Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension

Zeng, Liangwei; Mihalache, Dumitru; Malomed, Boris A.; Lu, Xiaowei; Cai, Yi; Zhu, Qifan; Li, Jingzhen

doi:10.1016/j.chaos.2020.110589

Cited by 66 publications

(20 citation statements)

References 76 publications

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“…It was checked that the numerical mesh with ∆x = ∆r = 0.1 and ∆z = 0.002 was sufficient for producing fully reliable results. It is relevant to mention that numerical methods for producing stationary and dynamical solutions to NLS/GP equations with a spatially varying nonlinearity coefficients were developed and used in many previous works [1,28,39,40,41,42], [48]- [62].…”

Section: Resultsmentioning

confidence: 99%

“…Nonlinear lattices [1] can also support many types of soliton families, including those of fundamental, dipole, and multipole types in regular [34,35,36], random [37] and defective [38] lattices, as well as in combined linear-nonlinear ones [39], and in nonlinear lattices embedded in a space of fractional dimension [40,41]. Vortex solitons in nonlinear lattices were predicted too [42].…”

Section: Introductionmentioning

confidence: 99%

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Flat-floor Bubbles, Dark Solitons, and Vortices Stabilized by Inhomogeneous Nonlinear Media

Zeng

Malomed

Mihalache

et al. 2021

Preprint

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We consider one- and two-dimensional (1D and 2D) optical or matter-wave media with a maximum of the local self-repulsion strength at the center, and a minimum at periphery. If the central area is broad enough, it supports ground states in the form of flat-floor “bubbles”, and topological excitations, in the form of dark solitons in 1D and vortices with winding number m in 2D. The ground and excited states are accurately approximated by the Thomas-Fermi expressions. The 1D and 2D bubbles, as well as vortices with m=1, are completely stable, while the dark solitons and vortices with m=2 have nontrivial stability boundaries in their existence areas. Unstable dark solitons are expelled to the periphery, while unstable double vortices split in rotating pairs of unitary ones. Displaced stable vortices precess around the central point.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Flat-floor Bubbles, Dark Solitons, and Vortices Stabilized by Inhomogeneous Nonlinear Media

Zeng

Malomed

Mihalache

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Nonlinear lattices [1] can also support many types of soliton families, including those of fundamental, dipole, and multipole types in regular [35][36][37], random [38] and defectcarrying [39] lattices, as well as in combined linear-nonlinear ones [40], and in nonlinear lattices embedded in a space of fractional dimension [41,42]. Vortex solitons in nonlinear lattices were predicted too [43].…”

Section: Introductionmentioning

confidence: 98%

Flat-floor bubbles, dark solitons, and vortices stabilized by inhomogeneous nonlinear media

Zeng,

Malomed,

Mihalache

et al. 2021

Preprint

Self Cite

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We consider one-and two-dimensional (1D and 2D) optical or matter-wave media with a maximum of the local self-repulsion strength at the center, and a minimum at periphery. If the central area is broad enough, it supports ground states in the form of flat-floor "bubbles", and topological excitations, in the form of dark solitons in 1D and vortices with winding number m in 2D. Unlike bright solitons, delocalized bubbles and dark modes were not previously considered in this setting. The ground and excited states are accurately approximated by the Thomas-Fermi expressions. The 1D and 2D bubbles, as well as vortices with m = 1, are completely stable, while the dark solitons and vortices with m = 2 have nontrivial stability boundaries in their existence areas. Unstable dark solitons are expelled to the periphery, while unstable double vortices split in rotating pairs of unitary ones. Displaced stable vortices precess around the central point.

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“…When nonlinear terms are introduced into the fractional Schrödinger equation, we are dealing with fractional NLSE. Such equations (both in 1D and higher dimensions) had been widely investigated theoretically both in 1D and higher dimensions, see [22,23] and references therein. The purpose of the present paper is to analyze analytically (using perturbation approach near α = 2 and variationally) and numerically the stabilization of the soliton texture in the simplest possible (without cubic terms, external potential, etc) model [9] by the introduction of the fractional derivatives in the corresponding NLSE.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of 1D Solitons By Fractional Derivatives In Systems With Quintic Nonlinearity

Stephanovich¹,

Olchawa²

2021

Preprint

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We study theoretically the properties of a soliton solution of the fractional Schrödinger equation with quintic nonlinearity. Under ”fractional” we understand the Schrödinger equation, where ordinary Laplacian (second spatial derivative in 1D) is substituted by its fractional counterpart with Lévy index α. We speculate that the latter substitution corresponds to phenomenological account for disorder in a system. Using analytical (variational and perturbative) and numerical arguments, we have shown that while in the case of Schrödinger equation with the ordinary Lapla-cian (corresponding to Lévy index α = 2), the soliton is unstable, even infinitesimal difference α from 2 immediately stabilizes the soliton texture. Our analytical and numerical investigations of ω(N) dependence (ω is soliton frequency and N its mass) show (within the famous Vakhitov-Kolokolov criterion) the stability of our soliton texture in the fractional α < 2 case. Direct numerical analysis of the linear stability problem of soliton texture also confirms this point. We show analytically and numerically that fractional Schrödinger equation with quintic nonlinearity admits the existence of (stable) soliton textures at 2/3 < α < 2, which is in accord with existing literature data. These results may be relevant to both Bose-Einstein condensates in cold atomic gases and optical solitons in the disordered media.

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Families of fundamental and multipole solitons in a cubic-quintic nonlinear lattice in fractional dimension

Cited by 66 publications

References 76 publications

Flat-floor Bubbles, Dark Solitons, and Vortices Stabilized by Inhomogeneous Nonlinear Media

Flat-floor Bubbles, Dark Solitons, and Vortices Stabilized by Inhomogeneous Nonlinear Media

Flat-floor bubbles, dark solitons, and vortices stabilized by inhomogeneous nonlinear media

Stabilization of 1D Solitons By Fractional Derivatives In Systems With Quintic Nonlinearity

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