Interaction Of Bright Solitons Research Articles

Solution of coupled nonlinear Schrödinger equations in focusing-defocusing medium by modified perturbation theory

The interaction of bright solitons of different orders and two different wavelengths propagating in the medium focusing for one wavelength and defocusing for the other is considered. The system of nonlinear Schrödinger equations is solved by means of perturbation theory. Application of an additional postulate to adjust both widths of the solitons and to modify the amplitude by a factor determined by the overlap integral greatly improves the accuracy of the description. The good accuracy of description is confirmed by numerical calculations. Full Text: PDF ReferencesY. Kivshar, G. P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals, (Amsterdam, Academic Press 2003). CrossRef F. Abdullaev, S. Darmanyan, P. Khabibullaev, Optical Solitons, (Springer-Verlag, Berlin, 1993) CrossRef G.I.A Stegema, D.N. Christodoulides, M. Segev, IEEE J. Selected Topics Quantum Electron. 6, (2000), 1419 CrossRef J. Yang, "Nonlinear Waves in Integrable and Nonintegrable Systems", (SIAM, Philadelphia 2010). CrossRef Y. Kivshar, B. Malomed, "Dynamics of solitons in nearly integrable systems", Rev. Mod. Phys. 61, 763 (1989). CrossRef P.G. Kevrekidis, D.J. Frantzeskakis, "Solitons in coupled nonlinear Schrödinger models: A survey of recent developments", Reviews in Physics 1 (2016), 140 CrossRef R. de la Fuente, A. Barthelemy, "Spatial soliton-induced guiding by cross-phase modulation", IEEE J. Quantum Electron. 28, 547 (1992). CrossRef H. T. Tran, R. A. Sammut, "Families of multiwavelength spatial solitons in nonlinear Kerr media", Phys. Rev. A 52, 3170 (1995). CrossRef S. Leble, B. Reichel, "Coupled nonlinear Schrödinger equations in optic fibers theory", Eur. Phys. J. Special Topics 173, 5 (2009). CrossRef M. Vijayajayanthi, T.Kanna, M. Lakshmanan, "Multisoliton solutions and energy sharing collisions in coupled nonlinear Schrödinger equations with focusing, defocusing and mixed type nonlinearities", Eur. Phys. J. Special Topics 173, 57 (2009). CrossRef S. V. Manakov, "On the theory of two-dimensional stationary self-focusing of electromagnetic waves ", Sov. Phys. JETP 38 (1973), 248 DirectLink J. Yang, Phys. Rev. E 65, 036606 (2002). CrossRef T.Kanna, M. Lakshmanan, "Exact Soliton Solutions, Shape Changing Collisions, and Partially Coherent Solitons in Coupled Nonlinear Schrödinger Equations", Phys. Rev. Lett. 86, 5043 (2001). CrossRef M. Jakubowski, K. Steiglitz, R. Squier, "State transformations of colliding optical solitons and possible application to computation in bulk media", Phys. Rev. E 58, 6752 (1998). CrossRef P. S. Jung, W. Krolikowski, U. A. Laudyn, M. Trippenbach, and M. A. Karpierz, "Supermode spatial optical solitons in liquid crystals with competing nonlinearities", Phys. Rev. A 95 (2017). CrossRef P. S. Jung, M. A. Karpierz, M. Trippenbach, D. N. Christodoulides, and W. Krolikowski, "Supermode spatial solitons via competing nonlocal nonlinearities", Photonics Lett. Pol. 10 (2018). CrossRef A. Ramaniuk, M. Trippenbach, P.S. Jung, D.N. Christodoulides, W.Krolikowski, G. Assanto, "Scalar and vector supermode solitons owing to competing nonlocal nonlinearities", Opt. Express 29, 8015 (2021) CrossRef

Vector bright soliton interactions of the two-component AB system in a baroclinic fluid

A two-component AB system, which describes the wave-packets in a baroclinic fluid, is investigated in this paper. Bright one- and two-soliton solutions are constructed via the Hirota method. We derive the degenerate and nondegenerate vector solitons associated with the one and two wave numbers, respectively, and the latter admit both the single- and double-hump profiles. Effects of the supercritical / subcritical coefficient μ and nonlinearity coefficient σ in that system on the wave packets A 1 and A 2 are analysed: Velocities of A 1 , A 2 and the wave-induced modification of the basic flow, B , are proportional to μ , amplitudes of A 1 and A 2 are inversely proportional to σ , and amplitude of B is proportional to μ . Via the asymptotic analysis, we find that (1) the interactions between the two degenerate solitons or two nondegenerate solitons can be elastic under certain conditions, otherwise the interactions are shape-changing with the total energy in each soliton (consisting of all the components) being conserved; (2) the interactions between the nondegenerate and degenerate solitons are shape-changing, and the nondegenerate solitons can remain the double-hump shape or change to the periodic oscillation after the interactions; (3) the energy redistribution between the A 1 and A 2 components occurs when the two degenerate solitons undergo the shape-changing interaction, and the nondegenerate soliton undergoes the interaction without any intensity redistribution.

