The nature of transverse instabilities to dark solitons and dispersive shock waves for the (2+1)-dimensional defocusing nonlinear Schrodinger equation / Gross-Pitaevskii (NLS / GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev-Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS / GP equation. The dispersion relation for shallow solitons is obtained asymptotically beyond the KP limit. This yields 1) the maximum growth rate and associated wavenumber of unstable perturbations; and 2) the separatrix between convective and absolute instabilities. The latter result is used to study the transition between convective and absolute instabilities of oblique dispersive shock waves (DSWs). Stationary and nonstationary oblique DSWs are constructed analytically and investigated numerically by direct simulations of the NLS / GP equation. The instability properties of oblique DSWs are found to be directly related to those of the dark soliton. It is found that stationary and nonstationary oblique DSWs have the same jump conditions in the shallow and hypersonic regimes. These results have application to controlling nonlinear waves in dispersive media.
We report the experimental observation of multiple dispersive waves emitted in the anomalous dispersion region of an optical fiber from a train of dark solitons. Each individual dispersive wave can be associated to one particular dark soliton of the train, using phase-matching arguments involving higher-order dispersion and soliton velocity. For a large number of dark solitons (>10), we observe the formation of a continuum associated with the efficient emission of dispersive waves.
For waveguides with two separate regions of anomalous dispersion, it is possible to create a quasi-stable two-color solitary wave. In this paper we consider how those waves interact with dispersive radiation, both generation of Cherenkov radiation and scattering of incident dispersive waves. We derive the analytic resonance conditions and verify them through numeric experiments. We also report incident radiation driving the internal oscillations of the soliton during the scattering process in case of an intense incident radiation. We generalize the resonance conditions for the case of an oscil…
We study resonant radiation generated by bound solitons in a twin-core fiber near zero-dispersion wavelength, in the presence of higher order dispersion terms. We propose a theoretical description of dispersive wave generation mechanism and derive resonance conditions. The presence of third order dispersion term leads to generation of polychromatic dispersive radiation and transition from the regime of center of mass oscillations to the regime of amplitude oscillations. Such a transition is not reproduced in the case of symmetric fourth order dispersion.
Ubiquitous nonlinear waves in dispersive media include localized solitons and extended hydrodynamic states such as dispersive shock waves. Despite their physical prominence and the development of thorough theoretical and experimental investigations of each separately, experiments and a unified theory of solitons and dispersive hydrodynamics are lacking. Here, a general soliton-mean field theory is introduced and used to describe the propagation of solitons in macroscopic hydrodynamic flows. Two universal adiabatic invariants of motion are identified that predict trapping or transmission of sol…
We present the results of asymptotic and numerical analysis of dissipative Kerr solitons in whispering gallery mode microresonators influenced by higher order dispersive terms leading to the appearance of a dispersive wave (Cherenkov radiation). Combining direct perturbation method with the method of moments we find expressions for the frequency, strength, spectral width of the dispersive wave and soliton velocity. Mutual influence of the soliton and dispersive wave was studied. The formation of the dispersive wave leads to a shift of the soliton spectrum maximum from the pump frequency (spect…
The problem of stability and spectrum of linear excitations of a soliton (kink) of the dispersive sine-Gordon and $φ^4$ - equations is solved exactly. It is shown that the total spectrum consists of a discrete set of frequencies of internal modes and a single band spectrum of continuum waves. It is indicated by numerical simulations that a translation motion of a single soliton in the highly dispersive systems is accompanied by the arising of its internal dynamics and, in some cases, creation of breathers, and always by generation of the backward radiation. It is shown numerically that a fast …
We present a numerical investigation of the propagation dynamics of a truncated Airy pulse in a highly dispersive and nonlinear medium by employing the split-step Fourier transform method and look, in particular, into the effects of fourth order dispersion (FOD) and cubic-quintic-septic nonlinearity on pulse evolution. Presence of FOD cancels the Airy pulse self-acceleration along with eclipsing the oscillatory tail during propagation in the linear regime. Further, we observe soliton shedding at low input pulse power in the presence of cubic and quintic nonlinearity and negative FOD. The emerg…
