Soliton Phenomena Research Articles

Breather degeneration and lump superposition for the (3 + 1)-dimensional nonlinear evolution equation

This paper is devoted to the study of a (3 + 1)-dimensional generalized nonlinear evolution equation for the shallow-water waves. The breather solutions with different structures are obtained based on the bilinear form with perturbation parameters. Some new lump solitons are found in the process of studying the degradation behavior of breather solutions, and we also study general lump soliton, lumpoff solution and superposition phenomenon between lump soliton and breather solution. Besides, some theorems about the superposition between lump soliton and [Formula: see text]-soliton ([Formula: see text] is a nonnegative integer) are given. Some examples, including lump-[Formula: see text]-exponential type, lump-[Formula: see text]-logarithmic type, higher-order lump-type [Formula: see text]-soliton, are given to illustrate the correctness of the theorems and corollaries described. Finally, some novel nonlinear phenomena, such as emergence of lump soliton, degeneration of breathers, fission and fusion of lumpoff, superposition of lump-[Formula: see text]-solitons, etc., are analyzed and simulated.

A study on the compatibility of the generalized Kudryashov method to determine wave solutions

In this article, we establish solitary wave solutions to the Estevez-Mansfield-Clarkson (EMC) equation and the coupled sine-Gordon equations which are model equations to analyze the formation of shapes in liquid drops, surfaces of negative constant curvature, etc. through contriving the generalized Kudryashov method. The extracted results introduce several types’ solitary waves, such as the kink soliton, bell-shape soliton, compacton, singular soliton, peakon and other sort of soliton for distinct valuation of the unknown parameters. The achieved analytic solutions are interpreted in details and their 2D and 3D graphs are sketched. The obtained solutions and the physical structures explain the soliton phenomenon and reproduce the dynamic properties of the front of the travelling wave deformation generated in the dispersive media. It shows that the generalized Kudryashov method is powerful, compatible and might be used in further works to found novel solutions for other types of nonlinear evolution equations ascending in physical science and engineering.

Manipulation of vector solitons in a system of inhomogeneous coherently coupled nonlinear Schrödinger models with variable nonlinearities

We investigate non-autonomous solitons in a general coherently coupled nonlinear Schrödinger (CCNLS) system with temporally modulated nonlinearities and with an external harmonic oscillator potential. This general CCNLS system encompasses three distinct types of CCNLS equations that describe the dynamics of beam propagation in an inhomogeneous Kerr-like nonlinear optical medium for different choices of nonlinear polarizations owing to the anisotropy of the medium. We identify a generalized similarity transformation to relate the considered model into the standard integrable homogeneous coupled nonlinear evolution equations with constant nonlinearities, accompanied by a constraint relation expressed in the form of the Riccati equation. With the help of a non-standard Hirota’s bilinearization method and exact soliton solutions, we explore the impact of varying nonlinearities and refractive index in the propagation and collisions analytically by reverse engineering. Interestingly, we show the emergence of several modulated solitonic phenomena such as periodic oscillation, amplification, compression, tunneling/cross-over, excitons, as well as their combined effect in the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity. Notably, we identify a tool to transform the nature of soliton collisions with certain type of inhomogeneous nonlinearities. The results could be of significant interest to the studies on management of nonlinear waves in the contexts like nonlinear optics and can also be extended to Bose–Einstein condensates and super-fluids.

Dynamics of Many-Body Photon Bound States in Chiral Waveguide QED

We theoretically study the few- and many-body dynamics of photons in chiral waveguides. In particular, we examine pulse propagation through a system of $N$ two-level systems chirally coupled to a waveguide. We show that the system supports correlated multi-photon bound states, which have a well-defined photon number $n$ and propagate through the system with a group delay scaling as $1/n^2$. This has the interesting consequence that, during propagation, an incident coherent state pulse breaks up into different bound state components that can become spatially separated at the output in a sufficiently long system. For sufficiently many photons and sufficiently short systems, we show that linear combinations of $n$-body bound states recover the well-known phenomenon of mean-field solitons in self-induced transparency. For longer systems, however, the solitons break apart through quantum correlated dynamics. Our work thus covers the entire spectrum from few-photon quantum propagation, to genuine quantum many-body (atom and photon) phenomena, and ultimately the quantum-to-classical transition. Finally, we demonstrate that the bound states can undergo elastic scattering with additional photons. Together, our results demonstrate that photon bound states are truly distinct physical objects emerging from the most elementary light-matter interaction between photons and two-level emitters. Our work opens the door to studying quantum many-body physics and soliton physics with photons in chiral waveguide QED.

