Stabilization of single- and multi-peak solitons in the fractional nonlinear Schrödinger equation with a trapping potential

Qiu, Yunli; Malomed, Boris A.; Mihalache, Dumitru; Zhu, Xing; Peng, Xi; He, Yan

doi:10.1016/j.chaos.2020.110222

Cited by 68 publications

(27 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, it is relevant to mention that the inclusion of the external potential, such as the periodic one given by Eq. ( 3), helps to suppress the collapse in the fractional NLSE, and thus to extend the range of the existence of solitons [67]. In the present case, this possibility can be illustrated by results following from Eqs.…”

Section: B Scaling Relations For Soliton Familiesmentioning

confidence: 52%

“…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%

See 1 more Smart Citation

Quadratic fractional solitons

Zeng,

Zhu,

Malomed

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: B Scaling Relations For Soliton Familiesmentioning

confidence: 52%

Section: Introductionmentioning

confidence: 99%

Quadratic fractional solitons

Zeng,

Zhu,

Malomed

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Second, while the smaller period of the initial corrugation can only delay, but not arrest, collapse, a recent study [34] has shown that a trapping potential can arrest it. This is consistent with the fact that collapse is caused by the interaction of the field over a sufficiently large range, and a suitable trapping potential is able to essentially truncate this range by localizing the field.…”

Section: Effects Of Nonzero Group Velocity (7) Of the Plane Wavementioning

confidence: 98%

Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion

2021

View full text Add to dashboard Cite

We analytically and numerically investigate the stability and dynamics of the plane wave solutions of the fractional nonlinear Schrödinger (NLS) equation, where the long-range dispersion is described by the fractional Laplacian (−Δ)α/2. The linear stability analysis shows that plane wave solutions in the defocusing NLS are always stable if the power α∈[1,2] but unstable for α∈(0,1). In the focusing case, they can be linearly unstable for any α∈(0,2]. We then apply the split-step Fourier spectral (SSFS) method to simulate the nonlinear stage of the plane waves dynamics. In agreement with earlier studies of solitary wave solutions of the fractional focusing NLS, we find that as α∈(1,2] decreases, the solution evolves towards an increasingly localized pulse existing on the background of a “sea” of small-amplitude dispersive waves. Such a highly localized pulse has a broad spectrum, most of whose modes are excited in the nonlinear stage of the pulse evolution and are not predicted by the linear stability analysis. For α≤1, we always find the solution to undergo collapse. We also show, for the first time to our knowledge, that for initial conditions with nonzero group velocities (traveling plane waves), an onset of collapse is delayed compared to that for a standing plane wave initial condition. For defocusing fractional NLS, even though we find traveling plane waves to be linearly unstable for α<1, we have never observed collapse. As a by-product of our numerical studies, we derive a stability condition on the time step of the SSFS to guarantee that this method is free from numerical instabilities.

show abstract

“…The experimental implementation of the FSE in condensed-matter [56,57] and optical [58] setups, where nonlinearity is a natural feature, has drawn interest to the possibility of existence of solitons in fractional dimensions [59][60][61][62]. In particular, "accessible solitons" [63,64] and self-trapped states of vectorial [65], gap [66], nonlocal [67], vortical [68], and multi-peak types [69] have been predicted in FSE models, as well as soliton clusters [70,71], symmetry breaking of solitons [72,73], coupled solitons [74] and dissipative solitons in a fractional complex-Ginzburg-Landau model [75]. In the case of the ubiquitous cubic (Kerr) self-focusing, the solitons are unstable at α ≤ 1, as the combination of such values with the Kerr nonlinearity gives rise to the collapse.…”

Section: Introductionmentioning

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Stabilization of single- and multi-peak solitons in the fractional nonlinear Schrödinger equation with a trapping potential

Cited by 68 publications

References 34 publications

Quadratic fractional solitons

Quadratic fractional solitons

Dynamics of Plane Waves in the Fractional Nonlinear Schrödinger Equation with Long-Range Dispersion

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

Contact Info

Product

Resources

About