The paper is devoted to numerical study of stability of nonlinear localized modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii equation (1D GPE) with periodic potential and repulsive interparticle interactions. We use the Evans function approach combined with the exterior algebra formulation in order to detect and describe weak oscillatory instabilities. We show that the simplest ("fundamental") gap solitons in the first and in the second spectral gaps can undergo oscillatory instabilities for certain values of the frequency parameter (i.e., the chemical potential). The …
The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow that interacts with edge wave packets or edge solitons. A conjecture about existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In case of localized simple wave pulses propagating through a quiescent medium this theory provided a new approach to derivation of an asymptotic formula for the number of…
Nonlinear interactions in focusing media between traveling solitons and the dispersive shocks produced by an initial discontinuity are studied using the one-dimensional nonlinear Schrodinger equation. It is shown that, when solitons travel from a region with nonzero background towards a region with zero background, they always pass through the shock structure without generating dispersive radiation. However, their properties (such as amplitude, velocity and shape) change in the process. A similar effect arises when solitons travel from a region with zero background towards a region with nonzer…
Dispersive shock waves and solitons are fundamental nonlinear excitations in dispersive media, but dispersive shock wave studies to date have been severely constrained. Here we report on a novel dispersive hydrodynamics testbed: the effectively frictionless dynamics of interfacial waves between two high contrast, miscible, low Reynolds' number Stokes fluids. This scenario is realized by injecting from below a lighter, viscous fluid into a column filled with high viscosity fluid. The injected fluid forms a deformable pipe whose diameter is proportional to the injection rate, enabling precise co…
The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a moving kink emitting radiation and breathers are studied analytically. Conditions of the breather excitation and its dynamical properties are specified. Processes of soliton complex formation are studied analytically and numerically in relation to the strength of the dispersion, soliton velocity, and distance between solitons. It is shown that moving bound soliton complexes with internal structu…
The possibility of tailoring the guidance properties of optical fibers along the same direction as the evolution of the optical field allows to explore new directions in nonlinear fiber optics. The new degree of freedom offered by axially-varying optical fibers enables to revisit well-established nonlinear phenomena, and even to discover novel short pulse nonlinear dynamics. Here we study the impact of meter-scale longitudinal variations of group velocity dispersion on the propagation of bright solitons and on their associated dispersive waves. We show that the longitudinal tailoring of fiber …
We show that the number of solitons produced from an arbitrary initial pulse of the simple wave type can be calculated analytically if its evolution is governed by a generalized nonlinear Schrödinger equation provided this number is large enough. The final result generalizes the asymptotic formula derived for completely integrable nonlinear wave equations like the standard NLS equation with the use of the inverse scattering transform method.
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…
Instabilities of vortex-ring-bright coherent structures in harmonically trapped two-component three-dimensional Bose-Einstein condensates are studied numerically within the coupled Gross-Pitaevskii equations and interpreted analytically. Interestingly, the filled vortex core with a sufficiently large amount of the bright component is observed to reduce the parametric interval of stability of the vortex ring. We have identified the mechanisms of several linear instabilities and one nonlinear parametric instability in this connection. Two of the linear instabilities are qualitatively different f…
We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The passed wave either retains the form of a soliton (though of different parameters) or becomes a bi-soliton. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapour saturation/condensation area. Some rough e…
The Galilean transformation is a universal operation connecting the coordinates of a dynamical system, which move relative to each other with a constant speed. In the context of exact solutions of the universal nonlinear Schrödinger equation (NLSE), inducing a Galilean velocity (GV) to the pulse involves a frequency shift to satisfy the symmetry of the wave equation. As such, the Galilean transformation has been deemed to be not applicable to wave groups in nonlinear dispersive media. In this paper, we demonstrate that in a wave tank generated Galilean transformed envelope and Peregrine solito…
