Search

The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrödinger equations. Starting from the two-dimensional extension of the well known AKNS q , r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q , r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends on whether the nonlocality is in space or time. A so-called quasi-variance identity is derived and its relationship to blow-up is discussed. Numerical simulations reveal that this quantity reaching zero in finite time does not (in general) guarantee collapse. An alternative approach to the study of wave collapse is presented via the study of transverse instability of line soliton solutions. In particular, the linear stability problem for perturbed solitons is formulated for the nonlocal RT and PT symmetric nonlinear Schrödinger (NLS) equations. Through a combination of numerical and analytical approaches, the stability spectrum for some stationary one soliton solutions is found. Direct numerical simulations agree with the linear stability analysis which predicts filamentation and subsequent blow-up. [ABSTRACT FROM AUTHOR]

We propose a nonlinear quantum walk model inspired in a photonic implementation in which the polarization state of the light field plays the role of the coin-qubit. In particular, we take profit of the nonlinear polarization rotation occurring in optical media with Kerr nonlinearity, which allows to implement a nonlinear coin operator, one that depends on the state of the coin-qubit. We consider the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac equation. The analysis of this continuum limit allows us to gain some insight into the existence of different solitonic structures, such as bright and dark solitons. We illustrate several properties of these solitons with numerical calculations, including the effect on them of an additional phase simulating an external electric field. [ABSTRACT FROM AUTHOR]

We numerically investigate the influence of higher-order dispersion on the dynamics of quartic solitons in a passively mode-locked fiber laser. The simulation results show that the output pulse characteristics of the quartic soliton fiber laser are not only affected by the intra-cavity net dispersion but also related to the position of the single-mode fiber (SMF). For different intra-cavity net higher-order dispersion values, the quartic soliton undergoes multiple alternations between the stable single soliton state and pulsating state in the fiber laser, and the number and intensity of Kelly sidebands in the output pulse spectrum are significantly different. For different positions of the SMF, the dynamics of the quartic soliton are also different. In addition, we also observe two types of novel pulsation-creeping behavior and time shift. These findings provide new insights into the output characteristics of quartic solitons in fiber lasers. [ABSTRACT FROM AUTHOR]

The Ablowitz–Ladik equations, hereafter called A L + and A L − , are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of A L ± in the periodic setting, showing in particular that the background solution of A L − is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of A L ± describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of A L − are always singular on curves of spacetime, if they live on a background of sufficiently large amplitude, and we construct a different continuous limit describing this regime: a NLS equation with a nonlinear and weak dispersion. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution (i) evolves according to A L + into a recurrence of the above exact solutions, in the case of one and two unstable modes, and (ii) evolves according to A L − into a singularity in finite time if the amplitude of the background is greater than 1. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of beating solitons in a two-component Bose–Einstein condensate with tunable Rabi coupling strength. Our results demonstrate that the balance between transition and beating effects permits the emergence of a novel family of asymmetric solitons in the symmetric physical settings. We derive the exact analytical solutions for them, which primarily consist of one bright soliton and one dark soliton element. The analytical solutions provide us with precise balance conditions required for the formation of asymmetric solitons. We also show that the degree of asymmetry can be effectively manipulated by adjusting the background density flow of dark soliton element, initial relative phase between two soliton elements, and their width. Furthermore, we discuss the oscillation behavior of asymmetric solitons in a harmonic potential, and the interaction between them. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend the store of models able to support integrable defects by investigating the two-dimensional Boussinesq nonlinear wave equation. As has been previously noted in many examples, insisting that a defect contributes to energy and momentum to ensure their conservation, despite the presence of discontinuities and the explicit breaking of translation invariance, leads to sewing conditions relating the two fields and their derivatives on either side of the defect. The manner in which several types of soliton solutions to the Boussinesq equation are affected by the defect is explored and reveals new effects that have not been observed in other integrable systems, such as the possibility of a soliton reflecting from a defect or of a defect decaying into one or two solitons. [ABSTRACT FROM AUTHOR]

We report the first experimental observation of spatial solitons with complex polarization states, called the Poincaré polarization solitons (PPSs) in lead glass with strongly nonlocal nonlinearity. The formations of PPSs with topological charge of l = 1, including the cylindrical elliptical-polarization soliton (CEPS) and the angularly-hybrid polarization soliton (AHPS), were observed. We showed that the annular profiles and the complex polarization distributions of the first-order PPSs can be remained. Based on the linear stability analysis, we proved that the first-order PPSs are fully stable and the second-order PPS can survive only when one of the two component vortices dominates. [ABSTRACT FROM AUTHOR]

