain these collision behaviors by a variational model which qualitatively reproduces this window sequence. The second discovery is that, when initial solitons are not orthogonally polarized, their collision can generate a different sequence of reflection windows. This window sequence is induced by the collision’s dependence on solitons’ phase differences. Our analytical formula for the locations of these phase-induced windows agrees well with our numerics. To our knowledge, this is the first report of phaseinduced window sequences in solitary-wave collisions in the literature.  2001 Elsevier Science B.V. All rights reserved.

Fractal dependence of vector-soliton collisions in birefringent fibers

Vector-soliton collisions in birefringent nonlinear optical fibers are investigated. The underlying mathematical model is the non-integrable coupled nonlinear Schrödinger equations. It is shown that the exit velocity versus collision velocity graph has a fractal structure. When we zoom into different positions of this fractal, we get structures which are either a copy, a horizontal reflection or vertical reflection of the original structure. Collision dynamics in the zoomed-in windows and that in the original graph follow simple and well-defined patterns as well. We explain this fractal dependence of the collision by a novel resonance mechanism between translational motion of vector solitons and radiation modes which cause internal oscillations inside a vector soliton.

Fractal Structure in the Collision of Vector Solitons

We study the collision of two orthogonally polarized and equal-amplitude vector solitons in the nonintegrable coupled nonlinear Schrodinger equations. We show that the separation velocity versus collision velocity graph has a fractal structure. When we zoom into this graph, we get a structure qualitatively identical to the original one. In addition, collision dynamics in the zoomed-in windows is intimately related to that in the original graph. We explain this fractal dependence of the collision by a resonance mechanism between the translational motion of vector solitons and internal oscillations inside a vector soliton.

Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics

The effect of small perturbations on the collision of vector solitons in the Manakov equations is studied in this paper. The evolution equations for the soliton parameters ~amplitude, velocity, polarization, position, and phases! throughout collision are derived. The method is based on the completeness of the bounded eigenstates of the associated linear operator in L2 space and a multiple-scale perturbation technique. These results are