Higher-order Solitons Research Articles

High-order solitary waves, fission, hybrid waves and interaction solutions in the nonlinear dissipative (2+1)-dimensional Zabolotskaya-Khokhlov model

The Zabolotskaya-Khokhlov model (ZKm) is a widely used nonlinear model in the fields of sound, ultrasound, and shock waves. The aims of this paper stems from its examination and rectification of earlier results concerning the N-soliton solutions of nonlinear dissipative (2+1)-dimensional ZKm. By recognizing and incorporating the non-zero values of the dispersion coefficient , this study addresses a significant omission in current research. The findings enhance the comprehension of higher-order soliton behaviors, encompassing bifurcation solitons, higher-order breathers, rogue waves, periodic lumps, and their interactions, which are crucial for both theoretical studies and practical applications in areas like nonlinear optics and fluid dynamics. Subsequent detailed numerical simulations are conducted to elucidate the complex behaviors of the obtained solutions. This thorough exploration provides crucial insights into the intricate patterns exhibited by the nonlinear dissipative (2+1)-dimensional ZKm under different conditions, enhancing our understanding of the underlying physical phenomena.

Higher-Order Soliton Solutions for the Derivative Nonlinear Schrödinger Equation via Improved Riemann–Hilbert Method

In this paper we discuss an improved Riemann–Hilbert method, by which arbitrary higher-order soliton solutions for the derivative nonlinear Schrödinger equation can be directly obtained. The explicit determinant form of a higher-order soliton which corresponds to one pth order pole is given. Besides the interaction related to one simple pole and the other one double pole is considered.

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.

Higher-order vortex solitons in Kerr nonlinear media with a flat-bottom potential

Higher-order vortex solitons in Kerr nonlinear media with a flat-bottom potential

Broadband flat supercontinuum generated by chalcogenide double-clad step-index fiber with special dispersion

To achave a broader supercontinuum in the mid-infrared band, a chalcogenide double-clad step-index fiber is proposed. Comprising As2Se3, AsSe2, and As2S5 glass, this fiber effectively confines light within the core, minimizing fiber dispersion fluctuations. By iteratively adjusting the dispersion curve of the fiber, enhancement of the supercontinuum spectrum generation is possible. Spectral broadening primarily results from four-wave mixing and self-phase modulation. Raman scattering within the pulse generates unstable higher-order solitons, which, upon breakup, contribute to spectrum broadening. This study explores the effects of pulse laser parameters and fiber length on supercontinuum generation, obtains regular cognition and offers theoretical analyses of the results. Through optimization of parameters, a mid-infrared supercontinuum with an 11 μm width is achieved. These findings underscore the potential of chalcogenide double-clad step-index fiber for generating a mid-infrared supercontinuum light source. The designed fiber structure should be helpful in supercontinuum sources, bio-molecule sensing, and spectroscopy.

A General Coupled Derivative Nonlinear Schrödinger System: Darboux Transformation and Soliton Solutions

In this work we present a general coupled derivative nonlinear Schrödinger system. We construct the corresponding N-fold Darboux transform and generalized Darboux transform. Under this construction, we give different soliton solutions and plot their figures describing the soliton characteristics and dynamical behaviors, including higher-order soliton and rouge wave solution etc.

Higher-order beyond-band discrete solitons in binary waveguide arrays

We study higher-order beyond-band discrete solitons (HOBBDSs) and quasi-HOBBDSs, which can be constructed by multiplying the solutions of fundamental single-peaked beyond-band discrete solitons by a soliton order parameter larger than unity. In the quasi-continuous regime when the HOBBDS peak amplitude is low (thus its width is large) and the soliton order parameter is a small integer number, HOBBDSs periodically evolve during propagation and their dynamics are similar to those of higher-order solitons governed by the nonlinear Schrödinger equation in an optical fiber, including the periodicity, pattern evolution, and independence of the period length on the soliton order parameter. If the soliton order parameter is still small but not an integer, then one can obtain the quasi-HOBBDSs whose profiles almost periodically evolve during propagation. The breathing length of quasi-HOBBDSs decreases if the soliton order parameter increases. Moreover, the breathing length of quasi-HOBBDSs is approximately inversely proportional to the square values of the peak amplitude of the fundamental beyond-band discrete solitons, just like what happens with the period length of the higher-order solitons governed by the nonlinear Schrödinger equation. If the fundamental beyond-band discrete solitons are intense enough and/or the soliton order parameter is large enough, then most of the energy of the beams is eventually trapped in a single waveguide.

Soliton in Bose-Einstein condensates with helicoidal spin-orbit coupling under a Zeeman lattice.