In this work, pure-quartic soliton (PQS)formation is investigated in the framework of a nonlinear Schrödinger equation with competing Kerr (cubic) and non-Kerr (quintic) nonlocal nonlinearities and quartic dispersion. In the process, the modulational instability (MI) phenomenon is activated under a suitable balance between the nonlocal nonlinearities and the quartic dispersion, both for exponential and rectangular nonlocal nonlinear responses. Interestingly, the maximum MI growth rate and bandwidth are reduced or can completely be suppressed for some specific values of the cubic and quintic nonlocality parameters, depending on the type of nonlocal response. The analytical results are confirmed via direct numerical simulations, where the instability supports the signature of pure-quartic dark and bright solitons. These results may provide a better understanding of PQS structures for their potential applications in the next generation of nonlinear optical devices. [ABSTRACT FROM AUTHOR]

Nonlinear waveguides with two distinct domains of anomalous dispersion can support the formation of molecule-like two-color pulse compounds. They consist of two tightly bound subpulses with frequency loci separated by a vast frequency gap. Perturbing such a two-color pulse compound triggers periodic amplitude and width variations, reminiscent of molecular vibrations. With increasing strength of perturbation, the dynamics of the pulse compound changes from harmonic to nonlinear oscillations. The periodic amplitude variations enable coupling of the pulse compound to dispersive waves, resulting in the resonant emission of multi-frequency radiation. We demonstrate that the location of the resonances can be precisely predicted by phase-matching conditions. If the pulse compound consists of a pair of identical subpulses, inherent symmetries lead to degeneracies in the resonance spectrum. Weak perturbations lift existing degeneracies and cause a splitting of the resonance lines into multiple lines. Strong perturbations result in more complex emission spectra, characterized by well separated spectral bands caused by resonant Cherenkov radiation and additional four-wave mixing processes. [ABSTRACT FROM AUTHOR]

The inverse scattering transform allows explicit construction of solutions to many physically significant nonlinear wave equations. Notably, this method can be extended to fractional nonlinear evolution equations characterized by anomalous dispersion using completeness of suitable eigenfunctions of the associated linear scattering problem. In anomalous diffusion, the mean squared displacement is proportional to t α , α > 0, while in anomalous dispersion, the speed of localized waves is proportional to A α , where A is the amplitude of the wave. Fractional extensions of the modified Kortewegâ€"deVries (mKdV), sine-Gordon (sineG) and sinh-Gordon (sinhG) and associated hierarchies are obtained. Using symmetries present in the linear scattering problem, these equations can be connected with a scalar family of nonlinear evolution equations of which fractional mKdV (fmKdV), fractional sineG (fsineG), and fractional sinhG (fsinhG) are special cases. Completeness of solutions to the scalar problem is obtained and, from this, the nonlinear evolution equation is characterized in terms of a spectral expansion. In particular, fmKdV, fsineG, and fsinhG are explicitly written. One-soliton solutions are derived for fmKdV and fsineG using the inverse scattering transform and these solitons are shown to exhibit anomalous dispersion. [ABSTRACT FROM AUTHOR]

We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Kortewegâ€"de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport. [ABSTRACT FROM AUTHOR]

We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotkaâ€"Volterra, Toda lattice (TL), and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional TL equations. Each of the delay-difference and delay-differential equations has the N -soliton solution, which depends on the delay parameter and converges to an N -soliton solution of a known soliton equation as the delay parameter approaches 0. [ABSTRACT FROM AUTHOR]

As a toy model for understanding the soliton resolution phenomenon we consider a characteristic initial boundary value problem for the 4 d equivariant Yangâ€"Mills equation outside a ball. Our main objective is to illustrate the advantages of employing outgoing null (or asymptotically null) foliations in analyzing the relaxation processes due to the dispersal of energy by radiation. In particular, within this approach it is evident that the endstate of evolution must be non-radiative (meaning vanishing flux of energy at future null infinity). In our toy model such non-radiative configurations are given by a static solution (called the half-kink) plus an alternating chain of N decoupled kinks and antikinks. We show numerically that the configurations N = 0 (static half-kink) and N = 1 (superposition of the static half-kink and the antikink which recedes to infinity) appear as generic attractors and we determine a codimension-one borderline between their basins of attraction. The rates of convergence to these attractors are analyzed in detail. [ABSTRACT FROM AUTHOR]

