Dark-dark Solitons Research Articles

Dark-soliton asymptotics for a repulsive nonlinear system in a baroclinic flow

In geophysical hydrodynamics, baroclinic instability denotes the process in which the perturbations draw the energy from the mean flow potential power. Researchers focus their attention on the baroclinic instability in the Earth's atmosphere and oceans for the meteorological diagnosis and prediction. Under investigation in this paper is a repulsive nonlinear system modeling the marginally unstable baroclinic wave packets in a baroclinic flow. With respect to the amplitude of the baroclinic wave packet and correction to the mean flow resulting from the self-rectification of the baroclinic wave, we present a Lax pair with the changeable parameters and then derive the N-dark-dark soliton solutions, where N is a positive integer. Asymptotic analysis on the N-dark-dark solitons is processed to obtain the algebraic expressions of the N-dark-dark soliton components. We find that the obtained phase shift of each dark-dark soliton component is relevant with the N − 1 spectral parameters. Furthermore, we take N = 3 as an example and graphically illustrate the 3-dark-dark solitons, which are consistent with our asymptotic-analysis results. Our analysis may provide the explanations of the complex and variable natural mechanisms of the baroclinic instability.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

Two-colour dissipative solitons and breathers in microresonator second-harmonic generation

Frequency conversion of dissipative solitons associated with the generation of broadband optical frequency combs having a tooth spacing of hundreds of giga-hertz is a topical challenge holding the key to practical applications in precision spectroscopy and data processing. The work in this direction is underpinned by fundamental problems in nonlinear and quantum optics. Here, we present the dissipative two-colour bright-bright and dark-dark solitons in a quasi-phase-matched microresonator pumped for the second-harmonic generation in the near-infrared spectral range. We also found the breather states associated with the pulse front motion and collisions. The soliton regime is found to be typical in slightly phase-mismatched resonators, while the phase-matched ones reveal broader but incoherent spectra and higher-order harmonic generation. Soliton and breather effects reported here exist for the negative tilt of the resonance line, which is possible only via the dominant contribution of second-order nonlinearity.

Interaction of chiral solitons in a Bose-Einstein Condensate under density-dependent gauge potential

The collision between two chiral solitons in a Bose-Einstein Condensate (BEC) under density-dependent gauge potential is studied. By means of Poincaré-Lighthill-Kuo (PLK) perturbation method, the different kinds of colliding chiral soliton pairs self-consistently excited upon the background of the system are obtained analytically. The coupling of the current nonlinearity induced by the gauge potential and mean-field atomic interaction results in rich collision dynamics of chiral soliton pairs. When the atomic interaction is repulsive, head-on collisions of dark-dark and dark-bright solitons are observed. However, when the atomic interaction is attractive, unidirectional chiral dark soliton pairs are excited, and overtaking collision between two chiral dark solitons takes place. The phase shifts following the collision of soliton pairs are derived analytically and confirmed numerically. Interestingly, due to the contribution of the density-dependent gauge potential, the phase shifts of the two interacting chiral solitons are different from each other. This is distinct from the case without the density-dependent gauge potential, where the phase shifts of the two colliding ordinary solitons are the same. Thus, phase shifts of interacting chiral soliton pairs can be used to identify the chiral properties of solitons in BEC under density-dependent gauge potential. Furthermore, both elastic and inelastic collisions of chiral solitons in the system are presented numerically. The phase diagram for occurrence of elastic and inelastic collisions of chiral soliton is obtained, which strongly depends on the strength of density-dependent gauge potential.

Coupled sn-cn, sn-dn, cn-dn Jacobi elliptic functions and solitons solutions in magneto-optic waveguides with quadratic-cubic nonlinearity

The current paper is dedicated to the study of the propagation of progressive solitons in magneto-optical waveguides that carry the quartic-cubic nonlinearity described by a coupled system of of nonlinear Schrödinger equations. In this direction, we take the help of a modified F-expansion method which allows us to widen the class of solutions to these coupled equations and permit the propagation of different waves in the two channels. First of all, we constructed the solutions in terms of Jacobi elliptic functions in the forms of and After that, we obtained the progressive soliton solutions when these functions approached their limiting values with regards to the modulus of ellipticity. The progressive solutions include bright-dark soliton, bright-bright soliton, dark-dark soliton, bright-front soliton, dark-front soliton. In addition, we present the figures 1–6 which graphically exhibit the representative structures of each explicit solution for some special parameter values.

