Multisoliton Interactions Research Articles

Dynamical and statistical features of soliton interactions in the focusing Gardner equation.

In this paper, the dynamical properties of soliton interactions in the focusing Gardner equation are analyzed by the conventional two-soliton solution and its degenerate cases. Using the asymptotic expressions of interacting solitons, it is shown that the soliton polarities depend on the signs of phase parameters, and that the degenerate solitons in the mixed and rational forms have variable velocities with the time dependence of attenuation. By means of extreme value analysis, the interaction points in different interaction scenarios are presented with exact determination of positions and occurrence times of high transient waves generated in the bipolar soliton interactions. Next, with all types of two-soliton interaction scenarios considered, the interactions of two solitons with different polarities are quantitatively shown to have a greater contribution to the skewness and kurtosis than those with the same polarity. Specifically, the ratios of spectral parameters (or soliton amplitudes) are determined when the bipolar soliton interactions have the strongest effects on the skewness and kurtosis. In addition, numerical simulations are conducted to examine the properties of multi-soliton interactions and their influence on higher statistical moments, especially confirming the emergence of the soliton interactions described by the mixed and rational solutions in a denser soliton ensemble.

Multisoliton interactions approximating the dynamics of breather solutions

AbstractNowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies.

Spiraling Laguerre-Gaussian solitons and arrays in parabolic potential wells.

Controllable trajectories of beams are one of the main themes in optical science. Here, we investigate the propagation dynamics of Laguerre-Gaussian (LG) solitons in parabolic potential wells and introduce off-axis and chirp parameters (which represent the displacement and the initial angle of beams) to make solitons sinusoidally oscillate in the x and y directions and undergo elliptically or circularly spiraling trajectories during propagation. Additionally, LG solitons with different orders and powers can be combined into soliton arrays of various shapes, depending on the off-axis parameter. Moreover, the soliton arrays can exhibit periodic converging, rotating, and other evolution behaviors, by the proper choice of the chirp parameter. A series of interesting examples demonstrate typical propagation scenarios. Our results may provide a new perspective on and stimulate further investigations of multisoliton interactions in potential wells and may find applications in optical communication and particle control.

Multi-soliton interactions, comb behavior, strange mixed breathers and other exotic rogue waves generated in a nonlinear left-handed transmission line

We establish a cubic–quintic-nonlinear Schrödinger equation (CQNLSE) including cubic- and quintic-nonlinearities, second-, third- and fourth-order dispersions in a left-handed nonlinear transmission line. We use the collective coordinates’ theory and the Gaussian Ansatz function with six coordinates to improve the comprehension of the system. Besides, the following innovations have been found when linear and nonlinear effects interact. (i) Some phenomena have emerged such as comb behavior, fragmentation of waves and wall of waves. (ii) Some solutions have also emerged such as the dark soliton, multi-peak waves, Kuznetsov–Ma breathers, periodical and non periodical mixed wave trains, Sasa–Satsuma waves, Peregrine waves, and crossed solitons. Otherwise, the internal perturbations leading to all aforementioned exotic rogue events are well studied in detail.

Hirota bilinear method and multi-soliton interaction of electrostatic waves driven by cubic nonlinearity in pair-ion–electron plasmas

Multi-soliton interaction of nonlinear ion sound waves in a pair-ion–electron (PIE) plasma having non-Maxwellian electrons including Kappa, Cairns, and generalized two spectral index distribution functions is studied. To this end, a modified Korteweg–de Vries (mKdV) equation is obtained to investigate the ion-acoustic waves in a PIE plasma at a critical plasma composition. The effects of temperature and density ratios and the non-Maxwellian electron velocity distributions on the overtaking interaction of solitons are explored in detail. The results reveal that both hump (positive peak) and dip (negative peak) solitons can propagate for the physical model under consideration. Two and three-soliton interactions are presented, and the novel features of interacting compressive and rarefactive solitons are highlighted. The present investigation may be useful in laboratory plasmas where PIE plasmas have been reported.