In this article, we prove that a sum of solitons and breathers of the modified Kortewegâ€"de Vries equation (mKdV) is orbitally stable. The orbital stability is shown in H 2. More precisely, we will show that if a solution of mKdV is close enough to a sum of solitons and breathers with distinct velocities at t = 0 in the H 2 sense, then it stays close to this sum of solitons and breathers for any time t â©ľ 0 in the H 2 sense, up to space translations for solitons or space and phase translations for breathers, provided the condition that the considered solitons and breathers are sufficiently decoupled from each other and that the velocities of the considered breathers are all positive, except possibly one. The constants that appear in this stability result do not depend on translation parameters. From this, we deduce the orbital stability of any multi-breather of mKdV, provided the condition that the velocities of the considered breathers are all positive, except possibly one (the condition about the decoupling of the considered solitons and breathers between each other is not required in this setting). The constants that appear in this stability result depend on translation parameters of the considered solitons and breathers. [ABSTRACT FROM AUTHOR]

We propose, via the Atiyahâ€"Manton approximation, a framework for studying skyrmions on R 3 using Atiyahâ€"Drinfeldâ€"Hitchinâ€"Manin (ADHM) data for Yangâ€"Mills instantons on R 4 . We provide a dictionary between important concepts in the Skyrme model and analogous ideas for ADHM data, and describe an efficient process for obtaining approximate Skyrme fields directly from ADHM data. We show that the approximation successfully describes all known skyrmions with charge B â©˝ 8 , with energies reproduced within 2% of the true minimisers. We also develop factorisation methods for studying clusters of instantons and skyrmions, generalising early work by Christâ€"Stantonâ€"Weinberg, and describe some relatively large families of explicit ADHM data. These tools provide a unified framework for describing coalesced highly-symmetric configurations as well as skyrmion clusters, both of which are needed to study nuclear systems in the Skyrme model. [ABSTRACT FROM AUTHOR]

By exploiting the theory of electromagnetic waves from Maxwell’s equations, the damped nonlinear Schrödinger (DNLS) equation is shown to govern the evolution of nonlinear periodic optical signals in a lossy optical fiber. These optical periodic pulses are mainly generated by the classical process of modulational instability (MI) in which nonlinearity is balanced by chromatic dispersion in the anomalous regime, with the linear loss generally suppressing the existence of soliton trains during propagation down the lossy fiber. When the periodic optical wave trains are subjected to weak external perturbations, this leads to the exposure of some internal modes of the system which are bound states solutions of the first order Lamé equation. These modes generally characterize various fundamental background excitations that co-propagate with the optical periodic signals in the fiber. Direct numerical simulations of the DNLS amplitude equation depict the exponential decrease in the amplitude and corresponding increase in the width of the wave trains during propagation. Power lasers are used in order to compensate for fiber losses; this is realized via time-division multiplexing of optical pulses which are periodically pumped into the lossy fiber at regular distances within the framework of a distributed amplification scheme. This leads to the regular energy restoration in the lossy fiber as a result of the interactions between the energized multiplexed light signals (generated by the power lasers) and the propagating damped optical pulses, hence ensuring effective transmission over long distances. [ABSTRACT FROM AUTHOR]

We present an analysis of 450 solitary wave pulses observed by the Langmuir Probe and Waves instrument on the Mars Atmosphere and Volatile EvolutioN spacecraft during its five passes around Mars on 2015 February 9. The magnitude and duration of these pulses vary between 1 and 25 mV mâˆ'1 and 0.2â€"1.7 ms, respectively. The ambient plasma conditions suggest that these pulses are quasi-parallel to the ambient magnetic field and can be considered electrostatic. These pulses are dominantly seen in the dawn (5â€"6 LT) and afternoon-dusk (15â€"18 LT) sectors at an altitude of 1000â€"3500 km. The frequencies of these electric field pulses are close to the ion plasma frequency (i.e., f pi ≤ f ef ≪ f pe), which suggests that their formation is governed by ion dynamics. The computer simulation performed for the Martian magnetosheath plasma hints that these pulses are ion-acoustic solitary waves generated by drifted ion and electron populations and their spatial scales are in the range of few ion Debye lengths (1.65â€"10 λ di). This is the first study to report and model solitary wave structures in the Martian magnetosheath. [ABSTRACT FROM AUTHOR]

