We demonstrate that the commonly known concept, which treats solitons as nonsingular solutions produced by the interplay of nonlinear self-attraction and linear dispersion, may be extended to include modes with a relatively weak singularity at the central point, which keeps their integral norm convergent. Such states are generated by self-repulsion, which should be strong enough, namely, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schroedinger equations (NLSEs), respectively. Although such solutions seem counterintuitive, we demonstrate that they admit a straightforward interpretation as a result of screening of an additionally introduced attractive delta-functional potential by the defocusing nonlinearity. The strength ("bare charge") of the attractive potential is infinite in 1D, finite in 2D, and vanishingly small in 3D. Analytical asymptotics of the singular solitons at small and large distances are found, entire shapes of the solitons being produced in a numerical form. Complete stability of the singular modes is accurately predicted by the anti-Vakhitov-Kolokolov criterion (under the assumption that it applies to the model), as verified by means of numerical methods. In 2D, the NLSE with a quintic self-focusing term admits singular soliton solutions with intrinsic vorticity too, but they are fully unstable. We also mention that dissipative singular solitons can be produced by the model with a complex coefficient in front of the nonlinear term.
Vortex solitons (dark solitons) are described in self-defocusing Kerr media whose phase line singularity is not parallel to the propagation direction, but is perpendicular or tilted almost arbitrary angles, depending on the medium linear dispersion. These solitons broaden the so-called spatiotemporal optical vortices recently observed in experiments to the nonlinear realm, opening new perspectives in practical applications of optical vortices.
We introduce a setting based on the one-dimensional (1D) nonlinear Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated by a singular function of the coordinate, |x|^{-a}. It may be additionally combined with the uniform self-defocusing (SDF) nonlinear background, and with a similar singular repulsive linear potential. The setting, which can be implemented in optics and BEC, aims to extend the general analysis of the existence and stability of solitons in NLSEs. Results for fundamental solitons are obtained analytically and verified numerically. The solitons feature a …
A model of the optical media with a spatially structured Kerr nonlinearity is introduced. The nonlinearity strength is modulated by a set of singular peaks on top of a self-focusing or defocusing uniform background. The peaks may include a repulsive or attractive linear potential too. We find that a pair of mutually symmetric peaks readily gives rise to the spontaneous symmetry breaking (SSB) of modes pinned to individual peaks, while antisymmetric pinned modes are always unstable, transforming into robust spatially odd breathers. Three- and five-peak structures support symmetric modes, with i…
We introduce a model of one- and two-dimensional (1D and 2D) optical media with the $χ^{(2)}$ nonlinearity whose local strength is subject to cusp-shaped spatial modulation, $χ^{(2)}\sim r^{-α}$, with $α>0$, which can be induced by spatially nonuniform poling. Using analytical and numerical methods, we demonstrate that this setting supports 1D and 2D fundamental solitons, at $α<1$ and $α<2$, respectively. The 1D solitons have a small instability region, while the 2D solitons have a stability region at $α<0.5$ and are unstable at $α>0.5$. 2D solitary vortices are found too. They are unstable, s…
We show the existence of stable two- and three-dimensional vortex solitons carrying multiple, spatially separated, single-charge topological dislocations nested around a vortex-ring core. Such new nonlinear states are supported by elliptical gain landscapes in focusing nonlinear media with two-photon absorption. The separation between the phase dislocations is dictated mostly by the geometry of gain landscape and it only slightly changes upon variation of the gain or absorption strength.
It was recently found that the electric local-field effect (LFE) can lead to strong coupling of atomic Bose-Einstein condensates (BECs) to off-resonant optical fields. We demonstrate that the magnetic LFE gives rise to a previously unexplored mechanism for coupling a (pseudo) spinor BEC or fermion gas to microwaves (MWs). We present a theory for the magnetic LFE, and find that it gives rise to a short-range attractive interaction between two components of the (pseudo) spinor, and a long-range interaction between them. The latter interaction, resulting from deformation of the magnetic field, is…
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.
We derive an exact solution to the local nonlinear Schrödinger equation (NLSE) using the Darboux transformation method. The new solution describes the profile and dynamics of a two-soliton molecule. Using an algebraically-decaying seed solution, we obtain a two-soliton solution with diverging peaks, which we denote as singular soliton molecule. We find that the new solution has a finite binding energy. We calculate the force and potential of interaction between the two solitons, which turn out to be of molecular-type. The robustness of the bond between the two solitons is also verified. Furthe…
We study collisions of moving nonlinear-Schrödinger solitons with a $\mathcal{PT}$-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solito…
On-chip manipulation of single resonance over broad background comb spectra of microring resonators is indispensable, ranging from tailoring laser emission, optical signal processing to non-classical light generation, yet challenging without scarifying the quality factor or inducing additional dispersive effects. Here, we propose an experimentally feasible platform to realize on-chip selective depletion of single resonance in microring with decoupled dispersion and dissipation, which are usually entangled by Kramer-Kroning relation. Thanks to the existence of non-Hermitian singularity, unsplit…
We show that ring-like localized gain landscapes imprinted in focusing cubic (Kerr) nonlinear media with strong two-photon absorption support new types of stable higher-order vortex solitons containing multiple phase singularities nested inside a single core. The phase singularities are found to rotate around the center of the gain landscape, with the rotation period being determined by the strength of the gain and the nonlinear absorption.
We consider soliton formation in thermal nonlinear media bounded by rectangular cross-sections and uncover a new class of nonlinear stationary topological state. Specifically, we find that stationary higher-order vortex states in standard shapes do not exist, but rather they take the form of multiple, spatially separated single-charge singularities nested in an elliptical beam. Double-charge states are found to be remarkably robust despite their shape asymmetry and the phase-singularity splitting. States with higher topological charges are found to be unstable.
The stabilization of one-dimensional solitons by a nonlinear lattice against the critical collapse in the focusing quintic medium is a challenging issue. We demonstrate that this purpose can be achieved by combining a nonlinearlatticeandsaturationofthequinticnonlinearity. Thesystemsupportsthreespeciesofsolitons, namely, fundamental (even-parity) ones and dipole (odd-parity) modes of on- and off-site-centered types. Very narrow fundamental solitons are found in an approximate analytical form, and systematic results for very broad unstable and moderately broad partly stable solitons, including t…
The interaction of matter-wave solitons with a potential barrier is a fundamentally important problem, and the splitting and subsequent recombination of the soliton by the barrier is the essence of soliton matter-wave interferometry. We demonstrate the three-dimensional (3D) character of the interactions in the case relevant to ongoing experiments, where the number of atoms in the soliton is sufficiently large that the soliton is close to collapse. The mean-field description is quite accurate, but the proximity to the collapse threshold makes the use of the 1D Gross-Pitaevskii equation (GPE) i…
Asymptotic soliton-like and asymptotic peakon-like solutions of the modified Camassa-Holm equation with variable coefficients and singular perturbation
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…