Dynamics of pulsating solitons derived from asymmetrical dispersive waves.

Pulsating soliton (PS) as a local structure of nonlinear systems has induced substantial interest in nonlinear photonics and ultrafast lasers. However, the interaction mechanism between PSs has not been fully studied. Here, the vital role of the asymmetric dispersive wave (DW) in PSs interaction is investigated for the first time. Based on the complex Ginzburg-Landau equation (CQGLE), we find that the asynchronous pulsating soliton molecule (PSM) composed of strong PSs and weak PSs will produce frequency shift due to the asymmetric DW, and the state of the PS can be transferred through the DW during the collision between PSs and PSM. Moreover, we firstly characterize the PS with asymmetric DW in experiment, and observe the drift of PSM, which agree with our simulation that the asymmetric DW can cause the frequency shift of PSMs. Our results provide new insights into the multi soliton interaction of nonlinear systems.

Singular solitons interaction of dust ion acoustic waves in the framework of Korteweg de Vries and Modified Korteweg‐de Vries equations with (r,q) distributed electrons

AbstractTwo nonlinear evolution equations (NLEEs), namely, Korteweg‐de Vries (KdV) and modified Korteweg‐de Vries (MKdV) are derived from the dust hydrodynamic model of collisionless, unmagnetized dusty plasma with electrons following double spectral velocity distribution, that is, (r,q) distribution. Regular as well as singular analytic solutions of derived KdV equations and MKdV equations are obtained separately with the aid of the Hirota Bilinear method. Two singular soliton solutions of both KdV and MKdV equations are obtained separately. The efficiency and interactive approach of the heuristic Hirota Bilinear method leads to multiple singular soliton solutions for its beautiful algebraic technique which generates solitons solutions as well as singular solitons explicitly. Then the significance of multi singular soliton interaction has been studied for KdV singular solitons and for MKdV singular solitons in the critical parameter set.

Integrability and the multi-soliton interactions of non-autonomous Zakharov–Kuznetsov equation

Integrability and the multi-soliton interactions of non-autonomous Zakharov–Kuznetsov equation

Synthesis and dissociation of soliton molecules in parallel optical-soliton reactors

Mode-locked lasers have been widely used to explore interactions between optical solitons, including bound-soliton states that may be regarded as “photonic molecules”. Conventional mode-locked lasers normally, however, host at most only a few solitons, which means that stochastic behaviours involving large numbers of solitons cannot easily be studied under controlled experimental conditions. Here we report the use of an optoacoustically mode-locked fibre laser to create hundreds of temporal traps or “reactors” in parallel, within each of which multiple solitons can be isolated and controlled both globally and individually using all-optical methods. We achieve on-demand synthesis and dissociation of soliton molecules within these reactors, in this way unfolding a novel panorama of diverse dynamics in which the statistics of multi-soliton interactions can be studied. The results are of crucial importance in understanding dynamical soliton interactions and may motivate potential applications for all-optical control of ultrafast light fields in optical resonators.

Rogue Waves With Rational Profiles in Unstable Condensate and Its Solitonic Model

In this brief report we study numerically the spontaneous emergence of rogue waves in 1) modulationally unstable plane wave at its long-time statistically stationary state and 2) bound-state multi-soliton solutions representing the solitonic model of this state. Focusing our analysis on the cohort of the largest rogue waves, we find their practically identical dynamical and statistical properties for both systems, that strongly suggests that the main mechanism of rogue wave formation for the modulational instability case is multi-soliton interaction. Additionally, we demonstrate that most of the largest rogue waves are very well approximated–simultaneously in space and in time–by the amplitude-scaled rational breather solution of the second order.