We address the existence and stability of one-dimensional (1D) holes and kinks and two-dimensional (2D) vortex-holes nested in extended binary Bose mixtures, which emerge in the presence of Leeâ€"Huangâ€"Yang (LHY) quantum corrections to the mean-field energy, along with self-bound quantum droplets. We consider both the symmetric system with equal intra-species scattering lengths and atomic masses, modeled by a single (scalar) LHY-corrected Grossâ€"Pitaevskii equation (GPE), and the general asymmetric case with different intra-species scattering lengths, described by two coupled (spinor) GPEs. We found that in the symmetric setting, 1D and 2D holes can exist in a stable form within a range of chemical potentials that overlaps with that of self-bound quantum droplets, but that extends far beyond it. In this case, holes are found to be always stable in 1D and they transform into pairs of stable out-of-phase kinks at the critical chemical potential at which localized droplets turn into flat-top states, thereby revealing the connection between localized and extended nonlinear states. In contrast, we found that the spinor nature of the asymmetric systems may lead to instability of 1D holes, which tend to break into two gray states moving in the opposite directions. Remarkably, such instability arises due to spinor nature of the system and it affects only holes nested in extended modulationally-stable backgrounds, while localized quantum droplet families remain completely stable, even in the asymmetric case, while 1D holes remain stable only close to the point where they transform into pairs of kinks. We also found that symmetric systems allow fully stable 2D vortex-carrying single-charge states at moderate amplitudes, while unconventional instabilities appear also at high amplitudes. Symmetry also strongly inhibits instabilities for double-charge vortex-holes, which thus exhibit unexpectedly robust evolutions at low amplitudes. [ABSTRACT FROM AUTHOR]

In the present work we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schrödinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension d = 2, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent p, there exists a progressively narrowing range of stable frequencies. Finally, above the critical p all solutions are unstable. The picture is rather similar in the anisotropic case, with the difference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor b and nonlinearity power p. [ABSTRACT FROM AUTHOR]

We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to darkâ€"bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x) and the ‘mixed parity’ cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions. [ABSTRACT FROM AUTHOR]

Cellular automata have recently attracted a lot of attention as testbeds to explore the emergence of many-body quantum chaos and hydrodynamics. We consider the Rule 54 model, one of the simplest interacting integrable models featuring two species of quasiparticles (solitons), in the presence of an integrability-breaking perturbation that allows solitons to backscatter. We study the onset of thermalization and diffusive hydrodynamics in this model, compute perturbatively the diffusion constant of tracer particles, and comment on its relation to transport coefficients. [ABSTRACT FROM AUTHOR]

A vortex-bright (VB) soliton can precess around a fixed point. Here, we find numerically that the fixed point and the associated precessional orbits can be shifted by applying a constant driving force on the bright component, the displacement is proportional to the force with a minus sign. This robust dynamics is then discussed theoretically by treating the VB soliton as an effective point particle. By appropriately tuning the force, the VB soliton can be guided following an arbitrary trajectory with very weak density modulations, including that it can be repeatedly pinned and released at will. This finding opens a highly flexible and controllable approach of engineering the dynamics of vortical structures in Boseâ€"Einstein condensates. [ABSTRACT FROM AUTHOR]

We address the dynamics of two-dimensional (2D) truncated Airy waves (TAWs) and three-component solitons in the system of two fundamental-frequency (FF) and second-harmonic (SH) fields, coupled by quadratic ( χ (2) ) terms. The system models secondâ€"harmonic-generating optical media and atomic-molecular mixtures in Boseâ€"Einstein condensates. In addition to stable solitons, the system maintains truncated-Airy-waves states in either one of the FF components, represented by exact solutions, which are stable, unlike the Airy waves in the degenerate (two-component) χ (2) system. It is also possible to imprint vorticity onto the 2D Airy modes. By means of systematic simulations, we examine interactions between TAWs originally carried by different FF components, which are bending in opposite directions, through the SH field. The interaction leads to fusion of the input into a pair of narrow solitons. This is opposed to what happens in the 1D system, where the interacting Aity waves split into a large number of solitons. The interaction of truncated Aity waves carrying identical imprinted vorticities creates an additional pair of solitons, while opposite vorticities create a set of small-amplitude ‘debris’ in the output. Slowly moving solitons colliding with a heavy TAW bounce back, faster ones are absorbed by it, and collisions are quasi-elastic for fast solitons. Soliton-soliton collisions lead to merger into a single mode, or elastic passage, for lower and higher velocities, respectively. [ABSTRACT FROM AUTHOR]

In plasma wakefield accelerators, the wave excited in the plasma eventually breaks and leaves behind slowly changing fields and currents that perturb the ion density background. We study this process numerically using the example of a Facility for Advanced aCcelerator Experimental Tests (FACET) experiment where the wave is excited by an electron bunch in the bubble regime in a radially bounded plasma. Four physical effects underlie the dynamics of ions: (1) attraction of ions toward the axis by the fields of the driver and the wave, resulting in formation of a density peak, (2) generation of ion-acoustic solitons following the decay of the density peak, (3) positive plasma charging after wave breaking, leading to acceleration of some ions in the radial direction, and (4) plasma pinching by the current generated during the wave-breaking. The interplay of these effects results in the formation of various radial density profiles, which are difficult to produce in any other way. [ABSTRACT FROM AUTHOR]