We investigate the existence and stability of higher-order bright solitons, stripe solitons, and bright-dark solitons in a Bose-Einstein condensate with helicoidal spin-orbit coupling under a Zeeman lattice using numerical methods. The higher-order bright solitons that exist in the first-finite energy gap are stable except near the edge. The stripe solitons with parity-time symmetry and pseudospin-parity symmetry have partially overlapping norm curves; they are stable in the lower edge of the first-finite energy gap. Additionally, the bright-dark solitons discovered in the system not only exist within energy gaps but also embed within energy bands as they have periodic backgrounds. These findings offer insights into the diversity and behavior of solitons within energy bands and contribute to a deeper understanding of their distribution and dynamics.

Multiple solitons with fission and multi waves interaction solutions of a (3+1)-dimensional combined pKP-BKP integrable equation

The potential Kadomtsev-Petviashvili (pKP) equation delineates the development of small-amplitude, nonlinear, long waves characterized by a gradual variation in the transverse coordinate. The B-type KP equation outlines the relationships among exponentially localized shapes and was employed as a representation for shallow water waves and plasma physics. In this paper, we consider the combined pKP-BKP integrable equation. We discuss the multiple solitons of a newly proposed (3+1)-dimensional combined pKP-BKP integrable equation. We use the Hirota bilinear (HB) form of the considered equation to deduce fission process in higher order solitons with different orders. Moreover, the breather dynamics and its interaction with other solitons are investigated via HB. The lump solution and its interaction with first order and fourth order kink soliton is studied.

Generalized cosine-Gaussian breathing solitons with transverse cross modulation in nonlocal nonlinear media

Based on the nonlocal nonlinear Schrödinger equation, the propagation characteristics of transverse cross-modulated cosine-Gaussian (TCMCG) breathing solitons in strongly nonlocal media are studied. A series of mathematical expressions are provided to describe the propagation dynamics characteristics of the TCMCG breathing solitons, including the soliton width, soliton width change rate, soliton width extreme values, change rates of soliton width extreme values, phase wavefront curvature and phase wavefront curvature extreme values, etc. The research results indicate that by modulating TCMCG breathing solitons, generalized high-order spatial solitons can be formed with periodic changes in light intensity modes but unchanged in beam width. The cosine function parameter has a significant impact on the propagation dynamics characteristics, which appears in the initial expression and plays a cross modulation role in the transverse spatial light intensity. The effect of the initial incident power on soliton width extreme values and phase wavefront curvature extreme values is also discussed.

Supercontinuum Generation by Controlling Pitch in Photonic Crystal Fibers

The influence of varying the distance between air holes (pitch) on the geography of solitone propagation through a photonic crystal fiber has been tested. The study depends on the Split-Step Fourier method and the results quantified using MATLAB. The first-order solitone was tested with the change in pitch, and it was found that there is a clear decay in the amplitude of the resulting pulse with an increase in pitch. When increasing the pitch in the case of second-order solitons, it was noticed that the pulse would split into multiple-order solitons down to higher-order solitons with the increase in pitch. In the case of third-order solitons, solitonic fission leads to the supercontinuum generation with increasing the pitch, where the supercontinuum generation was reached in this way depending on a very small energy source compared to the high energies used to generate this type of spectrum by the previous methods. In this study, I observed that when the pitch values increased in the third-order soliton, this result led to the use of supercontinuum generation (ScG) which has many applications such as medical and industrial applications and has an important role in modern communication systems. Keywords: Photonic Crystal Fiber, Split-Step Fourier, Soliton Pulse, Pitch, Supercontinuum Generation.

Optimize Nonlinear Effects on Fundamental and High-order Soliton in Photonic Crystal Fiber

Nonlinear effects in optical fibers are mainly caused by two sources: inelastic scattering behaviour or the intensity sensitivity of the medium's refractive index. The propagation process in photonic crystal fibers is more complex than the propagation process of first-order solitons, second-order solitons, and third-order solitons. This article discusses the effects of propagation on first-, second- and third-order solitons. A popular approach to supercontinuum generation through soliton fission is the higher-order soliton technique for spectral generation.

Two-dimensional modulation instability and soliton clusters in nonlocal media with competing cubic–quintic nonlinearities

Using linear-stability analysis, two-dimensional modulation instability (MI) of plane wave is studied in nonlocal media with competing cubic–quintic (CQ) nonlinearities, which shows that MI can be effectively eliminated by strong nonlocality and competing quintic nonlinearity. Furthermore, propagation properties of higher-order soliton clusters, i.e., Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) solitons are also investigated. Bifurcated solutions of these solitons are obtained analytically with variational approach. We also demonstrate in detail the propagation dynamics of the HG and LG solitons with split-step Fourier transform.

