Stability Of Solitons Research Articles

Three-component soliton states in spinor F = 1 Bose-Einstein condensates with PT-symmetric generalized Scarf-II potentials

Abstract Introducing PT-symmetric generalized Scarf-II potentials into the three-coupled nonlinear Gross-Pitaevskii equations offers a new way to seeking stable soliton states in quasi-one-dimensional spin-1 Bose-Einstein condensates. In scenarios where the spin-independent parameter c_0 and the spin-dependent parameter c_2 vary, we use both analytical and numerical methods to investigate the three-coupled nonlinear Gross-Pitaevskii equations with PT-symmetric generalized Scarf-II potentials. We obtain analytical soliton states and find that simply modulating c_2 may change the analytical soliton states from unstable to stable. Additionally, we obtain numerically stable double-hump soliton states propagating in the form of periodic oscillations, exhibiting distinct behavior in energy exchange. For further investigation, we discuss the interaction of numerical double-hump solitons with Gaussian solitons and observe the transfer of energy among the three components. These findings may contribute to a deeper understanding of solitons in Bose-Einstein condensates with PT-symmetric potentials and may hold significance for both theoretical understanding and experimental design in related physics experiments.

Elliptic Hermite-Gaussian soliton and transformations in nonlocal media induced by linear anisotropy.

Elliptic Hermite-Gaussian (HG) soliton clusters in nonlocal media with anisotropic diffractions are studied comprehensively. The relations among solitons parameters, diffraction indices, and the degree of nonlocality are derived analytically with the Lagrangian method. Stable elliptic HG soliton clusters can be obtained when linear diffraction is anisotropic. When the solitons are launched with an initial orientation angle, we also demonstrate numerically mode transformations between HG and Laguerre-Gaussian (LG) solitons induced by linear anisotropy. Our results will enrich the soliton phenomenon with linear anisotropic diffraction and may lead to novel applications in all-optical switching, interconnection, etc.

Wavelength-Switchable Ytterbium-Doped Mode-Locked Fiber Laser Based on a Vernier Effect Filter

A wavelength-switchable ytterbium-doped mode-locked fiber laser is reported in this article. Two Mach–Zehnder interferometers (MZIs, denoted as MZI1, MZI2) with close free spectral ranges (FSRs) are connected in series to form a Vernier effect sensor. By utilizing the filtering effect of the Vernier effect sensor, the wavelength-switchable output of an ytterbium-doped mode-locked fiber laser is realized. When the 3 dB bandwidth of the Vernier effect filter is set to be 5.31 nm around 1073.42 nm, stable dissipative solitons are obtained. Stretching MZI1 horizontally, the central wavelengths of the pulses can be switched among 1073.42 nm, 1055.38 nm, and 1036.22 nm, with a total tunable central wavelength range of 37.2 nm. When the 3 dB bandwidth of the Vernier effect filter is set to be 4.07 nm, stable amplifier similaritons are obtained. Stretching MZI1 horizontally, the central wavelengths of the pulses are switchable among 1072.71 nm, 1060.15 nm, 1048.92 nm, and 1037.26 nm, with a total tunable central wavelength range of 35.15 nm. Compared with traditional fiber interference filters, the Vernier effect filter has a higher sensitivity, making wavelength switching more convenient and providing a wider tuning range for the ytterbium-doped mode-locked fiber laser.