We theoretically investigate dynamics of two dark solitons in a polariton condensate under nonresonant pumping, based on driven dissipative Grossâ€"Pitaevskii equations coupled to the rate equation. The equation of motion of the relative center position of two-dark soliton is obtained analytically by using the Lagrangian approach. In particular, the analytical expression of the effective potential between two dark solitons is given. The resulting equation of motion captures how the open-dissipative character of a polariton Boseâ€"Einstein condensate affects properties of dynamics of two-dark soliton, i.e., two-dark soliton relax by blending with the background at a finite time. We further simulate the relative motion of two dark solitons numerically with the emphasis on how two-soliton motion is manipulated by the initial velocity, in excellent agreement with the analytical results. The prediction of this work is sufficient for the experimental observations within current facilities. [ABSTRACT FROM AUTHOR]

Recently, research of solitons in Boseâ€"Einstein condensates has become a popular topic. Here, we mainly study exact analytical solutions of Grossâ€"Pitaevskii equations describing spin-orbit coupled spin-1 Boseâ€"Einstein condensates. To begin with, we show the analytical relation between different types of one-dimensional spin-orbit coupling and Zeeman effect. In addition, we find a transformation that can simplify the three-component Grossâ€"Pitaevskii equations with spin-orbit coupling into the nonlinear Schrödinger equation. The abundant stripe phase and dynamic characteristics of the system are investigated. [ABSTRACT FROM AUTHOR]

Capacitively coupled plasma is investigated kinetically utilizing the particle-in-cell technique. The argon (Ar) plasma is generated via two radio frequencies. The plasma bulk density increases by increasing the voltage amplitude of the high frequency (⩾13.56 MHz), which is much greater than the ion plasma frequency. The intermediate radio frequencies (≈1 MHz), which are comparable to the ion plasma frequency, cause a considerable broadening of the ion energy distribution, i.e. ions gain energies higher and lower than the time-averaged energy. The good agreement between published experimental results and our theoretical calculations via the ensemble-in-spacetime model confirms the modulation of ions around time-averaged values. Intermediate frequencies allow ions to partially respond to the instantaneous electric field. The response of ions to the instantaneous electric field is investigated semi-analytically. The dispersion relation of the plasma sheath and bulk are derived. Stable ion acoustic modes are found. Ion-acoustic modes have two different velocities and carry energy from the sheath edge to the electrode. In addition, intermediate frequencies excite solitons in the plasma sheath. The results may help to explain the ion density, flux, and energy modulation, and, consequently, the broadening of the ion energy distribution. [ABSTRACT FROM AUTHOR]

We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation. What is more, each soliton molecule can be transformed into an asymmetric soliton by changing the parameter Ď•. In addition, the collision between soliton molecules (or asymmetric soliton) and several soliton solutions is observed. Finally, some related pictures are presented. [ABSTRACT FROM AUTHOR]

The interaction between optical solitons is of great significance for studying interaction between light and matter and development of all-optical devices, and is conducive to the design of integrated optical path. Optical soliton interactions for the nonlinear Schrödinger equation are investigated to improve the communication quality and system integration. Solutions of the equation are derived and used to analyze the interaction of two solitons. Some suggestions are put forward to weaken their interactions. [ABSTRACT FROM AUTHOR]

It is known that the interplay of the spinâ€"orbit-coupling (SOC) and mean-field self-attraction creates stable two-dimensional (2D) solitons (ground states) in spinor Boseâ€"Einstein condensates. However, SOC destroys the system’s Galilean invariance, therefore moving solitons exist only in a narrow interval of velocities, outside of which the solitons suffer delocalization. We demonstrate that the application of a relatively weak moving optical lattice (OL), with the 2D or quasi-1D structure, makes it possible to greatly expand the velocity interval for stable motion of the solitons. The stability domain in the system’s parameter space is identified by means of numerical methods. In particular, the quasi-1D OL produces a stronger stabilizing effect than its full 2D counterpart. Some features of the domain are explained analytically. [ABSTRACT FROM AUTHOR]

We investigate the existence and dynamical stability of multipole gap solitons in Boseâ€"Einstein condensate loaded in a deformed honeycomb optical lattice. Honeycomb lattices possess a unique band structure, the first and second bands intersect at a set of so-called Dirac points. Deformation can result in the merging and disappearance of the Dirac points, and support the gap solitons. We find that the two-dimensional honeycomb optical lattices admit multipole gap solitons. These multipoles can have their bright solitary structures being in-phase or out-of-phase. We also investigate the linear stabilities and nonlinear stabilities of these gap solitons. These results have applications of the localized structures in nonlinear optics, and may helpful for exploiting topological properties of a deformed lattice. [ABSTRACT FROM AUTHOR]