Widely tunable femtosecond soliton generation in a fiber-feedback optical parametric oscillator

Optical parametric oscillators (OPOs) are uniquely versatile—albeit complex—platforms for the generation of tunable coherent light in difficult-to-access spectral regions. A synchronously pumped χ ( 2 ) femtosecond OPO producing sub-100-fs pulses, by means of optical soliton formation in a single-mode fiber-feedback cavity, is presented for the first time. This approach removes the requirement for active stabilization and bulk dispersion compensation elements associated with traditional OPOs, while reducing the physical footprint by a factor of 3. Transform-limited pulses of 80–120 fs are generated for both normal and anomalous dispersion, while a new type of spectral sideband formation is observed and studied. Experimental results are supported by detailed simulations based on nonlinear pulse propagation theory, which serve to enrich the understanding of femtosecond pulse evolution in OPOs perturbed by strong Kerr nonlinearity. The OPO provides a unique platform for the study of intracavity soliton phenomena across a wide wavelength range.

Nonstandard bilinearization and interaction phenomenon for PT-symmetric coupled nonlocal nonlinear Schrödinger equations

We develop a nonstandard bilinearization procedure to generate more general bright soliton, breather soliton and quasi-rogue wave solutions for the PT-symmetric coupled nonlocal nonlinear Schrödinger (CNNLS) equations. We achieve some novel solutions by bilinearizing both the CNNLS equations and their associated auxiliary equations in a novel way. The soliton equations are written into bilinear operator forms by means of auxiliary equations, then the 1-soliton solution and 2-soliton solution of PT-symmetric CNNLS equations are constructed. Some novel interaction properties of the PT-symmetric two-soliton solutions are derived, which can present the potential applications to the soliton wave phenomena in nonlocal wave models.

Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation

In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator—Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations; however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities.

Some discrete soliton solutions and interactions for the coupled Ablowitz–Ladik equations with branched dispersion

The coupled Ablowitz-Ladik lattice equations are the integrable discretizations of the Schrödinger equation, which can be used to model the propagation of an optical field in a tight binding waveguide array. In this paper, the discrete N-fold Darboux transformation(DT) is used to derive the discrete breather and bright soliton solutions of coupled Ablowitz–Ladik equations. Soliton interaction structures of obtained solutions are shown graphically. Based on 4 × 4 discrete Lax pairs, the transformation matrix T of DT is constructed. Then, we derive novel discrete one-soliton and two-soliton with the zero and nonzero seed solutions. And the dynamic features of breather and bright solutions are displayed, some soliton interaction phenomena are shown in the coupled Ablowitz-Ladik lattice equations. These results may be useful to explain some nonlinear wave phenomena in certain electrical and optical systems.

Dark and multi-dark solitons in the three-component nonlinear Schrödinger equations on the general nonzero background**Project supported by the National Natural Science Foundation of China (Grant No. 11771151), the Guangdong Natural Science Foundation of China (Grant No. 2017A030313008), the Guangzhou Science and Technology Program of China (Grant No. 201904010362), and the Fundamental Research Funds for the Central Universities of China (Grant

We exhibit some new dark soliton phenomena on the general nonzero background for a defocusing three-component nonlinear Schrödinger equation. As the plane wave background undergoes unitary transformation SU(3), we obtain the general nonzero background and study its modulational instability by the linear stability analysis. On the basis of this background, we study the dynamics of one-dark soliton and two-dark-soliton phenomena, which are different from the dark solitons studied before. Furthermore, we use the numerical method for checking the stability of the one-dark-soliton solution. These results further enrich the content in nonlinear Schrödinger systems, and require more in-depth studies in the future.

Determinant solutions and asymptotic state analysis for an integrable model of transient stimulated Raman scattering

In this paper, the transient stimulated Raman scattering equations, which describe the stimulated Raman scattering in the transient limit, are investigated by Darboux transformation. Considering the trivial initial solutions, the explicit N-soliton solutions in determinant form for the transient stimulated Raman scattering equations are presented. Based on the explicit solutions formula, the one- and two-soliton solutions are given explicitly. The asymptotic state analysis of the two-soliton solutions is conducted. And different soliton propagation phenomena are discussed and illustrated including but not limited to the flat-top soliton, the two-hump soliton and bound two soliton with periodic structure, which is good for understanding the interaction properties under different wave parameters.