Dark-soliton behaviors arising from a coupled nonlinear Schrödinger system

The principal objective of this work focuses on achieving a class of advanced and impressive N-dark–dark solitons estimations to the coupled nonlinear Schrödinger system (CNLSS) that represents the propagation of optical pluses in the model-locked fiber lasers. The achieved results will perfectly describe different nonlinear coherent structures for the behavior of the resultant solitons to this model as well as ottosecond pulses in optical fibers that play fundamental rule for all recent telecommunications. The N-dark-dark solitons arise when nonlinearities are all removed, or when there is mixture between both focusing and defocusing nonlinearities. If various focusing and defocusing solitons components are collide with each other, the energies will emerge. The N–dark-dark solitons for this model will be established by using five significant schemas that were checked before to extracting the bright, dark and other types of solitons for many nonlinear problems that arise in many branches of science. The proposed methods that are bided for this goal are the extended direct algebraic method (EDAM), the (G′G) -expansion method, the extended simple equation method (ESEM), the Paul-Painleve approach method (PPAM) and the Riccati-Bernoulli Sub-ODE method (RBSODM). The suggested methods have been checked before for many other NLPDE and continuously achieve distinct results. The achieved results will add future studies for all recent applications of the internet communications.

Dark-dark solitons, soliton molecules and elastic collisions in the mixed three-level coupled Maxwell–Bloch equations

We study the integrable three-level coupled Maxwell–Bloch equations with the mixed focusing-defocusing case. The n-fold Darboux transformation is constructed on basis of a linear 3×3 matrix eigenvalue problem. As an application, the n-dark-dark soliton solution in a simple determinant form is presented and the asymptotic behavior of the n-dark-dark soliton solution is rigorously given. In particular, under the resonant mechanism, the unusual dark-dark soliton molecules which represent the soliton bound states are analytically demonstrated. The double- and triple-dark-dark soliton molecules that are composed of two and three solitons propagating with identical velocities are graphically shown, respectively. The elastic collisions between a double- or triple-dark-dark soliton molecule and a usual dark-dark soliton, and the elastic collision of two double-dark-dark soliton molecules are discussed via the standard asymptotic analysis method.

Chirped optical soliton propagation in birefringent fibers modeled by coupled Fokas-Lenells system

The propagation of ultrashort light pulses in a birefringent optical fiber exhibiting spatiotemporal dispersion, cross- and self-phase modulation, self-steepening, and group-velocity dispersion effects is addressed. The evolution of light pulses in such system is described by the coupled Fokas-Lenells equations which offer an accurate description of pulse dynamics in the femtosecond range when certain terms of the next asymptotic order beyond those necessary for the nonlinear Schrödinger equation are sustained. We report the first analytical demonstration of the propagation of chirped solitons in a birefringent fiber medium governed by the coupled Fokas-Lenells equations. The formation of those nonlinearly chirped solitons in the optical material may be attributed to the presence of self-steepening process. The results show that the frequency chirp associated with each of the two field components is directly proportional to the total intensity of the pulse. The chirped solitons for the system including the dark-dark and bright-bright soliton pairs in the presence of all fiber parameters are retrieved. The chirps accompanying the soliton pairs are also determined. The existence constraints of these chirped solitons are presented. In addition, the stability of the chirped solutions with respect to the finite perturbations is studied numerically.

Evolution from Periodic Intensity Modulations to Dissipative Vector Solitons in A Single-Mode Fiber Laser

We investigated—both experimentally and numerically—the operation of a weakly birefringent cavity fiber laser under different net cavity dispersion values. Experimentally, we found that under coherent cross-polarization coupling, either in-phase or anti-phase low frequency intensity modulations between the two orthogonal polarization components of the laser emission could be obtained. The evolution of the periodic intensity modulations in the fiber laser under different operation conditions was studied. In this paper, we show that under suitable conditions, they can be shaped into a train of bright-bright, dark-dark, or dark-bright vector solitons.

