Anti-dark Solitons Research Articles

Dynamics of optical solitons and conservation laws of a new (2+1)-dimensional integrable nonlinear evolution equation in deep water oceanic waves

An integrable extension of the famous Schrödinger equation in (2[Formula: see text]+[Formula: see text]1) dimension, named Kundu–Mukherjee–Naskar (KMN) equation, governing the evolution of ion-acoustic wave in magnetized plasma and oceanic rogue waves is considered, and dark/black as well as gray optical soliton solutions are constructed by using a complex envelope ansatz approach with appropriate conditions for the existence of solitons. Also, a new class of combined gray and black optical soliton solutions is obtained by applying Chupin Liu’s theorem, and it is found to be anti-dark solitons. Additionally, Gaussian wave solutions are derived. Further, the investigation of symmetry analysis, nonlinear self-adjointness and conservation laws (Cls) for the KMN equation are carried out. These results further enrich and deepen the understanding of the dynamics of a higher-dimensional soliton propagation.

Dynamics of solitons in the fourth-order nonlocal nonlinear Schrödinger equation

We consider the fourth-order nonlocal nonlinear Schrodinger equation and generate the Lax pair. We then employ Darboux transformation to generate dark and antidark soliton solutions. The highlight of the results is that one ends up generating a two-soliton solution characterized by one spectral parameter alone, a property which has never been witnessed so far.

Breathers, multi-peak solitons, breather-to-soliton transitions and modulation instability of the variable-coefficient fourth-order nonlinear Schrödinger system for an inhomogeneous optical fiber

Optical fiber communication system is one of the core supporting systems of the modern internet age. In this paper, under investigation is a variable-coefficient fourth-order nonlinear Schrödinger system, which describes the simultaneous propagation of the optical pulses in an inhomogeneous optical fiber. The first-order breathers are constructed. Breathers are converted into several types of the nonlinear waves, i.e., multi-peak solitons, antidark solitons, periodic waves and W-shaped solitons, under the transition condition. With the decrease of |λr+a2|, the peak number of a multi-peak soliton increases, where λr represents the real part of the spectral parameter and a is a real constant. Peak number reaches the maximum since the multi-peak soliton is transformed into a set of the periodic waves with λr+a2=0. Widths and velocities of the multi-peak solitons are related to the group velocity dispersion and fourth-order dispersion coefficients. We find that the condition of baseband modulational instability coincides with the transition condition when we use the rogue-wave eigenvalues, i.e., the transition between the rogue waves and multi-peak solitons can occur in the baseband modulational instability.

Quintic time-dependent-coefficient derivative nonlinear Schrödinger equation in hydrodynamics or fiber optics: bilinear forms and dark/anti-dark/gray solitons

Studies on the water waves contribute to the design of the related industries, such as the marine and offshore engineering, while the media with the negative refractive index can be applied as the carrier media in fiber optics. In consideration of the inhomogeneities of the media and nonuniformities of the boundaries in the real physical backgrounds, a quintic time-dependent-coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or medium with the negative refractive index is investigated in this paper. Bilinear forms and the N-soliton solutions with respect to the nonzero background, which are different from those in the existing studies, are derived under the certain constraints. Conditions for the dark/anti-dark/gray solitons are deduced due to the properties of the solitons derived via the asymptotic analysis. Effects of the dispersion coefficient $$\lambda (t)$$ , self-steepening coefficient $$\alpha (t)$$ , cubic nonlinearity $$\mu (t)$$ and quintic nonlinearity $$\nu (t)$$ on the interactions between the anti-dark and gray solitons under the certain condition are investigated. Interactions among the dark, anti-dark and gray solitons are discussed under two cases: when $${\alpha (t)}/{\lambda (t)}$$ and $${\mu (t)}/{\lambda (t)}$$ are the constants, whether the interaction is elastic or not depends on whether $$\lambda (t)$$ , $$\alpha (t)$$ and $$\mu (t)$$ are the constants or the functions of t; when $${\alpha (t)}/{\lambda (t)}$$ and $${\mu (t)}/{\lambda (t)}$$ are related to t, if the velocity of the soliton is a periodic function of t, the propagation of the corresponding soliton is periodic and the corresponding interaction is inelastic. Interactions among the three/four solitons are described to be elastic or inelastic based on the changes in the velocities and waveforms of the three/four solitons after the interactions.

