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 motion of two topological solitons leads to a formation of the bound soliton complex in the dispersive sine-Gordon system.
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 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…
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 effect of mutual interaction between dark solitons and dispersive waves is investigated numerically and analytically. The condition of the resonant scattering of dispersive waves on dark solitons is derived and compared against the results of numerical simulations. It is shown that the interaction with intense dispersive waves affects the dynamics of the soltons strongly changing their frequencies and accelerating or decelerating the solitons. It is also demonstrated that two dark solitons can form a cavity for dispersive weaves bouncing between the two dark solitons. The differences of th…
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 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.
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 have presented theoretical and numerical studies on interactions of dispersive waves with second order solitons. Disperive wave with considerable intensity resonantly colliding with the second order solitons can lead to acceleration/decceleration of the later with subsequent central frequency shifts, but still well preserve the oscillating structure of the 2-solitons. The 2-soliton creates an effective periodical refractive index profile and thus reflected and transmitted dispersive waves generate new spectral bands with sharp peak structures. When resonant dispersive waves interact with tw…
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…
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.
We introduce a mechanism to stabilize spatiotemporal solitons in Kerr nonlinear media, based on the dispersion of linear coupling between the field components forming the soliton states. Specifically, we consider solitons in a two-core guiding structure with inter-core coupling dispersion (CD). We show that CD profoundly affects properties of the solitons, causing the complete stabilization of the otherwise highly unstable spatiotemporal solitons in Kerr media with focusing nonlinearity. We also find that the presence of CD stimulates the formation of bound states, which however are unstable.
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…
We report on new types of two-component one-dimensional dark solitons (DSs) in a model of a dual-core waveguide with normal group-velocity dispersion and Kerr nonlinearity in both cores, the coupling between which is dispersive too. In the presence of the dispersive coupling, quiescent DSs supported by the zero-frequency background are always gray, being stable with the out-of-phase background, i.e., for opposite signs of the fields in the cores. On the contrary, the background with a nonzero frequency supports quiescent black solitons which may be stable for both out- and in-phase backgrounds…
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 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…
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…
In dispersive media, hydrodynamic singularities are resolved by coherent wavetrains known as dispersive shock waves (DSWs). Only dynamically expanding, temporal DSWs are possible in one-dimensional media. The additional degree of freedom inherent in two-dimensional media allows for the generation of time-independent DSWs that exhibit spatial expansion. Spatial oblique DSWs, dispersive analogs of oblique shocks in classical media, are constructed utilizing Whitham modulation theory for a class of nonlinear Schrödinger boundary value problems. Self-similar, simple wave solutions of the modulatio…
The dynamical behavior resulting from an initial discontinuity in focusing media is investigated using a combination of numerical simulations and Whitham modulation theory for the focusing nonlinear Schrodinger equation. Initial conditions with a jump in either or both the amplitude and the local wavenumber are considered. It is shown analytically and numerically that the space-time plane divides into expanding domains in which the solution is described by a slow modulation of genus-zero, genus-one or genus-two solutions, their precise arrangement depending on the specifics of the initial datu…
The dynamics of light beams in the nonlinear optical media with periodically modulated in the longitudinal direction parity-time distribution of the complex refractive index is investigated. The possibility of dynamical stabilization of PT-symmetric solitons is demonstrated.
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…
The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to nonconvex flux is studied in the framework of the modified Korteweg-de Vries (mKdV) equation, a canonical model for nonlinear internal gravity wave propagation in stratified fluids. The principal feature of the studied interaction is that both the solitary wave and the large-scale mean flow -- a rarefaction wave or a dispersive shock wave (undular bore) -- are described by the same dispersive hydrodynamic equation. A recent theoretical and experimental study of this new type of dynamic s…
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…
New resonant emission of dispersive waves by oscillating solitary structures is considered analytically and numerically. The resonance condition for the radiation is derived and it is demonstrated that the predicted resonances match the spectral lines observed in numerical simulations perfectly. The complex recoil of the radiation on the soliton dynamics is discussed.
The dynamics of coherent nonlinear wave groups is shown to be drastically different from the classical scenario of weakly nonlinear wave interactions. The coherent groups generate non-resonant (bound) waves which can be synchronized with other linear waves. By virtue of the revealed mechanism the groups may emit waves with similar or different lengths, which propagate in the same or opposite direction.