Excitation and manipulation of super cavity solitons in multi-stable passive Kerr resonators

We report on the theoretical analysis as well as the numerical simulations about the nonlinear dynamics of cavity solitons in a passive Kerr resonator operating in the multistable regime under the condition of a sufficiently strong pump. In this regime, the adjacent tilted cavity resonances might overlap, thus leading to the co-existence of combinatory states of temporal cavity solitons and the extended modulation instability patterns. The cavity in the regime of multistablity may sustain distinct families of cavity solitons, vividly termed as super cavity solitons with much higher intensity and broader spectra if compared with those in the conventional bi-stable regime. The description of such complex cavity dynamics in the multstable regime requires either the infinite-dimensional Ikeda map, or the derived mean-field coupled Lugiato-Lefever equations by involving multiple contributing cavity resonances. With the latter model, we revealed the existence of different orders of super cavity solitons, whose stationary solutions were obtained by using the Newton-Raphson algorithm. Along this line, with the continuation calculation, we have plotted the Hopf / saddle-node bifurcation curves, thus identifying the existing map of the stable and breathing (super) cavity solitons. With this defined parameter space, we have proposed an efficient method to excite and switch the super cavity solitons by adding an appropriate intensity (or phase) perturbation on the pump. Such deterministic cavity soliton manipulation technique is demonstrated to underpin the multi-level coding, which may enable the large capacity all-optical buffering based on the passive fiber ring cavities.

Solitons in one-dimensional non-Hermitian moiré photonic lattice

We study the excitation and stabilization of gap solitons and dipole solitons in a non-Hermitian moiré photonic lattice with self-focusing effect. The lattice is constituted by superimposition of a parity-time symmetric potential and an anti-parity-time symmetric potential with inconsistent frequencies. We find that the single gap solitons within semi-infinite bandgap can indeed stably persist, and their ss positions within the lattice can be precisely adjusted through controlling the oblique angles and propagation constants of incident solitons. Further study shows that the lattice enables dipole solitons with specific soliton energy to overcome inherent repulsion, facilitating their propagation within the lattice. This work unveils the existence of stable gap solitons supported by non-Hermitian moiré photonic lattices, emphasizing the pivotal roles of propagation constants and incident angles in regulating soliton propagation.

Optical soliton solutions of the stochastic resonant nonlinear Schrödinger equation with spatio temporal and inter-modal dispersion under generalized Kudryashov's law non-linearity

This article investigates optical soliton solutions within the stochastic resonant nonlinear Schrödinger equation (SRNLSE). This equation incorporates both spatio-temporal dispersion (STD) and inter-modal dispersion (IMD), along with multiplicative white noise and generalized Kudryashov's law non-linearity. By applying the extended auxiliary equation method, we identify a range of soliton solutions, including bright, dark, and singular solitons. Additionally, we derive solutions that take the form of Jacobi elliptic functions, Weierstrass elliptic functions, and periodic wave functions. The study provides significant insights into soliton dynamics within nonlinear optical systems affected by stochastic influences and complex dispersion interactions. Specifically, it highlights how the interplay of STD and IMD, coupled with the presence of multiplicative noise, shapes the behavior of solitons. Moreover, we delve into the effects of multiplicative noise on the exact solutions of the NLSE using the Maple software. Our analysis reveals that multiplicative noise, interpreted in the Ito sense, plays a crucial role in stabilizing the soliton solutions, particularly maintaining their stability around the zero state. This finding underscores the importance of noise in influencing the stability and dynamics of solitons in optical systems.

Chiral Bulk Solitons in Photonic Graphene with Decorated Boundaries

AbstractA chiral bulk soliton in a nonlinear photonic lattice with decorated boundaries, presenting a novel approach to manipulate photonic transport without extensive bulk modifications is proposed. Unlike traditional methods that rely on topological edge and corner modes, this strategy leverages the robust chiral propagation of bulk modes. By introducing nonlinearity into the system, a stable bulk soliton, akin to the topological valley Hall effects is found. The chiral bulk soliton exhibits remarkable stability; the energy does not decay even after a long‐distance propagation; and the corresponding Fourier spectrum confirms the absence of inter‐valley scattering indicating a valley‐locking property. The findings not only contribute to the fundamental understanding of nonlinear photonic systems but also hold significant practical implications for the design and optimization of photonic devices.

Modulational instability, generation, and evolution of rogue waves in the generalized fractional nonlinear Schrödinger equations with power-law nonlinearity and rational potentials.