We numerically study the interaction dynamics of two bright solitons with zero initial velocities in the one-dimensional dipolar Boseâ€"Einstein condensates. Under different dipolar strengths, the two bright solitons can merge into a breathing wave, and then split or propagate constantly after several oscillating periods. We quantitatively study the breathing frequency of wave after merging and the asymmetry property of solitons after splitting,and analyze their formation mechanism by the system’s energy evolution. Also, the change of initial phase difference brings distinct effects on the soliton interaction. Our results provide insight into the new dynamical phenomena in dipolar systems and enrich the understanding for interaction between dipolar solitons. [ABSTRACT FROM AUTHOR]

We analyze dynamical properties of the logarithmic Schrödinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected. [ABSTRACT FROM AUTHOR]

Detuning regulation based on the scanning pump wavelength plays an important role in the generation of temporal solitons in an optical microcavity. Field evolution inside a CaF2 microcavity during the detuning regulation process is demonstrated. It is found that a stable single soliton can be excited when modulation instability dominates in the microcavity during frequency tuning with an appropriate scanning speed and operational parameters. However, due to the excessive loss caused by tuning, the soliton eventually evolves into a DC distribution. The influence of different parameters on the light field after formation of the soliton is also studied. Other different parameters lead to different distributional forms of soliton evolution. For maintenance of the single soliton, the appropriate detuning parameter and pump power are changed suddenly after the soliton is generated, causing the soliton to remain stable. Moreover, in the single soliton region, a path within the parameter space of detuning and pump power is planned in order to compress the pulse width of the soliton, and the comb spectrum is broadened. The results of theoretical analysis are important for studying field and spectral characteristics during frequency detuning and for maintaining the temporal soliton in the microcavity. [ABSTRACT FROM AUTHOR]

We show that the quasiparticle kinetic theory for quantum and classical Calogero models reduces to the free-streaming Boltzmann equation. We reconcile this simple emergent behaviour with the strongly interacting character of the model by developing a Betheâ€"Lax correspondence in the classical case. This demonstrates explicitly that the freely propagating degrees of freedom are not bare particles, but rather quasiparticles corresponding to eigenvectors of the Lax matrix. We apply the resulting kinetic theory to classical Calogero particles in external trapping potentials and find excellent agreement with numerical simulations in all cases, both for harmonic traps that preserve integrability and exhibit perfect revivals, and for anharmonic traps that break microscopic integrability. Our framework also yields a simple description of multi-soliton solutions in a harmonic trap, with solitons corresponding to sharp peaks in the quasiparticle density. Extensions to quantum systems of Calogero particles are discussed. [ABSTRACT FROM AUTHOR]

The construction of Miura and Bäcklund transformations for A n mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in the well known sl (2) case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miura-gauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of A n . The construction of generalized gauge-Bäcklund transformation for the A n -KdV hierarchy is obtained as a composition of Miura and Bäcklund-gauge transformations for A n -mKdV hierarchy. The zero curvature representation provide a framework which is universal within all flows and generate systematically Bäcklund transformations for the entirely hierarchy. [ABSTRACT FROM AUTHOR]

We prove that any supersymmetric solution to five-dimensional minimal gauged supergravity with SU(2) symmetry, that is timelike outside an analytic horizon, is a Gutowski–Reall black hole or its near-horizon geometry. The proof combines a delicate near-horizon analysis with the general form for a Kähler metric with cohomogeneity-1 SU(2) symmetry. We also prove that any timelike supersymmetric soliton solution to this theory, with SU(2) symmetry and a nut or a complex bolt, has a Kähler base with enhanced U(1) × SU(2) symmetry, and we exhibit a family of asymptotically Ad solitons for p ⩾ 3 with a bolt in this class. [ABSTRACT FROM AUTHOR]

We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva et al (Ann. Phys., NY 403 198) (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a collective coordinates (CCs) theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the CCs. This Lagrangian differs only in the charge and mass with the Lagrangian of a CCs theory for the Gross–Neveu equation. Thus the soliton dynamics in the Alexeeva–Barashenkov–Saxena (ABS) spinor model is qualitatively the same as in the Gross–Neveu equation, but quantitatively it differs. These results of the CCs theory are confirmed by simulations, i.e. by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials. [ABSTRACT FROM AUTHOR]

Motivated by a recent development in the derivation of soliton solutions for initial-boundary value problems via the dressing method, we reconsider solutions to the nonlinear Schrödinger (NLS) equation on two half-lines connected via integrable defect conditions. The dressing method to construct soliton solutions is applied, while preserving the spectral boundary constraint with a time-dependent defect matrix. Moreover, N-soliton solutions are explicitly constructed and it is proven that the solitons are transmitted through the defect independently of one another. [ABSTRACT FROM AUTHOR]

