Dipole Solitons Research Articles

One-dimensional localized modes of spin-orbit-coupled Bose-Einstein condensates with spatially periodic modulated atom-atom interactions: Nonlinear lattices

Bose-Einstein condensates (BECs) provide a clear and controllable platform to study diverse intriguing emergent nonlinear effects that appear too in other physical settings, such as bright and dark solitons in mean-field theory as well as many-body physics. Various ways have been elaborated to stabilize bright solitons in BECs, three promising schemes among which are: optical lattices formed by counterpropagating laser beams, nonlinear managements mediated by Feshbach resonance, spin-orbit coupling engineered by dressing atomic spin states (hyperfine states of spinor atomic BECs) with laser beams. By combing the latter two schemes, we discover, from theory to calculations, that the two-component BECs with a spin-orbit coupling and cubic atom-atom interactions, whose nonlinear distributions exhibit a well-defined spatially periodic modulation (nonlinear lattice), can support one-dimensional localized modes of two kinds: fundamental solitons (with a single peak), and soliton pairs comprised of dipole solitons (anti-phase) or two-peak solitons (in-phase). The influence of three physical parameters: chemical potential of the system, strengths of both the Rashba spin-orbit coupling and atom-atom interactions, on the existence and stability of the localized modes is investigated based on linear-stability analysis and direct perturbed simulations. In particular, we demonstrate that the localized modes can be stable objects provided always that both the inter- and intraspecies interactions are attractive.

Gap solitons in parity-time-symmetric lattices with fractional-order diffraction

We investigate new types of gap solitons in a periodic parity-time (PT)-symmetric lattice with fractional-order diffraction. Both the fundamental and dipole solitons in the first and second gaps are discussed. It is found that fractional-order diffraction can not only stabilize low-power dipole PT solitons in the first gap under focusing nonlinearity, but also help to get stable dipole PT solitons in the second gap under defocusing nonlinearity. Additionally, the influence of the strength w i of the gain–loss component on the properties of solitons is also analyzed. It is shown that increasing w i is unfavorable to the stability of fractional fundamental solitons, especially for the second gap, while for fractional dipole solitons, the increase of w i may lead to their destabilization in the first gap, but stabilization in the second gap.

Elliptic fundamental, dipole and vortex solitons in nonlocal nonlinear media with linear anisotropic diffraction

Elliptic fundamental, dipole and vortex solitons in nonlocal nonlinear media with linear anisotropic diffraction are investigated analytically and numerically. In particular we do this for the whole range of degrees of nonlocality. Analytical solutions for the solitons are obtained with the variational approach. The dynamics of elliptic solitons are also numerically demonstrated with the split-step Fourier transform method. With the help of linear anisotropic diffraction, stable elliptic fundamental and dipole solitons can be obtained. However, elliptic vortex solitons are always unstable and transfer into spiraling elliptic vortex solitons.

Gap and embedded solitons in microwave-coupled binary condensates

It was recently found that, under the action of the spin-orbit coupling (SOC) and Zeeman splitting (ZS), binary BEC with intrinsic cubic nonlinearity supports families of gap solitons, provided that the kinetic energy is negligible in comparison with the SOC and ZS terms. We demonstrate that, also under the action of SOC and ZS, a similar setting may be introduced for BEC with two components representing different atomic states, resonantly coupled by microwave radiation, while the Poisson equation accounts for the feedback of the two-component atomic wave function onto the radiation. The microwave-mediated interaction induces an effective nonlinear trapping potential, which strongly affects the purport of the linear spectrum in this system. As a result, families of both gap and embedded solitons (those overlapping with the continuous spectrum) are found, being chiefly stable. The shape of the solitons features exact or broken skew symmetry. In addition to fundamental solitons (whose shape may or may not include a node), a family of dipole solitons is constructed too, which are even more stable than their fundamental counterparts. A nontrivial stability area is identified for moving solitons in the present system, which lacks Galilean invariance. Colliding solitons merge into a single one.