We consider the ultrashort light pulse propagation through an inhomogeneous monomodal optical fiber exhibiting higher-order dispersive effects. Wave propagation is governed by a generalized nonlinear Schrödinger equation with varying second-, third-, and fourth-order dispersions, cubic nonlinearity, and linear gain or loss. We construct a new type of exact self-similar soliton solutions that takes the structure of dipole via a similarity transformation connected to the related constant-coefficients one. The conditions on the optical fiber parameters for the existence of these self-similar stru…
Institute for Complex Systems, National Research Council (ISC-CNR)（国家研究委员会复杂系统研究所）；Department of Physics, University Sapienza（罗马大学物理系）
We study artificial neural networks with nonlinear waves as a computing reservoir. We discuss universality and the conditions to learn a dataset in terms of output channels and nonlinearity. A feed-forward three-layer model, with an encoding input layer, a wave layer, and a decoding readout, behaves as a conventional neural network in approximating mathematical functions, real-world datasets, and universal Boolean gates. The rank of the transmission matrix has a fundamental role in assessing the learning abilities of the wave. For a given set of training points, a threshold nonlinearity for un…
The peculiar crystal structure of one-dimensional molecular solids originates from packing of an array of molecules in which intermolecular interactions are dominantly dispersive, including hydrogen-bond, van der Waals and London-type forces. These forces are usually relatively weaker than covalent and ionic bondings such that long-range intermolecular interactions should play important role in dispersion properties of molecular crystals such as polymers and biomolecular chain structures. In this work the effects of long but finite-range intermolecular interactions on single-exciton dispersion…
We investigate propagating dark soliton solutions of the two-dimensional defocusing nonlinear Schrödinger / Gross-Pitaevskii (NLS/GP) equation that are transversely confined to propagate in an infinitely long channel. Families of single, vortex, and multi-lobed solitons are computed using a spectrally-accurate numerical scheme. The multi-lobed solitons are unstable to small transverse perturbations. However, the single-lobed solitons are stable if they are sufficiently confined along the transverse direction, which explains their effective one-dimensional dynamics. The emergence of a transvers…
A conceptually new notion of hydrodynamic optical soliton tunneling is introduced in which a dark soliton is incident upon an evolving, broad potential barrier that arises from an appropriate variation of the input signal. The barriers considered include smooth rarefaction waves and highly oscillatory dispersive shock waves. Both the soliton and the barrier satisfy the same one-dimensional defocusing nonlinear Schrödinger (NLS) equation, which admits a convenient dispersive hydrodynamic interpretation. Under the scale separation assumption of Whitham modulation theory the highly nontrivial non…
We present analytic threshold formulae applicable to both dispersive (time-domain) and diffractive (pattern-forming) instabilities in Fabry-Perot Kerr cavities of arbitrary finesse. We do so by extending the gain-circle technique, recently developed for counter-propagating fields in single-mirror-feedback systems, to allow for an input mirror. In time-domain counter-propagating systems walk-off effects are known to suppress cross-phase modulation contributions to dispersive instabilities. Applying the gain-circle approach with appropriately-adjusted cross-phase couplings extends previous resul…
In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann-Hilbert (RH) problem, and fractional multi-solitons of the integrable combined fractional higher-order mKdV (fhmKdV) hierarchy, including the fractional mKdV (fmKdV), fractional fifth-order mKdV (f5mKdV), fractional combined third-fifth-order mKdV (f35mKdV) equations, etc., which can be featured via completeness of squared scalar eigenfunctions of the ZS spectral problem. We construct a matrix RH problem to present three types of fractional N-solitons illustrating anomalous dispersions…
The scattering of weak dispersive waves (DW) on several equally spaced temporal solitons is studied. It is shown by systematic numerical simulations that the reflection of the DWs from the soliton trains strongly depends on the distance between the solitons. The dependence of the reflection and transmission coefficients on the inter-soliton distance and the frequency of the incident waves is studied in detail, revealing fascinating quasi-periodic behavior. The analogy between the observed nonlinear phenomena in temporal domain and usual Fabry-Perot and Bragg resonators is discussed.
