W-shaped Soliton Research Articles

Asymmetric W-shaped and M-shaped soliton pulse generated from a weak modulation in an exponential dispersion decreasing fiber**Project supported by the National Natural Science Foundation of China (Grant No. 11475135), the Fund from Shaanxi Province Science Association of Colleges and Universities (Grant No. 20160216), and Guangxi Provincial Education Department Research Project, China (Grant No. 2017KY0776).

We study localized waves on continuous wave background in an exponential dispersion decreasing fiber with two orthogonal polarization states. We demonstrate that asymmetric W-shaped and M-shaped soliton pulse can be generated from a weak modulation on continuous wave background. The numerical simulation results indicate that the generated asymmetric soliton pulses are robust against small noise or perturbation. In particular, the asymmetric degree of the asymmetric soliton pulse can be effectively controlled by changing the relative frequency of the two components. This character can be used to generate other nonlinear localized waves, such as dark–antidark and antidark–dark soliton pulse pair, symmetric W-shaped and M-shaped soliton pulse. Furthermore, we find that the asymmetric soliton pulse possesses an asymmetric discontinuous spectrum.

Rogue waves and W-shaped solitons in the multiple self-induced transparency system.

We study localized nonlinear waves on a plane wave background in the multiple self-induced transparency (SIT) system, which describes an important enhancement of the amplification and control of optical waves compared to the single SIT system. A hierarchy of exact multiparametric rational solutions in a compact determinant representation is presented. We demonstrate that this family of solutions contain known rogue wave solutions and unusual W-shaped soliton solutions. State transitions between the fundamental rogue waves and W-shaped solitons as well as higher-order nonlinear superposition modes are revealed in the zero-frequency perturbation region by the suitable choice for the background wavenumber of the electric field component. Particularly, it is found that the multiple SIT system can admit both stationary and nonstationary W-shaped solitons in contrast to the stationary results in the single SIT system. Moreover, the W-shaped soliton complex which is formed by a certain number of fundamental W-shaped solitons with zero phase parameters and its decomposition mechanism in the case of the nonzero phase parameters are shown. Meanwhile, some important characteristics of the nonlinear waves including trajectories and spectrum are discussed through the numerical and analytical methods.

Reverse Space-Time Nonlocal Sasa–Satsuma Equation and Its Solutions

The Sasa–Satsuma equation is an integrable high-order nonlinear Schrodinger (NLS) equation, and also is a complex modified KdV-type equation. It can describe the propagation of femtosecond pulses in optical fibers. Very recently, Ablowitz and Mussliman introduced a class of reverse space-time and reverse time nonlinear integrable equations, including the reverse space nonlocal NLS equation, the real and complex reverse space-time nonlocal mKdV, sine-Gordon, Davey–Stewartson equations, etc. So, what is nonlocal version of high-order NLS? In this paper, we introduce a reverse space-time nonlocal Sasa–Satsuma equation, i.e., a reverse space-time nonlocal high-order NLS equation, and derive its solutions with the binary Darboux transformation method. Periodic solutions, and some localized solutions, such as dark soliton, W-shaped soliton, M-shaped soliton and breather soliton of the reverse space-time nonlocal Sasa–Satsuma equation are constructed.

W-shaped soliton complexes and rogue-wave pattern transitions for the AB system

With the help of a general Nth-order rogue wave solution in a compact determinant form for the AB system, we illuminate that, by suitably choosing the wavenumber and frequency of the background wave of the component A, W-shaped soliton complex containing a fixed number of algebraic solitons merging or separating with each other exists in the component A, and rogue-wave pattern transition between the four-petaled structure and dark structure occurs in the component B. The more complicated rogue-wave pattern transitions due to the nonlinear superposition corresponding to the higher-order four-petaled rogue waves and higher-order dark rogue waves of fundamental, triangular and circular dynamical structures up to third order are demonstrated, respectively. The spectrum properties which are related to the rogue-wave pattern transitions are revealed.

Three types magnetic moment distribution of nonlinear excitations in a Heisenberg helimagnet

We study the nonlinear spin dynamics of an anisotropic Heisenberg helimagnet in a fourth-order integrable nonlinear Schrödinger equation. We demonstrate that there are three types of nonlinear spin excitations on a spin-wave background in the Heisenberg helimagnet, notably including anti-dark soliton, W-shaped soliton, and multi-peak soliton. The magnetic moment distribution that corresponds to each of these are characterized in detail. Additionally, the formation mechanism is clarified by the magnon density distribution.