Observation and analysis of multiple dark-antidark solitons in two-component Bose-Einstein condensates

We report on the static and dynamical properties of multiple dark-antidark solitons (DADs) in two-component, repulsively interacting Bose-Einstein condensates. Motivated by experimental observations involving multiple DADs, we present a theoretical study which showcases that bound states consisting of dark (antidark) solitons in the first (second) component of the mixture exist for different values of interspecies interactions. It is found that ensembles of few DADs may exist as stable configurations, while for larger DAD arrays, the relevant windows of stability with respect to the interspecies interaction strength become progressively narrower. Moreover, the dynamical formation of states consisting of alternating DADs in the two components of the mixture is monitored. A complex dynamical evolution of these states is observed, leading either to sorted DADs or to beating dark-dark solitons depending on the strength of the interspecies coupling. This study demonstrates clear avenues for future investigations of DAD configurations.

On Solutions of an Extended Nonlocal Nonlinear Schrödinger Equation in Plasmas

The parity-time symmetric nonlocal nonlinear Schrödinger equation with self-consistent sources (PTNNLSESCS) is used to describe the interaction between an high-frequency electrostatic wave and an ion-acoustic wave in plasmas. In this paper, the soliton solutions, rational soliton solutions and rogue wave solutions are derived for the PTNNLSESCS via the generalized Darboux transformation. We find that the soliton solutions can exhibit the elastic interactions of different type of solutions such as antidark-antidark, dark-antidark, and dark-dark soliton pairs on a continuous wave background. Also, we discuss the degenerate case in which only one antidark or dark soliton remains. The rogue wave solution is derived in some specially chosen situations.

New Exact Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

In this paper, coupled nonlinear Schrodinger equations with variable coefficients are studied, which can be used to describe the interaction among the modes in nonlinear optics and Bose-Einstein condensation. Some novel bright-dark solitons and dark-dark solitons are obtained by modified Sine-Gordon equation method. Moreover, some figures are provided to illustrate how the soliton solutions propagation is determined by the different values of the variable group velocity dispersion terms, which can be used to model various phenomena.

Bilinear Forms and Dark-Dark Solitons for the Coupled Cubic-Quintic Nonlinear Schrödinger Equations with Variable Coefficients in a Twin-Core Optical Fiber or Non-Kerr Medium**Supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11805020, 11272023 and 11471050, the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05),

Twin-core optical fibers are applied in such fields as the optical sensing and optical communication, and propagation of the pulses, Gauss beams and laser beams in the non-Kerr media is reported. Studied in this paper are the coupled cubic-quintic nonlinear Schrödinger equations with variable coefficients, which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium. Based on the integrable conditions, bilinear forms are derived, and dark-dark soliton solutions can be constructed in terms of the Gramian via the Kadomtsev-Petviashvili hierarchy reduction. Propagation and interaction of the dark-dark solitons are presented and discussed through the graphic analysis. With different values of the delayed nonlinear response effect b(z), where z represents direction of the propagation, the linear- and parabolic-shaped one dark-dark soltions can be derived. Interactions between the parabolic- and periodic-shaped two dark-dark solitons are presented with b(z) as the linear and periodic functions, respectively. Directions of velocities of the two dark-dark solitons vary with z and the amplitudes of the solitons remain unchanged can be observed. Interactions between the two dark-dark solitons of different types are displayed, and we observe that the velocity of one soliton is zero and direction of the velocity of the other soliton vary with z. We find that those interactions are elastic.

Multi-dark soliton solutions for the (2+1)-dimensional multi-component Maccari system

Based on reduction of the KP hierarchy, the general multi-dark soliton solutions in Gram type determinant forms for the (2[Formula: see text]+[Formula: see text]1)-dimensional multi-component Maccari system are constructed. Especially, the two component coupled Maccari system comprising of two component short waves and single-component long waves are discussed in detail. Besides, the dynamics of one and two dark-dark solitons are analyzed. It is shown that the collisions of two dark-dark solitons are elastic by asymptotic analysis. Additionally, the two dark-dark solitons bound states are studied through two different cases (stationary and moving cases). The bound states can exist up to arbitrary order in the stationary case, however, only two-soliton bound state exists in the moving case. Besides, the oblique stationary bound state can be generated for all possible combinations of nonlinearity coefficients consisting of positive, negative and mixed cases. Nevertheless, the parallel stationary and the moving bound states are only possible when nonlinearity coefficients take opposite signs.

Integrable discretizations of AB system and multi-soliton solutions

Different integrable discretizations of AB system are proposed in this paper. Discrete AB (dAB) system and two different versions of semi-discrete AB (sdAB) systems are obtained by discretization of associated Lax pair of continuous integrable AB system. Discrete multiple soliton solutions are constructed for dAB system and sdAB systems by applying Darboux transformation. In order to explain our construction we also computed discrete single and double soliton solutions for dAB and sdAB systems. Dynamics of one discrete bright and dark soliton solution and interaction of two discrete bright-bright and dark-dark soliton solutions for dAB system have been presented.