For a scalar integrable model, it is generally believed that the solitons interact with each other elastically, for instance, multi-bright solitons from the nonlinear Schrödinger equation and the Korteweg-de Vries equation, etc. We obtain double-valley dark solitons from the defocusing Hirota equation by the Darboux transformation. Particularly, we report a remarkable phenomenon for the inelastic interaction of the double-valley dark solitons, in contrast to the solitons interacting with each other elastically for a scalar integrable model in previous works. Furthermore, we give the explicit conditions for the elastic collision based on the asymptotic analysis results. It is shown that the double-valley dark solitons could also admit elastic interaction under the special parameters settings. [ABSTRACT FROM AUTHOR]

Solitonic excitations of the one-dimensional quantum droplets are obtained, which smoothly connect vacuum with the flat-top droplet, akin to compactons in classical liquids. These solitons are of the kink type, necessarily residing on a constant pedestal, determined by the mean-field (MF) repulsion and beyond mean field quantum correction and having exactly one-third of the uniform condensate amplitude. Akin to the kinks, the propagating modes occur in pairs and are phase-locked with the background. The lowest chemical potential and condensate amplitude at the flat-top boundary matches with the self-trapped quantum droplet. More general excitations of analogous kind are obtained through the Möbius transform, which connect the required solutions to elliptic functions in general. [ABSTRACT FROM AUTHOR]

In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov–Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in Dohnal and Dörfler, [2013 Multiscale Model. Simul. 11 162–191]. [ABSTRACT FROM AUTHOR]

A number of possible pulsar radio emission mechanisms are based on streaming instabilities in relativistically hot electron–positron pair plasmas. At saturation, the unstable waves can, in principle, form stable solitary waves, which could emit the observed intense radio signals. We searched for the proper plasma parameters that would lead to the formation of solitons, and investigated their properties and dynamics as well as the resulting oscillations of electrons and positrons that possibly lead to radio wave emission. We utilized a one-dimensional version of the relativistic particle-in-cell code ACRONYM initialized with an appropriately parameterized one-dimensional Maxwell–Jüttner particle distribution in velocity space to study the evolution of the resulting streaming instability in a pulsar pair plasma. We found that strong electrostatic superluminal L-mode solitons are formed for plasmas with normalized inverse temperatures ρ ≥ 1.66 or relative beam drift speeds with Lorentz factors γ > 40. The parameters of the solitons fulfill the conditions for wave emission. For appropriate pulsar parameters the resulting energy densities of superluminal solitons can reach 1.1 × 105 erg cm−3, while those of subluminal solitons reach only 1.2 × 104 erg cm−3. Estimated energy densities of up to 7 × 1012 erg cm−3 suffice to explain pulsar nanoshots. [ABSTRACT FROM AUTHOR]

We numerically study the breathing dynamics induced by collision between bright solitons in 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 in their different breathing amplitude, breathing frequency, and atom number. We clarify that the asymmetry of breathing frequency is directly induced by the asymmetric atom number, rather than initial phase difference. Moreover, a collision between breathing solitons can produce new after-two-collision breathing solitons, whose breathing amplitude can be adjusted and reaches the maximum (or minimum) when the peak–peak (or dip–dip) collision happens. [ABSTRACT FROM AUTHOR]

We relate the sub-Riemannian geometry on the group of rigid motions of the plane to 'bicycling mathematics'. We show that this geometry's geodesics correspond to bike paths whose front tracks are either non-inflectional Euler elasticae or straight lines, and that its infinite minimizing geodesics (or 'metric lines') correspond to bike paths whose front tracks are either straight lines or 'Euler's solitons' (also known as syntractrix or convicts' curves). [ABSTRACT FROM AUTHOR]

The formation and evolution of post-solitons has been discussed for quite some time both analytically and through the use of particle-in-cell (PIC) codes. It is however only recently that they have been directly observed in laser-plasma experiments. Relativistic electromagnetic (EM) solitons are localised structures that can occur in collisionless plasmas. They consist of a low-frequency EM wave trapped in a low electron number-density cavity surrounded by a shell with a higher electron number-density. Here we describe the results of an experiment in which a 100 TW Ti:sapphire laser (30 fs, 800 nm) irradiates a TMPTA foam target with a focused intensity. A third harmonic (nm) probe is employed to diagnose plasma motion for 25 ps after the main pulse interaction via Doppler-Spectroscopy. Both radiation-hydrodynamics and 2D PIC simulations are performed to aid in the interpretation of the experimental results. We show that the rapid motion of the probe critical-surface observed in the experiment might be a signature of post-soliton wall motion. [ABSTRACT FROM AUTHOR]

