We study $n$-pulse interaction in fast collisions of $N$ solitons of the cubic nonlinear Schrödinger (NLS) equation in the presence of generic weak nonlinear loss. We develop a reduced model that yields the contribution of $n$-pulse interaction to the amplitude shift for collisions in the presence of weak $(2m+1)$-order loss, for any $n$ and $m$. We first employ the reduced model and numerical solution of the perturbed NLS equation to analyze soliton collisions in the presence of septic loss $(m=3)$. Our calculations show that three-pulse interaction gives the dominant contribution to the collision-induced amplitude shift already in a full-overlap four-soliton collision, and that the amplitude shift strongly depends on the initial soliton positions. We then extend these results for a generic weak nonlinear loss of the form $G(|ψ|^{2})ψ$, where $ψ$ is the physical field and $G$ is a Taylor polynomial of degree $m_{c}$. Considering $m_{c}=3$, as an example, we show that three-pulse interaction gives the dominant contribution to the amplitude shift in a six-soliton collision, despite the presence of low-order loss. Our study quantitatively demonstrates that $n$-pulse interaction with high $n$ values plays a key role in fast collisions of NLS solitons in the presence of generic nonlinear loss. Moreover, the scalings of $n$-pulse interaction effects with $n$ and $m$ and the strong dependence on initial soliton positions lead to complex collision dynamics, which is very different from the one observed in fast NLS soliton collisions in the presence of cubic loss.
We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schrödinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of pertu…
We numerically study the breathing dynamics induced by collision between bright solitons in the one-dimensional Bose-Einstein condensates with strong dipole-dipole interaction. This breathing phenomenon is closely related to the after-collision short-lived attraction of solitons induced by the dipolar effect. The initial phase difference of solitons leads to the asymmetric dynamics after collision, which is manifested on their different breathing amplitude, breathing frequency, and atom number. We clarify that the asymmetry of breathing frequency is directly induced by the asymmetric atom numb…
Inspired by the well known sine-Gordon equation, we present a symmetric coupled system of two real scalar fields in $1+1$ dimensions. There are three different topological soliton solutions which be labelled according to their topological charges. These solitons can absorb some localized non-dispersive wave packets in collision processes. It will be shown numerically, during collisions between solitons, there will be an uncertainty which originates from the amount of the maximum amplitude and arbitrary initial phases of the trapped wave packets.
In this report, the various 1D single soliton and multi-soliton solutions of the Sine-Gordon equation are explored. First the topological kink solitons and their properties for the Sine-Gordon, as well as the $ϕ^{4}$ model are investigated. This includes energy and momentum conservation laws for both models, as well as static kinks and antikinks. The Bogomol'nyi equation and the importance of Lorentz invariance are also discussed. Then in the second half, the focus is on Sine-Gordon multi-kink solutions, which are found by a separation of variables method rather than using inverse scattering o…
During defect-antidefect scattering, bound modes frequently disappear into the continuous spectrum before the defects themselves collide. This leads to a structural, nonperturbative change in the spectrum of small excitations. Sometimes the effect can be seen as a hard wall from which the defect can bounce off. We show the existence of these spectral walls and study their properties in the $ϕ^4$ model with BPS preserving impurity, where the spectral wall phenomenon can be isolated because the static force between the antikink and the impurity vanishes. We conclude that such spectral walls shou…
In this paper, we discus the fascinating energy sharing collisions of multicomponent solitons in certain incoherently coupled and coherently coupled nonlinear Schrödinger type equations arising in the context of nonlinear optics.