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ, the Lévy index α, and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W-shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena.

Spontaneous symmetry breaking and vortices in a tri-core nonlinear fractional waveguide

We introduce a waveguiding system composed of three linearly-coupled fractional waveguides, with a triangular (prismatic) transverse structure. It may be realized as a tri-core nonlinear optical fiber with fractional group-velocity dispersion (GVD), or, possibly, as a system of coupled Gross–Pitaevskii equations for a set of three tunnel-coupled cigar-shaped traps filled by a Bose–Einstein condensate of particles moving by Lévy flights. The analysis is focused on the phenomenon of spontaneous symmetry breaking (SSB) between components of triple solitons, and the formation and stability of vortex modes. In the self-focusing regime, we identify symmetric and asymmetric soliton states, whose structure and stability are determined by the Lévy index of the fractional GVD, the inter-core coupling strength, and the total energy, which determines the system’s nonlinearity. Bifurcation diagrams (of the supercritical type) reveal regions where SSB occurs, identifying the respective symmetric and asymmetric ground-state soliton modes. In agreement with the general principle of the SSB theory, the solitons with broken inter-component symmetry prevail with the increase of the energy in the weakly-coupled system. Three-components vortex solitons (which do not feature SSB) are studied too. Because the fractional GVD breaks the system’s Galilean invariance, we also address mobility of the vortex solitons, by applying a boost to them.

Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations

Asymptotic stability of small solitons for one-dimensional nonlinear Schrödinger equations

Vortex light bullets in rotating Quasi-Phase-Matched photonic crystals

We present a methodology for the generation of stable vortex light bullets (LBs) in rotating Quasi-Phase-Matched (QPM) photonic crystals with quadratic nonlinearity. The photonic crystal is designed with a checkerboard structure, which is feasible to realize by using contemporary technological advancements. Within this framework, square- and rhombus-shaped LBs are observed, both of which are constructed as four-peak vortex mode. The control parameters include the effective phase mismatch, power, rotating frequency, and the size of checkerboard cells. These parameters play key roles in determining the distribution and stability domains of vortex LBs. In contrast to the stable vortex solitons observed in two-dimensional (2D) quadratic systems, the LBs investigated in the 3D rotating system exhibit narrower stability domains within the system’s parameter space. The rotating frequency results in the transition of LBs from quadrupole to traditional vortex modes. Potential applications of this research lie in the field of optical communications and information processing.

The theoretical study of high-order dissipation soliton molecules evolution in a passively mode-locked fiber laser

High-order solitons represent a special form of soliton. Due to the complexity of their spatial structure and dynamic evolution, they have important applications in communication systems such as wavelength division multiplexing and secure communication. However, compared to traditional solitons, research on the spectral evolution of high-order solitons is still insufficient. In this paper, through precise control of laser parameters, the dynamic evolution process from stable dissipative single soliton to complex dissipative 8-order soliton molecules in passively mode-locked fiber lasers was successfully simulated. This demonstrates the complexity of solitons interaction and reveals the regularity of soliton molecules order with variations in four parameters: the small signal gain coefficient, nonlinear coefficient, gain bandwidth and the polarization controller angle. Through deep analysis of the order of soliton molecules, pulse shape and spectral shape of dissipative soliton molecules, we showcase the potential of fine-tuning laser parameters to achieve high-order dissipative soliton molecules. This work provides important references for the optimization design and expanded applications of lasers, further driving the development of high-performance, high-efficiency fiber laser technology.