KP-KdV Hierarchy and Pseudo-Differential Operators

The study of KP-KdV equations are the archetype of integrable systems and are one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. One of the objectives of this paper is to study the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand-Levitan equations) used to solve it exactly. We study some generalities on the algebra of infinite order differential operators. The algebras of Virasoro and Heisenberg, nonlinear evolution equations such as the KdV, Boussinesq and KP play a crucial role in this study. We make a careful study of some connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato-Date-Jimbo-Miwa-Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations, the Gelfand-Dickey flows, the Heisenberg and Virasoro algebras are given.

Transient behaviors of pure soliton pulsations and soliton explosion in an L-band normal-dispersion mode-locked fiber laser.

As one of the most striking localized structures in dissipative systems, pulsating soliton has been widely studied in theory but rarely observed in experiments. Here, three typical types of soliton pulsations are experimentally demonstrated in an L-band normal-dispersion mode-locked fiber laser via the dispersive Fourier transform (DFT) technique. According to the distinctive features, they are classified as single-periodic pulsating soliton, double-periodic pulsating soliton and soliton explosion. These pulsations have common features such as energy oscillation, bandwidth breathing and temporal shift. However, the pulse is repeated every two oscillations for double-periodic pulsating soliton. When it comes to soliton explosion, because of the intermittent overdriven nonlinear effect induced by the extreme energy oscillation, the spectrum cracks into pieces at a periodic manner. To the best of our knowledge, it is the first time that both pure soliton pulsations and soliton explosion are observed experimentally in the same fiber laser. The results will enhance a more comprehensive understanding for the soliton pulsating phenomena.

M-shaped rational solitons and their interaction with kink waves in the Fokas–Lenells equation

In this paper, by utilizing logarithmic transformation and symbolic computation with the ansatz function technique, we investigated the rational solitons of the Fokas–Lenells equation. We obtained two types of M-shaped rational solitons and their dynamics are shown in figures by selecting the appropriate values of involved parameters. Furthermore, two types of interactions of M-shaped rational solitons with kink waves are investigated. We obtained the novel structures and very interesting interactional phenomena of M-shaped rational solitons with kink waves.

Periodic attenuating oscillation between soliton interactions for higher-order variable coefficient nonlinear Schrödinger equation

According to the change in the amplitude of the oscillation, it can be divided into equal-amplitude oscillation, amplitude-reduced oscillation (attenuating oscillation) and amplitude-increasing oscillation (divergence oscillation). In this paper, the periodic attenuating oscillation of solitons for a higher-order variable coefficient nonlinear Schrodinger equation is investigated. Analytic one- and two-soliton solutions of this equation are obtained by the Hirota bilinear method. By analyzing the soliton propagation properties, we study how to choose the corresponding parameters to control the soliton propagation and periodic attenuation oscillation phenomena. Results might be of significance for the study of optical communications including soliton control, amplification, compression and interactions.

Inverse scattering solutions and dynamics for a nonlocal nonlinear Gross–Pitaevskii equation with PT-symmetric external potentials

A generalized nonlocal nonlinear Gross–Pitaevskii(GP) equation is presented, which can be reduced to the nonlocal GP equation with self-induced PT-symmetric potential. We derive some novel non-autonomous soliton solutions of nonlocal GP equation via inverse scattering and similarity transformations. These obtained solutions have singularities and are not the standard soliton solutions, here they can be called soliton-like(SL) solutions. Then we consider some controllable behaviors of these non-autonomous wave solutions. The obtained results are different from the solutions of the local nonlinear GP equation. Some propagation phenomena are produced through manipulating non-autonomous SL waves, which can present the potential applications to the soliton wave phenomena in nonlocal wave models.

Effects of linear Zeeman splitting on the dynamics of bright solitons in spin-orbit coupled Bose-Einstein condensates