General soliton solutions to a reverse‐time nonlocal nonlinear Schrödinger equation

AbstractGeneral soliton solutions to a reverse‐time nonlocal nonlinear Schrödinger (NLS) equation are discussed via a matrix version of binary Darboux transformation. With this technique, searching for solutions of the Lax pair is transferred to find vector solutions of the associated linear differential equation system. From vanishing and nonvanishing seed solutions, general vector solutions of such linear differential equation system in terms of the canonical forms of the spectral matrix can be constructed by means of triangular Toeplitz matrices. Several explicit one‐soliton solutions and two‐soliton solutions are provided corresponding to different forms of the spectral matrix. Furthermore, dynamics and interactions of bright solitons, degenerate solitons, breathers, rogue waves, and dark solitons are also explored graphically.

Interacting signal packets in a lossless nonlinear transmission network with linear dispersion

In the small amplitude limit, we use the reductive perturbation method and the continuum limit approximation to derive a coupled nonlinear Schrö dinger (CNLS) equation describing the dynamics of two interacting signal packets in a discrete nonlinear electrical transmission line (NLTL) with linear dispersion. With the help of the derived CNLS equations, we present and analyze explicit expressions for the instability growth rate of a purely growing modulational instability (MI). We establish that the phenomenon of the MI can be observed only for “small” nonzero modulation wavenumbers. Also, we point out the effects of the linear dispersive element, as well as of the frequencies of the signal packets, on the instability growth rate. It is shown that the linear dispersion and the frequencies of signal packets can be well used to control the instability domain. Through the CNLS equations, we analytically investigate the propagation of solitary waves in the network. Our analytical studies show four types of interaction of signal packets propagating in the network: bright–bright, dark–dark, bright–dark and dark–bright soliton interactions.

Interaction of solitons with impurities in anisotropic ferromagnetic chains

The interaction of bright solitons with impurities in the on-site anisotropy of a ferromagnetic chain is studied. This type of impurities leads to linear and nonlinear perturbing terms in the corresponding nonlinear Schrödinger equation. We have found static soliton-impurity bound states and considered the conditions for their formation. For large values of the anisotropy the character of the impurities is mainly linear. The interaction of propagating solitons with the impurities is also investigated. We have obtained that the solitons can be transmitted, trapped or reflected. The behavior depends on the soliton velocity and width, magnetic parameters and the impurity strength.

Bright soliton interactions in a $$\mathbf (2 +\mathbf 1) $$ ( 2 + 1 ) -dimensional fourth-order variable-coefficient nonlinear Schrödinger equation for the Heisenberg ferromagnetic spin chain

Interactions of bright solitons in the Heisenberg ferromagnetic spin chain, governed by a $$(2+1)$$ -dimensional nonlinear Schrodinger equation with variable coefficients, are investigated theoretically. Analytical soliton solutions are derived by means of the Hirota bilinear method. Different scenarios of soliton propagation and interactions are illustrated. The influence of relevant parameters and variable coefficients of different function types on the soliton propagation and interactions is discussed. Solitons of different shapes in propagation, whose pulse widths and oscillation periods vary with the phase shift, are presented. The method for controlling the size of the phase shift is proposed. Furthermore, the fission solitons and the bound solitons, produced after their collisions, are displayed. Methods for controlling the number and amplitudes of fission solitons, the distance between solitons and the intensity of their interactions are put forward. Results in this paper will contribute to the effective control of solitons in the Heisenberg ferromagnetic spin chain system.

Vector bright soliton interactions of the coupled Sasa–Satsuma equations in the birefringent or two-mode fiber

Investigated in this paper are the vector bright soliton interactions of the coupled Sasa–Satsuma equations, which describe the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber. Lax pair and the corresponding Darboux-dressing transformation are constructed. Performing the iterative formula, analytic bright one-, two- and even N-soliton solutions can be obtained. Through classifying different parameter conditions, we present six cases of the interactions between the vector two solitons via the graphic analysis. During the interaction between the two solitons, in addition to the energy exchanges, the singular-I soliton always keeps its sech-type profile, while the singular-II and nonsingular solitons may be converted to each other.