W-shaped, bright and kink solitons in the quadratic-cubic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials

An extended non-linear Schrödinger equation (NLSE) combining quadratic and cubic Non-linearities, which appears as an approximate model of a relatively dense quasi-one-dimensional Bose–Einstein condensate (BEC), is considered. In particular, we focus on the most physically important situation where the external potential and the quadratic-cubic non-linearities are dependent on both time and spatial coordinates. We use the similarity transformation technique to construct novel exact solutions for such NLSEs with modulating coefficients. We first present the general theory related to the quadratic-cubic model and then apply it to calculate explicitly soliton solutions of W-shaped, bright and kink type. The dynamic behaviors of solitons in different non-linearities and potentials that are of particular interest in applications to BECs are analysed.

Characteristics of nonautonomous W-shaped soliton and Peregrine comb in a variable-coefficient higher-order nonlinear Schrödinger equation

In this paper, we study the nonautonomous characteristics of the W-shaped solitons on constant backgrounds for a variable-coefficient higher-order nonlinear Schrödinger (vc-HNLS) equation. Several types of solitons are obtained in exponentially, linearly and periodically fluctuating decreasing dispersion profiles, respectively. We also show the compression effect of the third-order dispersion (TOD) coefficient on the W-shaped soliton. We further analyze the formation and spatiotemporal characteristics of the Peregrine combs (PCs) when TOD coefficient is of the periodic form. Moreover, the PC can be converted into the Peregrine wall if the modulation amplitude is equal to 1.

Symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime.

We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a single pulse formed by a nonlinear superposition of a periodic wave and a single-peak (W-shaped or antidark) soliton. In particular, a phase diagram for different types of nonlinear excitations on a continuous wave background, including the unusual multipeak soliton, the W-shaped soliton, the antidark soliton, the periodic wave, and the known breather rogue wave, is established based on the explicit link between exact solution and modulation instability analysis. Numerical simulations are performed to confirm the propagation stability of the multipeak solitons with symmetric and asymmetric structures. Further, we unveil a remarkable shape-changing feature of asymmetric multipeak solitons. It is interesting that these shape-changing interactions occur not only in the intraspecific collision (soliton mutual collision) but also in the interspecific interaction (soliton-breather interaction). Our results demonstrate that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.

Optical solitons and conservation laws with quadratic-cubic nonlinearity

We consider an extended nonlinear Schrödinger equation with competing quadratic-cubic nonlinearity. The model appears as an approximate form of a relatively dense quasi-one-dimensional Bose–Einstein condensate with repulsive contact interactions between atoms and a long-range dipole–dipole attraction between them. Exact analytical soliton solutions for the model are derived by utilizing the method of undetermined coefficients. The solutions comprise bright, dark, and singular solitons. Furthermore, we present a new structure which describes the evolution of two different type solitons in the form of W-shaped soliton solution and a bright soliton solution on a continuous-wave background. The constraint conditions for the existence of these solutions are also exhibited. The numerical evolution of the obtained solutions is also presented.

Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects.

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.

Stationary nonlinear waves, superposition modes and modulational instability characteristics in the AB system

We study the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultra-short pulses in nonlinear optics. We show that the breather can be converted into different types of stationary nonlinear waves on constant backgrounds, including the multi-peak soliton, M-shaped soliton, W-shaped soliton and periodic wave. We also investigate the nonlinear interactions between these waves, which display some novel patterns due to the non-propagating characteristics of the solitons: (1) Two antidark solitons can produce a W-shaped soliton instead of a higher-order antidark one, (2) The interaction between an antidark soliton and a W-shaped soliton can not only generate a higher-order antidark soliton, but also form a W-shaped solion pair, (3) The interactions between an oscillation W-shaped soliton and an oscillation M-shaped soliton show the multipeak structures. We find that the transition occurs at a modulational stability region in a low perturbation frequency region.

Rational solitary wave and rogue wave solutions in coupled defocusing Hirota equation

We derive and study a general rational solution of a coupled defocusing Hirota equation which can be used to describe evolution of light in a two-mode fiber with defocusing Kerr effect and some certain high-order effects. We find some new excitation patterns in the model, such as M-shaped soliton, W-shaped soliton, anti-eye-shaped rogue wave and four-petaled flower rogue wave. The results are compared with the solutions obtained in other coupled systems like vector nonlinear Schrödinger equation, coupled focusing Hirota and Sasa–Satsuma equations. We explain the new characters by modulational instability properties. This further indicates that rational solution does not necessarily correspond to rogue wave excitation dynamics and the quantitative relation between nonlinear excitations and modulational instability should exist.

W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation.

We study rational solutions of continuous wave backgrounds with the critical frequencies of the Sasa-Satsuma equation, which can be used to describe the evolution of the optical field in a nonlinear fiber with some high-order effects. We find a striking dynamical process that two W-shaped solitons are generated from a weak modulation signal on the continuous wave backgrounds. This provides a possible way to obtain stable high-intensity pulses from a low-intensity continuous wave background. The process involves both modulational instability and modulational stability regimes, in contrast to the rogue waves and W-shaped solitons reported before which involve modulational instability and stability, respectively. Furthermore, we present a phase diagram on a modulational instability spectrum plane for the fundamental nonlinear localized waves obtained already in the Sasa-Satsuma equation. The interactions between different types of nonlinear localized waves are discussed based on the phase diagram.