Gap solitons and their stabilities in a quasi one-dimensional Bose-Einstein condensate under three-body interaction

We study the gap solitons and their stability properties in a Bose-Einstein condensation (BEC) under three-body interaction loaded in a Jacobian elliptic sine potential, which can be described by a cubic-quintic Gross-Pitaevskii equation (GPE) in the mean-field approximation. Firstly, the GPE is transformed into a stationary cubic-quintic nonlinear Schrödinger equation (NLSE) by the multi-scale method. A class of analytical solution of the NLSE is presented to describe the gap solitons. It is shown analytically that the amplitude of the gap soliton decreases as the two-body or three-body interaction strength increases. Secondly, many kinds of gap solitons, including the fundamental soliton and the sub-fundamental soliton, are obtained numerically by the Newton-Conjugate-Gradient (NCG) method. There are two families of fundamental solitons: one is the on-site soliton and the other is the off-site soliton. All of them are bifurcated from the Bloch band. Both in-phase and out-phase dipole solitons for off-site solitons do exist in such a nonlinear system. The numerical results also indicate that the amplitude of the gap soliton decreases as the nonlinear interaction strength increases, which accords well with the analytical prediction. Finally, long-time dynamical evolution for the GPE is performed by the time-splitting Fourier spectrum method to investigate the dynamical stability of gap solitons. It is shown that the on-site solitons are always dynamically stable, while the off-site solitons are always unstable. However, both stable and unstable in-phase or out-phase dipole solitons, which are not bifurcated from the Bloch band, indeed exist. For a type of out-phase soliton, there is a critical value <inline-formula><tex-math id="M1">\begin{document}$ q_c$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191278_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191278_M1.png"/></alternatives></inline-formula> when the chemical potential <i>μ</i> is fixed. The solitons are linearly stable as <inline-formula><tex-math id="M2">\begin{document}$ q>q_c$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191278_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191278_M2.png"/></alternatives></inline-formula>, while they are linearly unstable for <inline-formula><tex-math id="M3">\begin{document}$ q<q_c$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191278_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="1-20191278_M3.png"/></alternatives></inline-formula>. Therefore, the modulus <i>q</i> plays an important role in the stability of gap solitons. One can change the dynamical behavior of gap solitons by adjusting the modulus of external potential in experiment. We also find that there exists a kind of gap soliton, in which the soliton is dynamically unstable if only the two-body interaction is considered, but it becomes stable when the three-body interaction is taken into account. This indicates that the three-body interaction has influence on the stability of gap solitons.

1D Solitons in Saturable Nonlinear Media with Space Fractional Derivatives

AbstractTwo decades ago, standard quantum mechanics entered into a new territory called space‐fractional quantum mechanics, in which wave dynamics and effects are described by the fractional Schrödinger equation. Such territory is now a key and hot topic in diverse branches of physics, particularly in optics driven by the recent theoretical proposal for emulating the fractional Schrödinger equation. However, the light‐wave propagation in saturable nonlinear media with space fractional derivatives is yet to be clearly disclosed. Here, such nonlinear optics phenomenon is theoretically investigated based on the nonlinear fractional Schrödinger equation with nonlinear lattices—periodic distributions of either focusing cubic (Kerr) or quintic saturable nonlinearities—and the existence and evolution of localized wave structures allowed by the model are addressed. The model upholds two kinds of one‐dimensional soliton families, including fundamental solitons (single peak) and higher‐order solitonic structures consisting of two‐hump solitons (in‐phase) and dipole ones (anti‐phase). Notably, the dipole solitons can be robust stable physical objects localized merely within a single well of the nonlinear lattices—previously thought impossible. Linear‐stability analysis and direct simulations are executed for both soliton families, and their stability regions are acquired. The predicted solutions can be readily observed in optical experiments and beyond.

Dynamics of Multipole Solitons and Vortex Solitons in PT-Symmetric Triangular Lattices with Nonlocal Nonlinearity

We investigate the evolution dynamics of solitons with complex structures in the PT-symmetric triangular lattices with nonlocal nonlinearity, including dipole solitons, six-pole solitons, and vortex solitons. Dipole solitons can be linearly stable with a small degree of gain/loss, while six-pole solitons can only be stable when both the degree of gain/loss and the degree of nonlocality are small. For unstable solitons, some humps will decay quickly or new hotspots will appear during propagation. According to the existence range of dipole solitons, the multipole solitons tend to exist in PT-symmetric triangular lattices whose nonlocal nonlinearity is intermediate. We also consider the vortex solitons with high topological charges in the same triangular lattices and find that their profiles are codetermined by the propagation constant, degree of nonlocality, and topological charge.

Investigation of dynamics of nematicons in liquid crystals by extended sinh-Gordon equation expansion method

This paper secures new nematicons in liquid crystals from its governing equation by extended sinh-Gordon equation expansion method with the aid of symbolic computational package Maple. The two laws of nonlinearity, namely Kerr and parabolic laws are studied. As outcomes, some new explicit complex hyperbolic and complex trigonometric function solutions are derived regarding dark, bright, combined dark–bright, singular, combined singular optical, periodic wave, dipole soliton solutions and others. To show the real physical meaning of the derived results, three dimensional graphs with contours and two dimensional graphs are prepared under the choice of proper values of the arbitrary parameters. The derived solutions have been checked back into their corresponding equations with the symbolic computational package Maple. All the combined solutions are found to be new, which may play an important role in the context of liquid crystals. The applied method is found to provide a powerful tool for extracting exact solitary wave solutions and overcome the difficulties of the solitary wave ansatz method.