The interaction of an oblique line soliton with a one-dimensional dynamic mean flow is analyzed using the Kadomtsev–Petviashvili II (KPII) equation. Building upon previous studies that examined the transmission or trapping of a soliton by a slowly varying rarefaction or oscillatory dispersive shock wave (DSW) in one space and one time dimension, this paper allows for the incident soliton to approach the changing mean flow at a nonzero oblique angle. By deriving invariant quantities of the soliton–mean flow modulation equations—a system of three (1 + 1)-dimensional quasilinear, hyperbolic equations for the soliton and mean flow parameters—and positing the initial configuration as a Riemann problem in the modulation variables, it is possible to derive quantitative predictions regarding the evolution of the line soliton within the mean flow. It is found that the interaction between an oblique soliton and a changing mean flow leads to several novel features not observed in the (1 + 1)-dimensional reduced problem. Many of these interesting dynamics arise from the unique structure of the modulation equations that are nonstrictly hyperbolic, including a well-defined multivalued solution interpreted as a solution of the (2 + 1)-dimensional soliton–mean modulation equations, in which the soliton interacts with the mean flow and then wraps around to interact with it again. Finally, it is shown that the oblique interactions between solitons and DSW solutions for the mean flow give rise to all three possible types of two-soliton solutions of the KPII equation. The analytical findings are quantitatively supported by direct numerical simulations. [ABSTRACT FROM AUTHOR]

Solitons are ubiquitous phenomena that appear, among others, in the description of tsunami waves, fiber-optic communication and ultracold atomic gases. The latter systems turned out to be an excellent playground for investigations of matter-wave solitons in a quantum world. This tutorial provides a general overview of the ultracold contact interacting Bose and Fermi systems in a one-dimensional space that can be described by the renowned Lieb–Liniger and Yang–Gaudin models. Both the quantum many-body systems are exactly solvable by means of the Bethe ansatz technique, granting us a possibility for investigations of quantum nature of solitonic excitations. We discuss in details a specific class of quantum many-body excited eigenstates called yrast states and show that they are strictly related to quantum dark solitons in the both considered Bose and Fermi systems. [ABSTRACT FROM AUTHOR]

Using the mean-field normalized Lugiato–Lefever equation, we theoretically investigate the dynamics of cavity soliton and comb generation in the presence of Raman effect and the third-order dispersion. Both of them can induce the temporal drift and frequency shift. Based on the moment analysis method, we analytically obtain the temporal and frequency shift, and the results agree with the direct numerical simulation. Finally, the compensation and enhancement of the soliton spectral between the Raman-induced self-frequency shift and soliton recoil are predicted. Our results pave the way for further understanding the soliton dynamics and spectral characteristics, and providing an effective route to manipulate frequency comb. [ABSTRACT FROM AUTHOR]

We consider the Sasa–Satsuma (SS) and nonlinear Schrödinger (NLS) equations posed along the line, in 1 + 1 dimensions. Both equations are canonical integrable U(1) models, with solitons, multi-solitons and breather solutions Yang (2010 SIAM Mathematical Modeling and Computation). For these two equations, we recognize four distinct localized breather modes: the Sasa–Satsuma for SS, and for NLS the Satsuma–Yajima, Kuznetsov–Ma and Peregrine breathers. Very little is known about the stability of these solutions, mainly because of their complex structure, which does not fit into the classical soliton behavior Grillakis et al (1987 J. Funct. Anal. 74 160–97). In this paper we find the natural H2 variational characterization for each of them. This seems to be the first known variational characterization for these solutions; in particular, the first one obtained for the famous Peregrine breather. We also prove that Sasa–Satsuma breathers are H2 nonlinearly stable, improving the linear stability property previously proved by Pelinovsky and Yang (2005 Chaos 15 037115). Moreover, in the SS case, we provide an alternative understanding of the SS solution as a breather, and not only as an embedded soliton. The method of proof is based in the use of a H2 based Lyapunov functional, in the spirit of Alejo and Muńoz (2013 Commun. Math. Phys. 324 233–62), extended this time to the vector-valued case. We also provide another rigorous justification of the instability of the remaining three nonlinear modes (Satsuma–Yajima, Peregrine and Kuznetsov–Ma), based in the study of their corresponding linear variational structure (as critical points of a suitable Lyapunov functional), and complementing the instability results recently proved e.g. in Muńoz (2017 Proyecciones (Antofagasta) 36 653–83). [ABSTRACT FROM AUTHOR]