Solitons as self-supported solitary waves are one of the most fundamental objects in nonlinear science. With the realization of Bose-Einstein condensate, matter-wave solitons have aroused enormous interest due to their potential applications in atomic transport and atomic interferometer. In recent years, the artificial spin-orbit coupling has been realized in ultracold atoms, thus providing a new platform to study the nonlinear matter wave solitons under a gauge field, and a variety of novel soliton phenomena have been successively predicted. In this paper, we analyze the effects of linear Zeeman splitting on the dynamics of bright-bright solitons in spin-orbit coupled two-component Bose-Einstein condensate, via the variational approximation and the numerical simulation of Gross-Pitaevskii (GP) equations. For the SU(2) spin-rotational invariant attractive atomic interaction in a uniform case without external trap, we take a hyperbolic secant function as the variational Ansatz for bright soliton in variational approximation, and derive the Euler-Lagrange equations describing the evolution of the Ansatz parameters. By solving the time-independent Euler-Lagrange equations, we find two stationary solitons each with a finite momentum for a weak spin-orbit coupling due to the linear Zeeman splitting. Linearizing the Euler-Lagrange equations around these stationary solitons, we further obtain a zero-energy Goldstone mode and an oscillation mode with frequency related to linear Zeeman splitting: the former indicates that the continuous translational symmetry of the stationary solitons will be broken under a perturbation, and the later shows that the stationary solitons will oscillate under a perturbation. Furthermore, by solving the time-dependent Euler-Lagrange equations, we also obtain the exact full dynamical solutions of Ansatz parameters, and observe that the linear Zeeman splitting affects the period and velocity of soliton's oscillation and linear motion, which may provide a new method to control the dynamics of solitons. All the variational calculations are also confirmed directly by the numerical simulation of GP equations.

Asymmetric and Single-Side Splitting of Dissipative Solitons in Complex Ginzburg–Landau Equations with an Asymmetric Wedge-Shaped Potential**Supported by the National Natural Science Foundation of China under Grant No 61665007, and the Natural Science Foundation of Jiangxi Province under Grant No 20161BAB202039.

We report some novel dynamical phenomena of dissipative solitons supported by introducing an asymmetric wedge-shaped potential (just as a sharp ‘razor’) into the complex Ginzburg–Landau equation with the cubic-quintic nonlinearity. The potentials corresponding to a local refractive index modulation with breaking symmetry can be realized in an active optical medium with respective expanding antiwaveguiding structures. Using the razor potential acting on a central dissipative soliton, possible outcomes of asymmetric and single-side splitting of dissipative solitons are achieved with setting different strengths and steepness of the potentials. The results can potentially be used to design a multi-route splitter for light beams.

Solitons interaction and integrability for a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves

Under investigation in this paper is a (2+1)-dimensional variable-coefficient Broer–Kaup system in water waves. Via the symbolic computation, Bell polynomials and Hirota method, the Bäcklund transformation, Lax pair, bilinear forms, one- and two-soliton solutions are derived. Propagation and interaction for the solitons are illustrated: Amplitudes and shapes of the one soliton keep invariant during the propagation, which implies that the transport of the energy is stable for the (2+1)-dimensional water waves; and inelastic interactions between the two solitons are discussed. Elastic interactions between the two parabolic-, cubic- and periodic-type solitons are displayed, where the solitonic amplitudes and shapes remain unchanged except for certain phase shifts. However, inelastically, amplitudes of the two solitons have a linear superposition after each interaction which is called as a soliton resonance phenomenon.

Dynamic evolution of vortex solitons for coupled Bose-Einstein condensates in harmonic potential trap

We studied the evolution of vortex solitons in two-component coupled Bose-Einstein condensates trapped in a harmonic potential. Using a two-dimensional coupled Gross-Pitaevskii equation model and a variational method, we theoretically derived the vortex soliton solution. Under an appropriate parametric setting, the derived vortex soliton radius was found to oscillate periodically. The derived quasi-stable states with typical nonlinear features are pictorially demonstrated and can be used to guide relevant experimental observations of vortex soliton phenomena in coupled ultracold atomic systems.

Extending topological surgery to natural processes and dynamical systems.

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromosomal crossover, DNA recombination and when cosmic magnetic lines reconnect, while 2-dimensional surgery happens in the formation of tornadoes, in the phenomenon of Falaco solitons, in drop coalescence and in the cell mitosis. Inspired by such phenomena, we introduce new theoretical concepts which enhance topological surgery with the observed forces and dynamics. To do this, we first extend the formal definition to a continuous process caused by local forces. Next, for modeling phenomena which do not happen on arcs or surfaces but are 2-dimensional or 3-dimensional, we fill in the interior space by defining the notion of solid topological surgery. We further introduce the notion of embedded surgery in S3 for modeling phenomena which involve more intrinsically the ambient space, such as the appearance of knotting in DNA and phenomena where the causes and effect of the process lies beyond the initial manifold, such as the formation of black holes. Finally, we connect these new theoretical concepts with a dynamical system and we present it as a model for both 2-dimensional 0-surgery and natural phenomena exhibiting a ‘hole drilling’ behavior. We hope that through this study, topology and dynamics of many natural phenomena, as well as topological surgery itself, will be better understood.