Coupling of optical spatial solitons in photorefractive multiple quantum well planar waveguide

A comprehensive theory predicting the existence of both incoherently coupled and coherently coupled grey and dark optical spatial solitons in multiple quantum well planar (MQW) waveguides is presented. In addition, we also show the existence of dark Manakov soliton pair and the bright-dark Manakov soliton pair in such Multiple Quantum well waveguides for the first time. In the case of a coherently coupled soliton pair, the mutual phase difference, the overlap of the optical mode on the waveguide, the external bias field and beam intensity ratio have a profound effect on the FWHM of the soliton pairs. As a general case, a theory for the interesting and hitherto uninvestigated case of coherently coupled multicomponent solitons has been presented and analyzed for the first time. The effect of the mutual phase difference, the overlap of the optical mode on the waveguide(overlap parameter), the external bias field and beam intensity ratio on the FWHM is studied for these coherently coupled multicomponent solitons also. The Manakov soliton solutions are found analytically and they exist in the limit of low total intensity of the beams. We find that the dark-dark and bright-dark Manakov soliton pairs exist in such MQW waveguides whereas bright-bright Manakov soliton pair does not exist. The Manakov solitons’ spatial profile depends upon the overlap parameter, the intensity of the components, and the external electric field.

A coupled focusing-defocusing complex short pulse equation: Multisoliton, breather, and rogue wave.

Nonlinear Schrödinger equation, short pulse equation, and complex short pulse equation have important applications in nonlinear optics. They can be derived from the Maxwell equations. In this paper, we investigate a coupled focusing-defocusing complex short pulse equation. The bright-bright, bright-dark, and dark-dark soliton solutions of the coupled focusing-defocusing complex short pulse equation are constructed. Furthermore, the breather solutions are derived from the dark soliton solution. The rogue wave solutions are also constructed. The dynamics and the asymptotic behavior of the soliton solutions are analyzed, which reveals a fact that the bright-bright soliton collision can be either elastic or inelastic, but the bright-dark and dark-dark soliton collision can only be elastic.

Dark-dark-soliton dynamics in two density-coupled Bose-Einstein condensates

We study the 1D dynamics of dark-dark solitons in the miscible regime of two density-coupled Bose-Einstein condensates having repulsive interparticle interactions within each condensate ($g>0$). By using an adiabatic perturbation theory in the parameter $g_{12}/{g}$, we show that, contrary to the case of two solitons in scalar condensates, the interactions between solitons are attractive when the interparticle interactions between condensates are repulsive $g_{12}>0$. As a result, the relative motion of dark solitons with equal chemical potential $\mu$ is well approximated by harmonic oscillations of angular frequency $w_r=(\mu/\hbar)\sqrt{({8}/{15}){g_{12}}/{g}}$. We also show that in finite systems, the resonance of this anomalous excitation mode with the spin density mode of lowest energy gives rise to alternating dynamical instability and stability fringes as a function of the perturbative parameter. In the presence of harmonic trapping (with angular frequency $\Omega$) the solitons are driven by the superposition of two harmonic motions at a frequency given by $w^2=(\Omega/\sqrt{2})^2+w_r^2$. When $g_{12}<0$, these two oscillators compete to give rise to an overall effective potential that can be either single well or double well through a pitchfork bifurcation. All our theoretical results are compared with numerical solutions of the Gross-Pitaevskii equation for the dynamics and the Bogoliubov equations for the linear stability. A good agreement is found between them.

Dark-dark solitons for a set of the coupled nonlinear Schrödinger equations in a birefringent fiber

In this paper, we investigate a set of the coupled nonlinear Schrödinger equations with the four-wave mixing term, which describes the optical solitons in a birefringent fiber. Via the Kadomtsev–Petviashvili hierarchy reduction, we obtain the nondegenerate N-dark-dark soliton solutions in the Gram determinant form. Based on such solutions, we get the nondegenerate dark-dark solitons. With the analytic and graphic analysis, soliton propagating on a periodic background is presented. Via the asymptotic analysis, we find that the interaction between the two dark-dark solitons is elastic, and such an elastic interaction is shown graphically. Moreover, we also present the three dark-dark and bound-state dark-dark solitons.