Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity

Li, Pengfei; Malomed, Boris A.; Mihalache, Dumitru

doi:10.1016/j.chaos.2020.109783

Cited by 87 publications

(39 citation statements)

References 102 publications

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“…In the case of the cubic-only self-focusing nonlinearity, LI is limited by values α > 1, as the collapse occurs at α ≤ 1, as mentioned above. However, the inclusion of the quintic term with the defocusing sign suppresses the collapse and makes it possible to consider values α < 1 as well [68], although the solutions tend to be unstable in the latter case, see Fig. 2(b) below.…”

Section: The Modelmentioning

confidence: 99%

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

Zeng

Mihalache

Malomed

et al. 2021

Chaos, Solitons & Fractals

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We construct families of fundamental, dipole, and tripole solitons in the fractional Schrödinger equation (FSE) incorporating self-focusing cubic and defocusing quintic terms modulated by factors cos 2 x and sin 2 x, respectively. While the fundamental solitons are similar to those in the model with the uniform nonlinearity, the multipole complexes exist only in the presence of the nonlinear lattice. The shapes and stability of all the solitons strongly depend on the Lévy index (LI) that determines the FSE fractionality. Stability areas are identified in the plane of LI and propagation constant by means of numerical methods, and some results are explained with the help of an analytical approximation. The stability areas are broadest for the fundamental solitons and narrowest for the tripoles.

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Section: The Modelmentioning

confidence: 99%

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

Zeng

Mihalache

Malomed

et al. 2021

Chaos, Solitons & Fractals

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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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“…Further, realization of the propagation dynamics of light beams governed by this equation was proposed [50], and its extension for the model including the PT symmetry was put forward too [51]. Many types of optical solitons produced by fractional NLSEs have been theoretically investigated [52]- [79], such as "accessible solitons" [54,55], GSs [59][60][61][62][63], solitary vortices [64,65], multipole and multi-peak solitons [66][67][68][69], soliton clusters [70], symmetrybreaking solitons [73][74][75] as well as solitons in couplers [77][78][79].…”

Section: Introductionmentioning

confidence: 99%

Quadratic fractional solitons

Zeng,

Zhu,

Malomed

et al. 2021

Preprint

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We introduce a system combining the quadratic self-attractive or composite quadratic-cubic nonlinearity, acting in the combination with the fractional diffraction, which is characterized by its Lévy index α. The model applies to a gas of quantum particles moving by Lévy flights, with the quadratic term representing the Lee-Huang-Yang correction to the mean-field interactions. A family of fundamental solitons is constructed in a numerical form, while the dependence of its norm on the chemical potential characteristic is obtained in an exact analytical form. The family of quasi-Townes solitons, appearing in the limit case of α = 1/2, is investigated by means of a variational approximation. A nonlinear lattice, represented by spatially periodical modulation of the quadratic term, is briefly addressed too. The consideration of the interplay of competing quadratic (attractive) and cubic (repulsive) terms with a lattice potential reveals families of single-, double-, and triple-peak gap solitons (GSs) in two finite bandgaps. The competing nonlinearity gives rise to alternating regions of stability and instability of the GS, the stability intervals shrinking with the increase of the number of peaks in the GS.

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Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity

Cited by 87 publications

References 102 publications

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

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

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

Quadratic fractional solitons

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