Patterns of water in light

The intricate patterns emerging from the interactions between soliton stripes of a two-dimensional defocusing nonlinear Schrödinger (NLS) model with a non-local nonlinearity are considered. We show that, for sufficiently strong non-locality, the model is asymptotically reduced to a Kadomtsev–Petviashvilli-II (KPII) equation, which is a common model arising in the description of shallow water waves, as such patterns of water may indeed exist in light (this non-local NLS finds applications in nonlinear optics, modelling beam propagation in media featuring thermal nonlinearities, in plasmas, and in nematic liquid crystals). This way, approximate antidark soliton solutions of the NLS model are constructed from the stable KPII line solitons. By means of direct numerical simulations, we demonstrate that non-resonant and resonant two- and three-antidark NLS stripe soliton interactions give rise to wave configurations that are found in the context of the KPII equation. Thus, our study indicates that patterns which are usually observed in water can also be found in optics.

Phase shift, oscillation and collision of the anti-dark solitons for the (3+1)-dimensional coupled nonlinear Schrödinger equation in an optical fiber communication system

In the long-distance optical fiber communication system, we consider the (3+1)-dimensional coupled nonlinear Schrodinger equation with perturbation functions, which controls the transverse effects, because of the complexity of nonlinear phenomena. Anti-dark two-soliton solutions are derived by the Hirota method. The perturbation function representing the dissipation rate is discussed. Five different transmission structures of solitons are presented. Besides, their propagation and collision dynamics are analyzed. Control methods for phase shift and oscillation are also suggested. We hope results in this paper are of guiding significance to the research of phase shifters, optical logic devices and ultrashort pulse lasers.

Control of dark and anti-dark solitons in the (2+1)-dimensional coupled nonlinear Schrödinger equations with perturbed dispersion and nonlinearity in a nonlinear optical system

In this paper, we investigate the (2+1)-dimensional coupled nonlinear Schrodinger equations with perturbed dispersion and nonlinearity, which govern the transverse effects in a nonlinear optical system. Using symbolic calculation, the vector one- and two-soliton solutions are obtained via the Hirota method. By choosing the perturbation $$\alpha (t)$$ of the dispersion rate of soliton transmission as different functions, we observe different dark and anti-dark soliton structures. Among other, the parabolic dark soliton, m-shaped and w-shaped anti-dark solitons, two kinds of s-shaped anti-dark solitons with different curvatures and an anti-dark soliton with a peak are displayed. Moreover, the effects of other free parameters on the phase shift and pulse width, and collision of solitons are discussed. These results are of potential significance for the study of ultrashort pulse lasers and optical logic switches.

Baseband modulation instability, rogue waves and state transitions in a deformed Fokas–Lenells equation

We study a deformed Fokas–Lenells equation which is related to the integrable derivative nonlinear Schrodinger hierarchy with higher-order nonholonomic constraint. The baseband modulation instability as an origin of rogue waves is displayed. The explicit rogue wave solutions are obtained via the Darboux transformation. Typical rogue wave patterns such as the standard rogue wave, dark rogue wave and twisted rogue wave pair in three different components of the deformed Fokas–Lenells equation are presented. Besides, the state transitions between rogue waves and solitons are analytically found when the modulation instability growth rate tends to zero in the zero-frequency perturbation region. The explicit soliton solutions under the special parameter condition are given. The anti-dark and W-shaped solitons in their respective components are shown.