Dipole and Combo Optical Solitons in Birefringent Fibers in the Presence of Four-Wave Mixing

The ultimate theme of this study is to develop dipole and combo optical solitons in birefringent fibers (BF) along with the effect of four-wave mixing (4WM). Two types of rough mediums are used which are Kerr law and parabolic law. Ansatz method of Choudhuri is applied to obtain dark in the bright (dipole) soliton solutions providing bright background for the propagation of optical dark pulse. Ansatz method of Li is applied to obtain combo soliton solutions providing bright solitary wave and dark solitary wave solutions. These results also exist to hold in non-Kerr media with higher order dispersion.

New combo and dipole solitons in multiple-core couplers with dual dispersion

This paper is concerned with combo and dipole solitons in nonlinear directional optical couplers using Kerr law media. The governing model is multiple-core couplers with group velocity dispersion (GVD) and spatio-temporal dispersion (STD) described by a nonlinear Schrödinger's equation (NLSE). The Ansatz method is used to construct these combo and dipole solitons. The necessary conditions for the existence of these solitons is also given.

Spatial solitons in a nonlocal fused coupler

We study soliton-modes of optical pulses and their stabilities in a nonlocal nonlinear fused dual-core planar waveguide. Taking the different parameters, we find the symmetric, antisymmetric and asymmetric bright soliton pairs and dipole soliton pairs. During the long-distance propagation, the amplitudes of some solitons adding the white noise exhibit periodic oscillations and the solitons except asymmetric soliton can propagate steadily. We also find that the phases and profiles of solitons in the nonlocal coupler are determined by the propagation constant, the amplitude of the optical field and the region of the fused segment.

Chirped and dipole soliton in nonlinear negative-index materials

In this article, we find chirped and dipole soliton for nonlinear negative-index materials with quadratic-cubic nonlinearity. We will use sub-ODE method [32] to find chirped soliton and ansatz method of Choudhuri and Porsezian [41] to get dipole soliton.

Optical solitons in media with focusing and defocusing saturable nonlinearity and a parity-time-symmetric external potential.

We report results for solitons in models of waveguides with focusing or defocusing saturable nonlinearity and a parity-time ([Formula: see text])-symmetric complex-valued external potential of the Scarf-II type. The model applies to the nonlinear wave propagation in graded-index optical waveguides with balanced gain and loss. We find both fundamental and multipole solitons for both focusing and defocusing signs of the saturable nonlinearity in such [Formula: see text]-symmetric waveguides. The dependence of the propagation constant on the soliton's power is presented for different strengths of the nonlinearity saturation, S The stability of fundamental, dipole, tripole and quadrupole solitons is investigated by means of the linear-stability analysis and direct numerical simulations of the corresponding (1+1)-dimensional nonlinear Schrödinger-type equation. The results show that the instability of the stationary solutions can be mitigated or completely suppressed, increasing the value of SThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Propagation properties of dipole-managed solitons through an inhomogeneous cubic–quintic–septic medium

The presence of inhomogeneities in an optical nonlinear material may significantly change the physical features of propagating envelopes. In this paper, we discuss the propagation of very short pulses in an inhomogeneous highly nonlinear single-mode fiber within the context of a higher-order nonlinear Schrödinger equation exhibiting a diversity of important physical effects and spatially inhomogeneous coefficients. Additional effects to the cubic model include distributed third- and fourth-order dispersion, self-steepening, self-frequency shift due to stimulated Raman scattering, quintic–septic non-Kerr nonlinearities, the time derivative of non-Kerr nonlinear terms, and gain or loss term. By adopting a complex amplitude ansatz solution that is composed of the product of bright and dark solitary waves, the exact dipole soliton solution is derived. The conditions on the inhomogeneous fiber parameters for the existence of dipole structures are also reported. It is shown that the soliton dynamics in the inhomogeneous fiber media can be effectively controlled by choosing the parameters associated with the group velocity and the gain or loss term appropriately. Different from the higher-order nonlinear Schrödinger equation with constant coefficients, dipole solitons in variable-coefficient model have shown novel and interesting features. Finally, the stability of the dipole solitons is discussed numerically under finite initial perturbations.

Stability and internal interaction of multipole solitons in nonlocal PT-symmetric lattices.