Nonlinear stability of smooth multi-solitons for the Dullin-Gottwald-Holm equation

In this work, the stability of exact smooth multi-solitons for the Dullin-Gottwald-Holm (DGH) equation is investigated for the first time. Firstly, through the inverse scattering approach, we derive the conservation laws in terms of scattering data and multi-solitons related to the discrete spectrum based on the Lax pair of the DGH equation. Notably, a new series Hamiltonian derived from bi-Hamilton structure are equivalent to the conservation laws, and can also be expressed in terms of scattering data. Thus we successfully connect scattering data (multi-solitons) to the recursion operator between Hamiltonian, and to the Lyapunov functional constructed from Hamiltonian. With this Lyapunov functional, it is identified that these smooth multi-solitons serve as non-isolated constrained minimizers, adhering to a suitable variational nonlocal elliptic equation. Moreover, the integrable properties of the recursion operator are presented, which is the crucial for spectral analysis in this work. Consequently, the investigation of dynamical stability transforms into a problem on the spectrum of explicit linearized systems. It is worth noting that, compared to previous works on Camassa-Holm equation and KdV-type equations, recursion operator derived from existing Hamiltonian for the DGH equation do not possess good integrable properties. We construct a new series of Hamiltonians with both analysable recursion operator and simple initial variation derivative (δH1/δu=m) to overcome this problem. Furthermore, orbital stability of the smooth double solitons is proved at last of the work.

Optical soliton stability in zig-zag optical lattices: comparative analysis through two analytical techniques and phase portraits

This article explores the examination of the widely employed zig-zag optical lattice model for cold bosonic atoms, which is commonly utilized to depict nonlinear wave in fluid mechanics and plasma physics. The focus is on obtaining soliton solutions in optics and investigating their physical properties. A wave transformation is initially applied to convert a partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are subsequently obtained through the application of two distinct methods, namely the generalized logistic equation method and the Sardar sub-equation method. These solutions include bright, dark, combined dark-bright, chirped type solitons, bell-shaped, periodic, W-shape, and kink solitons. In this paper, the solutions derived from two analytical approaches were compared to enhance the understanding of the behavior of the discussed nonlinear model. The obtained solutions have significant implications across various fields such as plasma physics, fluid dynamics, optics, and communication technology. Furthermore, 3D and 2D graphs are generated to depict the physical phenomena of the derived solutions by assigning appropriate constant parameters. The qualitative evaluation of the undisturbed planar system involves the analysis of phase portraits within bifurcation theory. Subsequently, the introduction of an outward force is carried out to induce disruption, and chaotic phenomena are unveiled. The detection of chaotic trajectory in the perturbed system is achieved through 3D plots, 2D plots, time scale plots, and Lyapunov exponents. Furthermore, stability analysis of the examined model is addressed under distinct initial conditions. Finally, the sensitivity assessment of the model under consideration is carried out using the Runge–Kutta method. The results of this study are innovative and have not been previously investigated for the system under consideration. The results obtained underscore the reliability, simplicity, and effectiveness of these techniques in analyzing a variety of nonlinear models found in mathematical physics and engineering disciplines.

Stability and instability nature of solitons in an optical fiber with four wave mixing effect

The investigation into modulational instability (MI) within the Kundu-Eckhaus (KE) equation, governing optical solitons, involves a thorough examination of the effects of self-phase modulation, cross-phase modulation, and intermodal dispersion. Special attention is given to understanding the influence of the four-wave mixing effect. The KE equation, which models birefringent fiber and includes terms related to intermodal dispersion, cross-phase modulation, and self-phase modulation, serves as the fundamental framework for this analytical study. Employing conventional linear stability analysis, the gain within the KE equation is determined. To shed light on the role of four-wave mixing in various scenarios, the gain spectrum is utilized as a tool to analyze the behavior of the KE equation under different conditions. This methodology seeks to provide insightful information about the intricate interactions that impact the modulational instability of solitonic pulses in an optical systems. After that, we have investigated the soliton solution by implementing the Jacobian elliptical function approach. Finally, our focus here is on linear stability analysis, which employs eigenvalue spectra to study solitons’ stability via direct numerical simulation.