Certain bright soliton interactions of the Sasa-Satsuma equation in a monomode optical fiber.

Under investigation in this paper is the Sasa-Satsuma equation, which describes the propagation of ultrashort pulses in a monomode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Based on the known bilinear forms, through the modified expanded formulas and symbolic computation, we construct the bright two-soliton solutions. Through classifying the interactions under different parameter conditions, we reveal six cases of interactions between the two solitons via an asymptotic analysis. With the help of the analytic and graphic analysis, we find that such interactions are different from those of the nonlinear Schrödinger equation and Hirota equation. When those solitons interact with each other, the singular-I soliton is shape-preserving, while the singular-II and nonsingular solitons may be shape preserving or shape changing. Such elastic and inelastic interaction phenomena in a scalar equation might enrich the knowledge of soliton behavior, which could be expected to be experimentally observed.

A uniformly accurate multiscale time integrator spectral method for the Klein–Gordon–Zakharov system in the high-plasma-frequency limit regime

A multiscale time integrator sine pseudospectral (MTI-SP) method is presented for discretizing the Klein–Gordon–Zakharov (KGZ) system with a dimensionless parameter 0<ε≤1, which is inversely proportional to the plasma frequency. In the high-plasma-frequency limit regime, i.e. 0<ε≪1, the solution of the KGZ system propagates waves with amplitude at O(1) and wavelength at O(ε2) in time and O(1) in space, which causes significantly numerical burdens due to the high oscillation in time. The main idea of the numerical method is to carry out a multiscale decomposition by frequency (MDF) to the electric field component of the solution at each time step and then apply the sine pseudospectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the MDF and the equation of the ion density component. The method is explicit and easy to be implemented. Extensive numerical results show that the MTI-SP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O(τ) for ε∈(0,1] with τ time step size and optimally with quadratic convergence rate at O(τ2) in the regime when either ε=O(1) or 0<ε≤τ. Thus the meshing strategy requirement (or ε-scalability) of the MTI-SP for the KGZ system in the high-plasma-frequency limit regime is τ=O(1) and h=O(1) for 0<ε≪1, which is significantly better than classical methods in the literatures. Finally, we apply the MTI-SP method to study the convergence rates of the KGZ system to its limiting models in the high-plasma-frequency limit and the interactions of bright solitons of the KGZ system, and to identify certain parameter regimes that the solution of the KGZ system will be blow-up in one dimension.

Polarization interactions in multi-component defocusing media

We study dark–bright soliton interactions in multi-component media such as nonlinear optical media in the defocusing regime and repulsive Bose–Einstein condensates. This is achieved using the recently developed formalism of the inverse scattering transform for the defocusing multi-component nonlinear Schrödinger equation with non-zero boundary conditions. We show that, generically, these interactions result in a non-trivial polarization shift for the bright components. We compute such polarization shift analytically and compare it to that in focusing two-component nonlinear Schrödinger systems.

Dark–bright soliton interactions for the coupled cubic-quintic nonlinear Schrödinger equations in fiber optics

The coupled cubic-quintic nonlinear Schrödinger equations in fiber optics, which describe the effects of quintic nonlinearity on the ultrashort optical pulse propagation in non-Kerr media, are investigated. Using symbolic computation, we derive the dark–bright soliton solutions of such equations. By virtue of asymptotic analysis and graphical analysis, the overtaking and head-on elastic interactions of the two bright and dark solitons are presented. Mixed bound-state solitons also arise when all of the solitons propagate in parallel, displaying the beating effects due to the oscillatory terms, or displaying breather-like structures

Formation and Interaction of Bright Solitons with Shape Changing in a DNA Model

I explore the collision of localized structures that arise from a general initial solutions in the Peyrard- Bishop model. By means of the semi-discrete approximation, it is shown that the amplitudes of waves are described by the the discrete nonlinear Schrodinger equation. The corresponding soliton solutions of this equation are obtained through the Hirota’s bilinearization method. These solutions include the one- as well as the two-soliton solutions. Particular attention is paid to the behaviors displayed by the two-soliton solution. Taking one of the soliton as a pump and the other as the bubble that describes the local opening of the two strands of DNA, I show that, the enhancement of the bubbles is due to energy transfer from the pump to the bubble within the collision process. It is also shown that the underlying solitons undergo fascinating shape changing (intensity redistribution) collision.