We consider a general multicomponent (2+1)-dimensional long-wave--short-wave resonance interaction (LSRI) system with arbitrary nonlinearity coefficients, which describes the nonlinear resonance interaction of multiple short waves with a long-wave in two spatial dimensions. The general multicomponent LSRI system is shown to be integrable by performing the Painlevé analysis. Then we construct the exact bright multi-soliton solutions by applying the Hirota's bilinearization method and study the propagation and collision dynamics of bright solitons in detail. Particularly, we investigate the head…
Multiple double-valley and single-valley dark solitons in the complex modified Korteweg-de Vries equation: shape-preserving collisions and shape-altering collisions
Shenzhen University（深圳大学）；Bishop Heber College（比숍希伯学院）；Hubei University of Science and Technology（湖北科技学院）
The possibility of generating stable anisotropic solitons in dipolar Bose-Einstein condensates (BECs) leads to a different scenario not possible in a nondipolar BEC with contact interaction. We study the statics and dynamics of anisotropic bright solitons in quasi-two-dimensional BECs consisting of polarized dipolar atoms.We study the collision dynamics of two such solitons at different velocities for different angles between the polarization and collision directions. The collision is found to be quasi elastic at large velocities. At small velocities the collision is inelastic leading to the f…
We develop a consistent semiclassical method to calculate the probability of topological soliton-antisoliton pair production in collisions of elementary particles. In our method one adds an auxiliary external field pulling the soliton and antisoliton in the opposite directions. This transforms the original scattering process into a Schwinger pair creation of the solitons induced by the particle collision. One describes the Schwinger process semiclassically and recovers the original scattering probability in the limit of vanishing external field. We illustrate the method in (1+1)-dimensional sc…
Recently, we have shown that the Manakov equation can admit a more general class of nondegenerate vector solitons, which can undergo collision without any intensity redistribution in general among the modes, associated with distinct wave numbers, besides the already known energy exchanging solitons corresponding to identical wave numbers. In the present comprehensive paper, we discuss in detail the various special features of the reported nondegenerate vector solitons. To bring out these details, we derive the exact forms of such vector one-, two- and three-soliton solutions through Hirota bil…
Synchronous collisions of solitons of the Korteweg -- de Vries equation are considered as a representative example of the interaction of a large number of solitons in a soliton gas. Statistical properties of the soliton field are examined for a model distribution of soliton amplitudes according to a power law. $N$-soliton solutions ($N \le 50$) are constructed with the help of a numerical procedure using the Darboux transformation and 100-digits arithmetic. It is shown that there exist qualitatively different patterns of evolving multisoliton solutions depending on the amplitude distribution. …
Synchronous collisions between a large number of solitons are considered in the context of a statistical description. It is shown that during the interaction of solitons of the same signs the wave field is effectively smoothed out. When the number of solitons increases and the sequence of their amplitudes decay slower, the focused wave becomes even smoother and the statistical moments get frozen for a long time. This quasi-stationary state is characterized by greatly reduced statistical moments and by the density of solitons close to some critical value. This state may be treated as the small-…
We study general collisions between chiral solitons in Bose-Einstein condensates subject to combined attractive and current-dependent interatomic interactions. A simple analysis based on the linear superposition of the solitons allows us to determine the relevant time and space scales of the dynamics, which is illustrated by extensive numerical simulations. By varying the differential amplitude, the relative phase, the average velocity, and the relative velocity of the solitons, we characterize the different dynamical regimes that give rise to oscillatory and interference phenomena. Apart from…
We construct single input logic gates using the energy sharing collisions of a minimal number of (three) bright optical solitons associated with the three soliton solution of the integrable Manakov system. As computation requires state changes to represent binary logic, here we make use of the state change of a particular soliton during its sequential collision with other two solitons for constructing single input gates. As a consequence, we clearly demonstrate the construction of various one-input logic gates such as COPY gate, NOT gate and ONE gate using energy sharing three soliton collisio…
Solitons are localised wave disturbances that propagate without changing shape, a result of a nonlinear interaction which compensates for wave packet dispersion. Individual solitons may collide, but a defining feature is that they pass through one another and emerge from the collision unaltered in shape, amplitude, or velocity. This remarkable property is mathematically a consequence of the underlying integrability of the one-dimensional (1D) equations, such as the nonlinear Schrödinger equation, that describe solitons in a variety of wave contexts, including matter-waves$^{1,2}$. Here we expl…
Ultracold gases provide an unprecedented level of control for the investigation of soliton dynamics and collisions. We present a scheme for deterministically preparing pairs of three-component solitons in a Bose-Einstein condensate. Our method is based on local spin rotations which simultaneously imprint suitable phase and density distributions. This enables us to observe striking collisional properties of the vector degree of freedom which naturally arises for the coherent nature of the emerging multi-component solitons. We find that the solitonic properties in the quasi-one-dimensional syste…
Ion acoustic (IA) solitons are accompanied by vortex-shaped nonlinear structures (e.g. hollows, plateaus or humps) in the electron distribution function, called electron holes, portraying trapped electrons. These structures appear as charged flexible clouds (shielded by the background plasma) in the phase space with their own inertia, depending on the number of trapped electrons. According to simulation studies, electron holes tend to merge in pairs until one accumulative hole remains in the simulation box. This tendency has been analytically and qualitatively explained in the frame of the ene…
The energy sharing collision of bright optical solitons in the Manakov system, governing pulse propagation in high birefringent fiber, is employed theoretically to realize optical logic gates. Especially, for the first time, we successfully construct (theoretically) the universal NOR gate and the OR gate from the energy sharing collisions of just four bright solitons which can be well described by the exact bright four-soliton solution of the Manakov system. This construction procedure has the important merits like realizing the two input gates with a minimal number of soliton collisions and p…
We perform a comparative beyond mean-field study of black and grey solitonic excitations in a finite ensemble of ultracold bosons confined to a one-dimensional box. An optimized density-engineering potential is developed and employed together with phase-imprinting to cleanly initialize grey solitons. Based on our recently developed Multi-Layer Multi-Configuration Time-Dependent Hartree Method for Bosons, we demonstrate an enhancement of the quantum fluctuations limited lifetime of the soliton contrast with increasing soliton velocity. A natural orbital analysis reveals a two-stage process unde…
We study interfacial waves in a system of two horizontal layers of immiscible inviscid fluids involved into horizontal vibrational motion. We analyze the linear and nonlinear stability properties of the solitons in the system and consider two-soliton collision scenarios. We describe the events of explosive formation of sharp peaks on the interface, which may presumably lead to the layer rapture, and find that beyond the vicinity of this peaks the system dynamics can be represented as a kinetics of a soliton gas.