Lax pair, breather-to-soliton conversions, localized and periodic waves for a coupled higher-order nonlinear Schrödinger system in a birefringent optical fiber

A coupled higher-order nonlinear Schrodinger system, which describes the ultrashort pulses in a birefringent optical fiber, is analytically investigated. For the complex envelopes of the electric field in the fiber, we construct a Lax pair which is different from that in the existing literature, and derive out the corresponding first- and second-order breather solutions. We present the first- and second-order breather-to-soliton conversion conditions, related to the strength of the higher-order linear and nonlinear effects. We find that the strength affects the peak numbers of solitons, and see the multi-peak soliton, W-shaped soliton, M-shaped soliton, anti-dark soliton and two kinds of periodic waves. From the second-order breather solutions under the first-order breather-to-soliton conversion condition, interactions between the breather and a W-shaped soliton, an M-shaped soliton or periodic waves are given. From the second-order breather solutions under the second-order breather-to-soliton conversion conditions, we present the interactions between the two M-shaped solitons, the two anti-dark solitons, a W-shaped soliton and an M-shaped soliton, or a W-shaped soliton and two kinds of the periodic waves. Those results might provide certain assistance for the studies on the birefringent optical fibers.

Manipulation of a stable dark soliton train in polariton condensate.

With a potential trap to mimic the finite-size effect, we predict the generation of dark soliton train in a nonresonant polariton condensate excited with the spatially homogeneous continuous wave (cw) field by exploiting the quantum interference in spatial domain. The number of solitons in the train is demonstrated to be controllable on demand in the domain of material parameters. We show that, similar to dark solitons in conservative systems, these nonlinear excitations have infinite lifetime and remain spatially localized even for the periodically oscillating and colliding state of dark soliton train, which do not depend on the parameters of the condensate. Especially, a final state of anti-dark soliton is observed for all kinds of dark soliton trains with the increase of pump power.

Excitation conditions of several fundamental nonlinear waves on continuous-wave background.

We study the excitation conditions of antidark solitons and nonrational W-shaped solitons in a nonlinear fiber with both third-order and fourth-order effects. We show that the relative phase can be used to distinguish antidark solitons and nonrational W-shaped solitons. The excitation conditions of these well-known fundamental nonlinear waves (on a continuous-wave background) can be clarified clearly by the relative phase and three previously reported parameters (background frequency, perturbation frequency, and perturbation energy). Moreover, the numerical simulations from the nonideal initial states also support these theoretical results. These results provide an important complement for the studies on relationship between modulation instability and nonlinear wave excitations, and are helpful for controllable nonlinear excitations in experiments.

On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation

We construct symmetry preserving and symmetry broken N-bright, dark and antidark soliton solutions of a nonlocal nonlinear Schrödinger equation. To obtain these solutions, we use appropriate eigenfunctions in Darboux transformation (DT) method. We present explicit one and two bright soliton solutions and show that they exhibit stable structures only when we combine the field and parity transformed complex conjugate field. Further, we derive two dark/antidark soliton solution with the help of DT method. Unlike the bright soliton case, dark/antidark soliton solution exhibits stable structure for the field and the parity transformed conjugate field separately. In the dark/antidark soliton solution case we observe a contrasting behaviour between the envelope of the field and parity transformed complex conjugate envelope of the field. For a particular parametric choice, we get dark (antidark) soliton for the field while the parity transformed complex conjugate field exhibits antidark (dark) soliton. Due to this surprising result, both the field and PT transformed complex conjugate field exhibit sixteen different combinations of collision scenario. We classify the parametric regions of dark and antidark solitons in both the field and parity transformed complex conjugate field by carrying out relevant asymptotic analysis. Further we present 2N-dark/antidark soliton solution formula and demonstrate that this solution may have 22N×22N combinations of collisions.

Localized waves for the mixed coupled Hirota equations in an optical fiber

Under investigation in this paper are the mixed coupled Hirota equations, which describe the simultaneous propagation of two ultrashort optical fields in an optical fiber. With respect to the two-component electric fields in the fiber, we construct the Lax pair, Nth-order Darboux transformation, and two types of the first-order and second-order localized wave solutions are derived. Based on such solutions, moving breathers, Akhmediev breathers, multi-peak solitons and anti-dark solitons are obtained. We find that the increase of the strength of the higher-order linear/nonlinear effects leads to the decrease of the velocity of the moving breather. Breather-to-soliton conversion is studied. We show that the moving breather can be converted to the anti-dark soliton. Elastic interactions between the breathers and dark solitons are studied. Additionally, inelastic interaction between the anti-dark soliton and dark soliton is shown.