Rotation of the trajectories of bright solitons and realignment of intensity distribution in the coupled nonlinear Schrödinger equation

We reconsider the collisional dynamics of bright solitons in the coupled nonlinear Schrödinger equation. We observe that apart from the intensity redistribution in the interaction of bright solitons, one also witnesses a rotation of the trajectories of bright solitons. The angle of rotation can be varied by suitably manipulating the self-phase-modulation (SPM) or cross-phase-modulation (XPM) parameters. The rotation of the trajectories of the bright solitons arises due to the excess energy that is injected into the dynamical system through SPM or XPM. This extra energy contributes not only to the rotation of the trajectories, but also to the realignment of intensity distribution between the two modes. We also notice that the angular separation between the bright solitons can also be maneuvered suitably. The above results, which exclude quantum superposition for the field vectors, may have wider ramifications in nonlinear optics, Bose-Einstein condensates, and left- and right-handed metamaterials.

Dark–dark and dark–bright soliton interactions in the two-component defocusing nonlinear Schrödinger equation

We use the Inverse Scattering Transform machinery to construct multisoliton solutions to the 2-component defocusing nonlinear Schrödinger equation. Such solutions include dark–dark solitons, which have dark solitonic behaviour in both components, as well as dark–bright soliton solutions, with one dark and one bright component. We then derive the explicit expressions of two soliton solutions for all possible cases: two dark–dark solitons, two dark–bright solitons, and one dark–dark and one dark–bright soliton. Finally, we determine the long-time asymptotic behaviours of these solutions, which allows us to obtain explicit expressions for the shifts in the phases and in the soliton centers due to the interactions.

Novel Multi Optical Trapping Tool Generation within Add/Drop Filter System Controlled by Light

We propose a novel system of multi optical trapping tool using a bright soliton pulses propagating within the add/drop optical filter system. The multiplexing signals with different wavelengths of the bright solition are controlled and amplified within the system. The dynamic behavior of bright soliton interaction is analyzed and described. The signal is controlled and tuned to be an optical probe which can be configured as the multi optical tweezers. By using some suitable parameters, we found that the tweezers widths in add/drop at the drop ports are about 18nm is achieved. Therefore, the generated multi optical tweezers can be amplified within the design system. In application, the optical tweezers train can be trapped light/atom, which can be transmitted and recovered by using the proposed system.

NOVEL OPTICAL TRAPPING TOOL GENERATION AND STORAGE CONTROLLED BY LIGHT

We propose a novel system of an optical trapping tool using a dark-bright soliton pulse-propagating within an add/drop optical filter. The multiplexing signals with different wavelengths of the dark soliton are controlled and amplified within the system. The dynamic behavior of dark bright soliton interaction is analyzed and described. The storage signal is controlled and tuned to be an optical probe which can be configured as the optical tweezer. The optical tweezer storage is embedded within the add/drop optical filter system. By using some suitable parameters, we found that the tweezers storage time of 1.2 ns is achieved. Therefore, the generated optical tweezers can be stored and amplified within the design system. In application, the optical tweezers can be stored and trapped light/atom, which can be transmitted and recovered by using the proposed system.

Interactions of bright solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations from optical fibres with symbolic computation

In nonlinear optical fibres, the evolution of two polarization envelopes is governed by a system of coupled nonlinear Schrödinger (CNLS) equations. In this paper, with the aid of symbolic computation, the analytical bright one- and two-soliton solutions of the (2+1)-dimensional CNLS equations under certain constraints are presented by employing the Hirota method. We have discussed the head-on and overtaking interactions which include elastic and inelastic collisions between two parallel bright solitons. In the interaction process, the intensities of solitons can exhibit various redistributions. We also point out that these properties have important physical applications in constructing various logic gates and nonlinear optical fibers.

Interaction of two one-dimensional Bose-Einstein solitons: chaos and energy exchange.

We analyze the bright soliton interactions of a Bose-Einstein condensate in quasi-one-dimensional traps putting an emphasis on integrability break down due to a trapping potential. In particular, we derive a simple analytical model, which describes well all major features of soliton dynamics including chaos and energy exchange between interacting solitons induced by the trapping potential.

Interactions in bright N-soliton trains propagating in birefringent optical fibres

The role of birefringence in bright soliton interactions in an optical fibre is investigated numerically. Groups containing up to five solitons have been used as the input condition, with an angle ψ being used to determine the amplitudes of the resulting partial pulses in the slow (u) and fast (v) polarisation states. The concept of displacement length, ξ1, is used to characterise the interactions and general plots of the variation of this quantity with group velocity birefringence parameter, δ, and interpulse separation, are presented. Both a minimum and a maximum are found. Unequal amplitudes and interpulse phase differences are also investigated. Amplitude variations are shown to be effective in improving ξ1 but phasing has a negative effect. Finally, shadow amplitudes for pairs of solitons in the u and v states are given as a general function of offset and interpulse parameters.