We demonstrate that pulses of linear physical systems, weakly perturbed by nonlinear dissipation, exhibit soliton-like behavior in fast collisions. The behavior is demonstrated for linear waveguides with weak cubic loss and for systems described by linear diffusion-advection models with weak quadratic loss. We show that in both systems, the expressions for the collision-induced amplitude shifts due to the nonlinear loss have the same form as the expression for the amplitude shift in a fast collision between two optical solitons in the presence of weak cubic loss. Our analytic predictions are c…
We study the dynamics of emission of radiation (small-amplitude waves) in fast collisions between two solitons of the nonlinear Schrödinger (NLS) equation in the presence of weak cubic loss. We calculate the radiation dynamics by a perturbation technique with two small parameters: the cubic loss coefficient $ε_{3}$ and the reciprocal of the group velocity difference $1/β$. The agreement between the perturbation theory predictions and the results of numerical simulations with the full coupled-NLS propagation model is very good for large $β$ values, and is good for intermediate $β$ values. Addit…
In this work, we report observations of phase transitions in 2D multistable mechanical metamaterials that are initiated by collisions of soliton-like pulses in the metamaterial. Analogous to first-order phase transitions in crystalline solids, we experimentally and numerically observe that the multistable metamaterials support phase transitions if the new phase meets or exceeds a critical nucleus size. If this criterion is met, the new phase subsequently propagates in the form of transition waves, converting the rest of the metamaterial to the new phase. More interestingly, we observe that the…
The theory of soliton gas had been previously developed for unidirectional integrable dispersive hydrodynamics in which the soliton gas properties are determined by the overtaking elastic pairwise interactions between solitons. In this paper, we extend this theory to soliton gases in bidirectional integrable Eulerian systems where both head-on and overtaking collisions of solitons take place. We distinguish between two qualitatively different types of bidirectional soliton gases: isotropic gases, in which the position shifts accompanying the head-on and overtaking soliton collisions have the s…
We investigate the focusing coupled PT-symmetric nonlocal nonlinear Schrodinger equation employing Darboux transformation approach. We find a family of exact solutions including pairs of Bright-Bright, Dark-Dark and Bright-Dark solitons in addition to solitary waves. We show that one can convert bright bound state onto a dark bound state in a two-soliton solution by selectively fine tuning the amplitude dependent parameter. We also show that the energy in each mode remains conserved unlike the celebrated Manakov model. We also characterize the behaviour of the soliton solutions in detail. We e…
Solitons are very effective in transporting energy over great distances and collisions between them can produce high energy density spots of relevance to phase transformations, energy localization and defect formation among others. It is then important to study how energy density accumulation scales in multi-soliton collisions. In this study, we demonstrate that the maximal energy density that can be achieved in collision of $N$ slowly moving kinks and antikinks in the integrable sine-Gordon field, remarkably, is proportional to $N^2$, while the total energy of the system is proportional to $N…
Although extreme or freak waves are repeatedly measured in the oceans, their origin is largely unknown. The interaction of different water waves is seen as one reason for their emergence. One way to consider nonlinear waves in deep water is to look at solutions of the nonlinear Schrödinger equation, which plays an important role in the determination of extreme waves. One specific solution is the soliton solution. Therefore the question arises, how nonlinear waves behave as they interact or collide. Using a relaxation pseudo spectral scheme for the computation of solutions of the nonlinear Schr…
We have studied the dynamic evolution of the collective excitations in Bose-Einstein condensates in a deep optical lattice with tunable three-body interactions. Their dynamics is governed by a high order discrete nonlinear Schrodinger equation (DNLSE). The dynamical phase diagram of the system is obtained using the variational method. The dynamical evolution shows very interesting features. The discrete breather phase totally disappears in the regime where the three-body interaction completely dominates over the two-body interaction. The soliton phase in this particular regime exists only when…
Magnetic solitons offer functionalities as information carriers in multiple spintronic and magnonic applications. However, their potential for nanoscale energy transport has not been revealed. Here we demonstrate that antiferromagnetic solitons, e.g. domain walls, can uptake, transport and release energy. The key for this functionality resides in their relativistic kinematics; their self-energy increases with velocity due to Lorentz contraction of the soliton and their dynamics can be accelerated up to the effective speed of light of the magnetic medium. Furthermore, their classification in ro…