Dark and antidark solitons for the defocusing coupled Sasa–Satsuma system by the Darboux transformation

In this article, we construct the N-fold Darboux transformation for the defocusing coupled Sasa–Satsuma system which describes the simultaneous propagation of two nonlinear waves in optical fibers with higher order effects. With the non-zero constant background as a seed, we derive the dark and antidark soliton solutions from the once-iterated formula. We find that this coupled system can exhibit the dark–dark, dark–antidark and antidark–dark vector solitons.

Hybrid solutions to Mel’nikov system

Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.

Deformation of dark solitons in a -invariant variable coefficients nonlocal nonlinear Schrödinger equation.

We derive dark and antidark soliton solutions of a parity-time reversal -invariant variable coefficients nonlocal nonlinear Schrödinger (NNLS) equation. We map the considered equation into a defocusing -invariant NNLS equation with a constraint between dispersion, nonlinearity, and gain/loss parameters. We show that the considered system is -invariant only when the dispersion and nonlinearity coefficients are even functions and gain/loss coefficient is an odd function. The characteristics of the constructed dark soliton solutions are investigated with four different forms of dispersion parameters, namely, (1) constant, (2) periodically distributed, (3) exponentially distributed, and (4) periodically and exponentially distributed dispersion parameter. We analyze in detail how the nonlocal dark soliton profiles get deformed in the plane wave background with these dispersion parameters.

Abundant exact solutions to the discrete complex mKdV equation by Darboux transformation

In this paper, an N-fold Darboux transformation is constructed for the discrete complex modified Korteweg-de Vries equation of focusing type, in terms of determinants. Through the obtained one-fold and two-fold Darboux transformations, a variety of new exact solutions, including an anti-dark soliton solution, a breather solution, a periodic solution, and a two-soliton solution, are derived from a nonzero constant and plane-wave seed solution. Via numerical simulation, a new kind of dynamical behavior of the two-soliton solution is exhibited, which tells that the two-soliton solution includes an anti-dark solitary wave and a w-shaped solitary wave.

Interactions of vector anti-dark solitons for the coupled nonlinear Schrödinger equation in inhomogeneous fibers

We investigate the interactions of vector anti-dark solitons for variable coefficients coupled nonlinear Schrodinger equation, which governs the propagation mechanisms of electromagnetic waves in inhomogeneous birefringent fibers. Based on the Hirota method, various interactions in the inhomogeneous system are studied analytically. The double-S structure interactions are presented in this paper for the first time. The influences of relevant parameters on S-shape solitons are discussed. Periodical transmitting anti-dark solitons are presented by analyzing the effects on their amplitude, phase and intensity, respectively. Results in this paper may be potential applications for designing the birefringence managed switching and dark soliton fiber lasers.

Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation

We study the linear stability analysis and nonlinear excitations on a continuous wave background for the fifth-order nonlinear Schrödinger equation. We find that there are a modulation stability (MS) quasi-elliptic ring and a MS curve in the modulation instability (MI) regime. The concrete expressions of different types of nonlinear waves are obtained by using Darboux transformation. And we present the relations and phase diagram between MI and these waves. We discover that the antidark (AD) soliton and nonrational W-shaped soliton can only exist on the resonance line of outside of the MS quasi-elliptic ring and the right of the MS curve. We perform numerical simulation to test the stability of the AD soliton. By introducing the perturbation energy, we further show that solitons in the MI regime are caused by both the fourth- and fifth-order effects.

General soliton solutions to a $$\varvec{(2+1)}$$ ( 2 + 1 ) -dimensional nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions

We investigate a $$(2+1)$$ -dimensional nonlocal nonlinear Schrodinger equation with the self-induced parity-time symmetric potential. By employing the Hirota’s bilinear method and the KP hierarchy reduction method, general soliton solutions to the $$(2+1)$$ -dimensional nonlocal NLS equation with zero and nonzero boundary conditions are derived. These solutions are given in forms of Gram-type determinants. We first construct general bright solitons with zero boundary condition by constraining the tau functions of two-component KP hierarchy. Furthermore, we derive general dark and antidark solitons with nonzero boundary from the tau functions of single-component KP hierarchy.