Generation of noise-like pulses and soliton rains in a graphene mode-locked erbium-doped fiber ring laser

We demonstrate the generation of noise-like pulses (NLPs) and soliton rains in a graphene saturable absorber mode-locked erbium-doped fiber laser. Typical NLPs are obtained at a proper pump power and in a cavity polarization state. The soliton rain operation with multiple solitons can be achieved by finely adjusting the cavity polarization state. In addition, distinctive multi-soliton interactions are observed and investigated, including the fundamental mode-locking and multiple pulses. The experimental results can help further understand nonlinear pulse dynamics in ultrafast optics.

Weak Multi-Phase Asymptotics for Nonintegrable Equations

We describe an approach that allows us to construct multi-soliton asymptotic solutions for essentially nonintegrable equations and analyze the type of wave collision. The general idea is demonstrated by the example of a generalization of Korteweg-de Vries equation with small dispersion in the cases of two and three waves. We also discuss the phenomenon of nonuniqueness that occurs in the framework of the weak description of a multi-soliton interaction.

Nonlinear waves behaviors for a coupled generalized nonlinear Schrödinger–Boussinesq system in a homogeneous magnetized plasma

Under investigation in this paper is a coupled generalized nonlinear Schrodinger–Boussinesq system, which describes the coupled upper-hybrid and magnetoacoustic modes in a homogeneous magnetized plasma for the bidirectional propagation near the magnetoacoustic speed. Based on the Hirota method, the expressions for the multi-soliton solutions are given. Effects of the group velocity, group dispersion coefficient for the upper-hybrid, and the properties of the magnetic field on the soliton are discussed. Based on the asymptotic analysis, interaction between two solitons is proved to be elastic through the asymptotic analysis. Position at which the maximal distortion occurs is obtained. Multi-soliton interaction is illustrated and investigated. Two prerequisites of the formation and features of the bound state are discussed. For the cases of three solitons, inelastic interaction occurs with phase shifts. Characteristics of the breather and its relation with the bound state and the breather are investigated. Interaction between the bound state (even the breather) and a single soliton is discussed for both cases that they are parallel or not.

Dynamics of rotating Laguerre-Gaussian soliton arrays.

Trajectory control of spatial solitons is an important subject in optical transmission field. Here we investigate the propagation dynamics of Laguerre-Gaussian soliton arrays in nonlinear media with a strong nonlocality and introduce two parameters, which we refer to as initial tangential velocity and displacement, to control the propagation path. The general analytical expression for the evolution of the soliton array is derived and the propagation properties, such as the intensity distribution, the propagation trajectory, the center distance, and the angular velocity are analyzed. It is found that the initial tangential velocity and displacement make the solitons sinusoidally oscillate in the x and y directions, and each constituent soliton undergoes elliptically or circularly spiral trajectory during propagation. A series of numerical examples is exhibited to graphically illustrate these typical propagation properties. Our results may provide a new perspective and stimulate further active investigations of multisoliton interaction and may be applied in optical communication and particle control.

Multi-soliton interaction of a generalized Schrödinger-Boussinesq system in a magnetized plasma

Under investigation in this paper is a generalized Schrodinger-Boussinesq system, which describes the stationary propagation of coupled upper-hybrid waves and magnetoacoustic waves in a magnetized plasma. Bilinear forms, one-, two- and three-soliton solutions are derived by virtue of the Hirota method and symbolic computation. Propagation and interaction for the solitons are illustrated graphically: Coefficients \( \beta_{1}^{}\) and \( \beta_{2}^{}\) can affect the velocities and propagation directions of the solitary waves. Amplitude, velocity and shape of the one solitary wave keep invariant during the propagation, implying that the transport of the energy is stable in the upper-hybrid and magnetoacoustic waves, and amplitude of the upper-hybrid wave is bigger than that of the magnetoacoustic wave. For the upper-hybrid and magnetoacoustic waves, head-on, overtaking and bound-state interaction between the two solitary waves are asymptotically depicted, respectively, indicating that the interaction between the two solitary waves is elastic. Elastic interaction between the bound-state soliton and a single one soliton is also displayed, and interaction among the three solitary waves is all elastic.