Two-dimensional flat-band solitons in superhoneycomb lattices

Abstract Flat-band periodic materials are characterized by a linear spectrum containing at least one band where the propagation constant remains nearly constant irrespective of the Bloch momentum across the Brillouin zone. These materials provide a unique platform for investigating phenomena related to light localization. Meantime, the interaction between flat-band physics and nonlinearity in continuous systems remains largely unexplored, particularly in continuous systems where the band flatness deviates slightly from zero, in contrast to simplified discrete systems with exactly flat bands. Here, we use a continuous superhoneycomb lattice featuring a flat band in its spectrum to theoretically and numerically introduce a range of stable flat-band solitons. These solutions encompass fundamental, dipole, multi-peak, and even vortex solitons. Numerical analysis demonstrates that these solitons are stable in a broad range of powers. They do not bifurcate from the flat band and can be analyzed using Wannier function expansion leading to their designation as Wannier solitons. These solitons showcase novel possibilities for light localization and transmission within nonlinear flat-band systems.

Gap solitons of spin–orbit-coupled Bose–Einstein condensates in a Jacobian elliptic sine potential

Gap solitons of quasi-one-dimensional Bose–Einstein condensate loaded in a Jacobian elliptic sine potential with spin–orbit-coupled are investigated theoretically. Under the mean-field approximation, the dynamical behaviors of such a system are described using the Gross–Pitaevskii equation (GPE). Firstly, we linearize the GPE to obtain the band-gap structures. Secondly, the Newton-Conjugate-Gradient (NCG) method is used to search for gap solitons. Thirdly, the dynamic stability of the gap solitons obtained by the NCG method was investigated using linear stability analysis and nonlinear dynamic evolution methods. It is found that in this nonlinear system, there exist both stable gap solitons and structurally rich unstable gap solitons. The external potential modulus and the strength of the spin–orbit coupling are important parameters that influence the stability of the gap solitons.

Three-dimensional solitons supported by the spin–orbit coupling and Rydberg–Rydberg interactions in [formula omitted]-symmetric potentials

Excited states (ESs) of two- and three-dimensional (2D and 3D) solitons of the semivortex (SV) and mixed-mode (MM) types, supported by the interplay of the spin–orbit coupling (SOC) and local nonlinearity in binary Bose–Einstein condensates, are unstable, on the contrary to the stability of the SV and MM solitons in their fundamental states. We propose a stabilization strategy for these states in 3D, combining SOC and long-range Rydberg–Rydberg interactions (RRI), in the presence of a spatially-periodic potential, that may include a parity-time (PT)-symmetric component. ESs of the SV solitons, which carry integer vorticities S and S+1 in their two components, exhibit robustness up to S=4. ESs of MM solitons feature an interwoven necklace-like structure, with the components carrying opposite fractional values of the orbital angular momentum. Regions of the effective stability of the 3D solitons of the SV and MM types (both fundamental ones and ESs), are identified as functions of the imaginary component of the PT-symmetric potential and strengths of the SOC and RRI terms.

Orbital stability of smooth solitons for the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved H1 and L1 quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space H3. Furthermore, the global well-posedness of the solution is established for certain initial data in Hs with s≥3.

Stability of bright solitons in optical system supported by cubic and quintic nonlinearities with PT -symmetric quartic harmonic complex potential

This work analyzed the existence and stability of optical waves in linear and nonlinear media with self-focusing cubic-quintic nonlinearity by way of interactions of nonlinearity, spatial diffraction, and parity-time ( PT ) symmetric complex potential defined by quartic harmonic real potential and Gaussian imaginary distribution. We demonstrated that the order of nonlinear function, the width and depth of the real potential, and the coefficient of diffraction modify the span of the complex potential parameter onto which PT -symmetry breakdown occurs. The region of stable PT -symmetry in the self-focusing medium improves as the order of the nonlinear function increases, as demonstrated by comparing the soliton solutions with cubic and quintic nonlinearities. We addressed the nonlinear evolution of the stable and unstable PT solitons under propagation. The linear stability of bright solitons is extensively studied and established the conditions for stable soliton for both PT− symmetric and broken PT− symmetric regions.