(INVITED) Vortex solitons: Old results and new perspectives

Malomed, Boris A.

doi:10.1016/j.physd.2019.04.009

Cited by 168 publications

(129 citation statements)

References 307 publications

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“…At x → 0, the universal (k-independent) asymptotic form of the singular soliton is U (x) ≈ (2/9) 1/6 |x| −1/3 (7) (as mentioned above, this asymptotic form was first found in Ref. [30]), and in the limit of |x| → ∞ its exponentially decaying tail is…”

Section: The Singular Soliton Created By the Septimal Nonlinearitymentioning

confidence: 61%

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Singular solitons

Sakaguchi

Malomed

2020

Phys. Rev. E

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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 Schrödinger 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 singularsoliton 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.

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Section: The Singular Soliton Created By the Septimal Nonlinearitymentioning

confidence: 61%

“…with constant U 0 . An approximate global form of the soliton can be produced by matching the asymptotic expressions (7) and (8):…”

Section: The Singular Soliton Created By the Septimal Nonlinearitymentioning

confidence: 99%

Singular solitons

Sakaguchi

Malomed

2020

Phys. Rev. E

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“…[4]. The vanishing saturating nonlinearity in this case does not sustain solitonic solutions with sufficiently large am- * Electronic address: vlashkin62@gmail.com plitudes in contrast to the ordinary saturating nonlinearity [1]. Under this, 2D vortex solitons with amplitudes (or, equivalently, nonlinear frequency shifts) below and above some critical value turn out to be stable while the 3D ones split into filaments due to the azimuthal instability.…”

Section: Introductionmentioning

confidence: 91%

Stable three-dimensional Langmuir vortex soliton

Lashkin

2020

Physics of Plasmas

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We present a numerical solution in the form of a three-dimensional (3D) vortex soliton in unmagnetized plasma in the model of the generalized Zakharov equations with saturating exponential nonlinearity. To find the solution with a high accuracy we use two-step numerical method combining the Petviashvili iteration procedure and the Newton-Kantorovich method. The vortex soliton with the topological charge m = 1 turns out to be stable provided the nonlinear frequency shift exceeds a certain critical value. The stability predictions are verified by direct simulations of the full dynamical equation.

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“…In addition to 2D fundamental discrete solitons which may be naturally expected in the unstaggered-staggered system, one may also look for compound modes in which one or both components are represented by discrete vortex solitons [31], which is the 2D analogue of the 1D twisted solitons. Vortex solitons have previously been found in various optical setups [32,33,34], see also recent reviews [35,36]. Experimental creation of discrete vortex solitons in self-focusing optically induced lattices was reported in Refs.…”

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confidence: 89%

“…The most sophisticated compound mode is built of vortical modes in both the unstaggered (u) and staggered (v) components, as per ansatz (36). Substituting the ansatz in Lagrangian (30), we obtain the respective effective Lagrangian,…”

Section: Vortical Unstaggered -Vortical Staggered Pairsmentioning

confidence: 99%

Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices

Gorder

Krause

Malomed

et al. 2020

Communications in Nonlinear Science and Numerical Simulation

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We study coupled unstaggered-staggered soliton pairs emergent from a system of two coupled discrete nonlinear Schrödinger (DNLS) equations with the self-attractive on-site self-phasemodulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction, on 1D and 2D lattice domains. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. While most work on DNLS systems addressed symmetric on-site-centered fundamental solitons, these models give rise to a variety of other excited states, which may also be stable. The simplest among them are antisymmetric states in the form of discrete twisted solitons, which have no counterparts in the continuum limit. In the extension to 2D lattice domains, a natural counterpart of the twisted states are vortical solitons. We first introduce a variational approximation (VA) for the solitons, and then correct it numerically to construct exact stationary solutions, which are then used as initial conditions for simulations to check if the stationary states persist under time evolution. Twocomponent solutions obtained include (i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton pairs. We also highlight a variety of other transient dynamical regimes, such as breathers and amplitude death. The findings apply to modeling binary Bose-Einstein condensates, loaded in a deep lattice potential, with identical or different atomic masses of the two components, and arrays of bimodal optical waveguides. keywords: discrete nonlinear Schrödinger equations; unstaggered-staggered lattice; variational 1 arXiv:2003.00591v1 [nlin.PS] 1 Mar 2020 approximation; solitons 1 IntroductionDiscrete nonlinear Schrödinger (DNLS) equations provide models for a great variety of physical systems [1]. A well-known implementation of the basic DNLS equation is provided by arrays of transversely coupled optical waveguides, as predicted in [2] and realized experimentally, in various optical settings [3,4,5,6]. A comprehensive review of nonlinear optics in discrete settings was given by Ref. [7]. Another realization of the DNLS equation in provided by Bose-Einstein condensates (BECs) loaded into deep optical-lattice potentials, which split the condensate into a chain of droplets trapped in local potential wells, which are tunnel-coupled across the potential barriers between them [8,9]. In the tight-binding approximation, this setting is also described by the DNLS version of the Gross-Pitaevskii (GP) equation [10,11,12,13,14].One-dimensional (1D) DNLS equations with self-attractive and self-repulsive on-site nonlinearity generate localized modes of unstaggered and staggered types, respectively. In the latter case, the on-site amplitudes alternate between adjacent sites of the lattice [1]. In the continuum limit, the unstaggered discrete solitons carry over into regular ones, while the staggered solitons correspond to gap solitons, which are supported...

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(INVITED) Vortex solitons: Old results and new perspectives

Cited by 168 publications

References 307 publications

Singular solitons

Singular solitons

Stable three-dimensional Langmuir vortex soliton

Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices

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