Abundant dynamical solitary waves solutions of M -fractional Oskolkov model

This work uses a truncated M-fractional derivative variant of the Oskolkov model to investigate the dynamic behavior of solitary wavefronts. The methods used in this framework produce a variety of solitary waveforms, such as bright and dark solitons. A suitable choice of the free parameters is used to investigate the geometrical structures for the wave solutions, which are further characterized by stable bright periodic and soliton waves. The coefficient of the highest-order derivative and the effects of fractionality are shown in the figures. Moreover, the graphics are arranged to highlight the characteristics of novel soliton wave propagation. The findings of this research demonstrate that the fractional Oskolkov model may accommodate fundamental and higher-order soliton behaviors, each of which has unique characteristics. The fractional form of the several dynamical solitary waves seen in the study represents their practical ramifications. These waves can be seen as transmission waves via a Kelvin-Voigt fluid.

Evolution of multi-solitons and interaction behaviors of lump to a (2+1) dimensional generalized shallow water wave model

In this paper, the trajectory equations of 1-lump before and after collision with high-order solitons and the degradation of some novel breather waves are studied in the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation(gCBS). Firstly, we derive N-solitons for the gCBS equation by the Hirota bilinear form. With the help of N-solitons, we obtain M-lump as well as high-order breather based on the long-wave limit technique and the parametric conjugate method. Secondly, we construct many hybrid waves, such as the hybrid wave between breather and lump. Thirdly, the interaction phenomenon of lump-N-solitons(N → ∞) is investigated, and the theory of its existence is given and proved. Besides, the different degeneracies of double and single breather are discussed. Finally, we also present a large number of two-dimensional and three-dimensional images to better illustrate these nonlinear evolutionary behaviors.

Periodic transmission theory of circularly symmetric multi-ring solitons in nonlinear Schrödinger equation

In this paper, the evolution characteristics of periodic transmission of circularly symmetric multi-ring solitons in optical nonlocal materials based on nonlinear Schrödinger equation are investigated in detail. The transmission expression of circularly symmetric multi-ring solitons has been derived. It was found that the number and size of rings in these solitons can be controlled by initial parameters. The transmission of circularly symmetric multi-ring solitons is similar to that of high-order temporal solitons in nonlinear fibers, exhibiting periodic variations. When the input energy is a specific value, the statistical width of circularly symmetric multi-ring solitons remains constant during transmission, otherwise it exhibits periodic changes, which can be considered as generalized breathing solitons. The influence of various parameters on the transmission characteristics has been analyzed in detail, and some important transmission characteristics have been intuitively demonstrated through numerical simulation.

Two-dimensional modulation instability and higher-order soliton clusters in nematic liquid crystals with competing re-orientational and thermal nonlocal nonlinearities

We investigate two-dimensional modulation instability (MI) and higher-order soliton clusters in nematic liquid crystals with competing re-orientational and thermal nonlocal nonlinearities. With linear-stability analysis, we show nonlocality can suppress effectively MI of two-dimensional plane wave. By employing variational (Lagrangian) approach, bifurcated solutions of higher-order solitons, i.e., Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) soliton clusters are obtained analytically. Propagation dynamics of both upper and lower branches of solitons are demonstrated numerically. Properties, such as stable propagation distance and mode transformation between HG and LG modes are also discussed in detail.

General high-order solitons and breathers with a periodic wave background in the nonlocal Hirota–Maccari equation

General high-order solitons and breathers with a periodic wave background in the nonlocal Hirota–Maccari equation

Higher-order soliton solutions for the Sasa–Satsuma equation revisited via bar{partial } method

In optics, the Sasa–Satsuma equation can be used to model ultrashort optical pulses. In this paper higher-order soliton solutions for the Sasa–Satsuma equation with zero boundary condition at infinity are analyzed by bar{partial } method. The explicit determinant form of a soliton solution which corresponds to a single p_{l}-th order pole is given. Besides the interaction related to one simple pole and the other one double pole is considered.

Abundant dynamical solitary waves through Kelvin-Voigt fluid via the truncated M-fractional Oskolkov model

This effort explores the dynamic behavior of solitary wave fronts using the Oskolkov model in a truncated M-fractional derivative form. This effort has been performed through the application of modified extended tanh and a novel form of modified Kudryashov techniques. All obtained solutions manifest in the form of sinusoidal, hyperbolic, and constant basis function solutions within the fractional model. Within this framework, the applied techniques yield various solitary wave structures, including bright and dark bell solitons, kinks, anti-kinks, linked lumps, and collisions of kinks. Additionally, they produce singular kink solitons and lump waves. The effects of fractionality and the coefficient of the highest-order derivative are depicted in Figs. 1 through 11. Moreover, all figures are organized to illustrate the properties of the innovative soliton wave propagations. The studies have also revealed that the fractional Oskolkov model can support both fundamental and higher-order solitons, each characterized by distinct properties and highly efficient behaviors. Various dynamical solitary waves obtained here can be described transmission waves through Kelvin-Voigt fluid with real happenings as it represented by fractional form.