Adiabatic interactions of Manakov solitons—Effects of cross-modulation

We investigate the asymptotic behavior of the Manakov soliton trains perturbed by cross-modulation in the adiabatic approximation. The multisoliton interactions in the adiabatic approximation are modeled by a generalized Complex Toda chain (GCTC). The cross-modulation requires special treating for the evolution of the polarization vectors of the solitons. The numerical predictions of the Manakov system are compared with the perturbed GCTC. For certain set of initial parameters GCTC describes very well the long-time evolution of the Manakov soliton trains.

Bäcklund Transformation and Soliton Solutions for a (3+1)-Dimensional Variable-Coefficient Breaking Soliton Equation

Abstract In this article, a (3+1)-dimensional variable-coefficient breaking soliton equation is investigated. Based on the Bell polynomials and symbolic computation, the bilinear forms and Bäcklund transformation for the equation are derived. One-, two-, and three-soliton solutions are obtained via the Hirota method. N-soliton solutions are also constructed. Propagation characteristics and interaction behaviors of the solitons are discussed graphically: (i) solitonic direction and position depend on the sign of the wave numbers; (ii) shapes of the multisoliton interactions in the scaled space and time coordinates are affected by the variable coefficients; (iii) multisoliton interactions are elastic for that the velocity and amplitude of each soliton remain unchanged after each interaction except for a phase shift.

Lax pair, rogue-wave and soliton solutions for a variable-coefficient generalized nonlinear Schrödinger equation in an optical fiber, fluid or plasma

In this paper, a variable-coefficient generalized nonlinear Schrodinger equation, which can be used to describe the nonlinear phenomena in the optical fiber, fluid or plasma, is investigated. Lax pair, higher-order rogue-wave and multi-soliton solutions, Darboux transformation and generalized Darboux transformation are obtained. Wave propagation and interaction are analyzed: (1) The Hirota and Lakshmanan–Porsezian–Daniel coefficients affect the propagation velocity and path of each one soliton; three types of soliton interaction have been attained: the bound state, one bell-shape soliton’s catching up with the other and two bell-shape soliton head-on interaction. Multi-soliton interaction is elastic. (2) The Hirota and Lakshmanan–Porsezian–Daniel coefficients affect the propagation direction of the first-step rogue waves and interaction range of the higher-order rogue waves.

Solitons, Bäcklund Transformation, Lax Pair, and Infinitely Many Conservation Law for a (2+1)-Dimensional Generalised Variable-Coefficient Shallow Water Wave Equation

Abstract Under investigation in this article is a (2+1)-dimensional generalised variable-coefficient shallow water wave equation, which describes the interaction of the Riemann wave propagating along the y axis with a long-wave propagating along the x axis in a fluid, where x and y are the scaled space coordinates. Bilinear forms, Bäcklund transformation, Lax pair, and infinitely many conservation law are derived based on the binary Bell polynomials. Multi-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the multi-soliton interaction in the scaled space and time coordinates. (ii) Positions of the solitons depend on the sign of wave numbers after each interaction. (iii) Interaction of the solitons is elastic, i.e. the amplitude, velocity, and shape of each soliton remain invariant after each interaction except for a phase shift.

Asymptotic behavior of Manakov solitons: Effects of potential wells and humps

We consider the asymptotic behavior of the soliton solutions of Manakov’s system perturbed by external potentials. It has already been established that its multisoliton interactions in the adiabatic approximation can be modeled by the complex Toda chain (CTC). The fact that the CTC is a completely integrable system, enables us to determine the asymptotic behavior of the multisoliton trains. In the present study we accent on the 3-soliton initial configurations perturbed by sech-like external potentials and compare the numerical predictions of the Manakov system and the perturbed CTC in different regimes. The results of conducted analysis show that the perturbed CTC can reliably predict the long-time evolution of the Manakov system.