Breather-to-soliton and rogue wave-to-soliton transitions in a resonant erbium-doped fiber system with higher-order effects

Under investigation in this paper is the higher-order nonlinear Schrodinger and Maxwell–Bloch system which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order effects including the fourth-order dispersion and quintic non-Kerr nonlinearity. The breather and rogue wave (RW) solutions are shown that they can be converted into various soliton solutions including the multi-peak soliton, periodic wave, antidark soliton, M-shaped soliton, and W-shaped soliton. In addition, under different values of higher-order effect, the locus of the eigenvalues on the complex plane which converts breathers or RWs into solitons is calculated.

Different types of nonlinear localized and periodic waves in an erbium-doped fiber system

We study nonlinear waves on a plane-wave background in an erbium-doped fiber system, which is governed by the coupled nonlinear Schrödinger and the Maxwell–Bloch equations. We find that prolific different types of nonlinear localized and periodic waves do exist in the system, including multi-peak soliton, periodic wave, antidark soliton, and W-shaped soliton (as well as the known bright soliton, breather, and rogue wave). In particular, the dynamics of these waves can be extracted from a unified exact solution, and the corresponding existence conditions are presented explicitly. Our results demonstrate the structural diversity of the nonlinear waves in this system.

Transition, coexistence, and interaction of vector localized waves arising from higher-order effects

We study vector localized waves on continuous wave background with higher-order effects in a two-mode optical fiber. The striking properties of transition, coexistence, and interaction of these localized waves arising from higher-order effects are revealed in combination with corresponding modulation instability (MI) characteristics. It shows that these vector localized wave properties have no analogues in the case without higher-order effects. Specifically, compared to the scalar case, an intriguing transition between bright–dark rogue waves and w-shaped–anti-w-shaped solitons, which occurs as a result of the attenuation of MI growth rate to vanishing in the zero-frequency perturbation region, is exhibited with the relative background frequency. In particular, our results show that the w-shaped–anti-w-shaped solitons can coexist with breathers, coinciding with the MI analysis where the coexistence condition is a mixture of a modulation stability and MI region. It is interesting that their interaction is inelastic and describes a fusion process. In addition, we demonstrate an annihilation phenomenon for the interaction of two w-shaped solitons which is identified essentially as an inelastic collision in this system.

Multi-Solitons and Combined Solitons with Self-Similar Behaviors in a Birefringent Fiber with Higher-Order Effects* *Supported by the Scientific Research Fund of Zhejiang Provincial Education Department under Grant No. Y201120994 and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY13A05001

A coupled variable-coefficient higher-order nonlinear Schrödinger equation in birefringent fiber is studied, and analytical multi-soliton, combined bright and dark soliton, W-shaped and M-shaped soliton solutions are obtained. Nonlinear tunnelling of these combined solitons in dispersion barrier and dispersion well on an exponential background is discussed, and the decaying or increasing, even lossless tunnelling behaviors of combined solitons are decided by the decaying or increasing parameter.

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UUτyy − UyUτy + U2Uτ + 3Uy = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations of the smooth and nonsmooth soliton solutions for the Novikov equation with cubic nonlinearity. These solutions contain peaked soliton, smooth soliton, W-shaped soliton and periodic solutions. Our work extends some previous results.

Localized Waves on Continuous Wave Background in a Two-Mode Nonlinear Fiber with High-Order Effects

We study on rational solutions on nonzero background of coupled Sasa-Satsuma equations through Darboux transformation method, which take into account third order dispersion, the term with self-frequency shift, and the term describing self-steepening corrections to the cubic nonlinearity. We find there are some new types of localized waves in the coupled system, such as dark-antidark soliton pair, W-shaped soliton, dark W-shaped soliton, and the combined localized waves of them. The results indicate that there are abundant novel localized waves in the two-mode fiber with these high-order effects, which would inspire experimental realization in the related physical systems.

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation.

We investigate the solution in rational form for the Sasa-Satsuma equation on a continuous background which describes a nonlinear fiber system with higher-order effects including the third-order dispersion, Kerr dispersion, and stimulated inelastic scattering. The W-shaped soliton in the system is obtained analytically. It is found that the height of hump for the soliton increases with decreasing the background frequency in certain parameter regime. The maximum height of the soliton can be three times the background's height and the corresponding profile is identical with the one for the well-known eye-shaped rogue wave with maximum peak. The numerical simulations indicate that the W-shaped soliton is stable with small perturbations. Particularly, we show that the W-shaped soliton corresponds to a stable supercontinuum pulse by performing exact spectrum analysis.