We investigate the existence, stability and internal interaction of two-dimensional multipole solitons in defocusing PT symmetric nonlocal nonlinear media. Compared with nonlocal fundamental solitons in PT-symmetric lattices, the multipole solitons reveal the novel internal interaction such as the oscillation of the gravity center. We consider the change of the gravity center of dipole solitons during propagation, and find that these dipole solitons have "breather-like" or "competition" behavior in different nonlinearities. We also demonstrate that dipole solitons are easier to be stabilized under the intermediate nonlocality. Moreover, two more complicated multipole solitons are studied and it is found that there also exist the novel internal interaction for the multipole solitons.

Dipole and combo solitons in DWDM systems

In this article, we find the dipole soliton and the combo soliton solutions in DWDM systems. The governing model is Schrödinger–Hirota equation with spatio-temporal dispersions having group velocity dispersions. We use ansatz method of Choudhuri [20] to get dipole soliton solutions in DWDM systems. We also use ansatz method of Li [16] to get combo soliton.

Multi-pole dark solitons in nonlocal and cubic-quintic nonlinear medium

In this paper, we mainly simulate the characteristics of the ground state dark soliton and the multipole dark soliton in the nonlocal and cubic-quintic nonlinear medium. Firstly, the influences of the degree of nonlocality on the amplitude and the width of the dark soliton in the self-defocusing cubic-and self-focusing quantic-nonlinear medium are studied. Secondly, we find the nonlinear parameters affecting the amplitude values of solitons, but the refractive index induced by the light beam is always a fixed value. The numerical results show that the ground state dark soliton can be propagated stably alone the z axis, and the stable states of the dipole soliton and the dark tri-pole and quadru-pole solitons are stable. However, some quadru-pole dark soliton is unstable after propagating the remote distance. Furthermore, we also discuss the characteristics of the ground state dark soliton and the dark dipole soliton in the local cubic-nonlinear and nonlocal quantic nonlinear media. Both the amplitude and the beam width of the dark ground state soliton and dark dipole soliton are also affected by the degree of nonlocality and nonlinearity. Two boundary values of the induced refractive index change with the variations of the three nonlinear parameters. The dark soliton and the dipole dark soliton are more stable in the self-focusing cubic nonlinear and the nonlocal self-defocusing quantic nonlinear medium than those in the self defocusing cubic nonlinear and nonlocal self-focusing quantic nonlinear medium. The powers of single dark soliton and dark tri-pole soliton decrease monotonically with the increase of propagation constant when the cubic-quintic nonlinearities are certain values and these degrees of nonlocalities are taken different values. Furthermore, we also analyze linear stabilities of various nonlocal spatial dark solitons. And we find that the dipole dark soliton is unstable when the propagation constant is in the region[-0.9,-1.0]. These properties of linear stabilities of other multi-pole dark solitons are the same as their propagation properties.

Scalar and vector multipole and vortex solitons in the spatially modulated cubic–quintic nonlinear media

We derive scalar and vector multipole and vortex soliton solutions in the spatially modulated cubic–quintic nonlinear media, which is governed by a (3+1)-dimensional N-coupled cubic–quintic nonlinear Schrodinger equation with spatially modulated nonlinearity and transverse modulation. If the modulation depth \(q=1\), the vortex soliton is constructed, and if \(q=0\), the multipole soliton, including dipole, quadrupole, hexapole, octopole and dodecagon solitons, is constructed, respectively, when the topological charge \(k=1\)–5. If the topological charge \(k=0\), scalar solitons can be obtained. Moreover, the number of layers for the scalar and vector multipole and vortex solitons is decided by the value of the soliton order number n.

Solitary wave solutions for quintic complex Ginzburg–Landau model

The article studies dark in the bright (dipole) soliton solutions and the solitary wave (combo) solutions for quintic complex Ginzburg–Landau model (CGLQ). The model is generated from cubic complex Ginzburg–Landau equation (CGLC), derived from nonlinear Schrödinger equation (NLSE). We use ansatz method of Li [21] to get combo soliton solutions for (CGLQ) model. To get the dipole soliton solution, we use the ansatz method of Choudhuri [20].

Dipole solitons in an extended nonlinear Schrödinger's equation with higher-order even and odd terms

We investigate the extended nonlinear Schrödinger equation with higher-order odd and even terms. The model includes additional higher-order dispersion and nonlinear terms that are most important for applications in fiber optics, in the case of the Heisenberg spin chain, and for ocean waves. Special exact solutions in the form of a dipole soliton is obtained, by adopting a complex amplitude ansatz composed of the product of bright and dark solitary waves. The conditions on the system parameters for the existence of this localized structure are also reported. The derived solution characteristically exists due to a balance among physical effects of different nature. Numerical results and discussions are also presented.