Soliton dynamics in a fractional complex Ginzburg-Landau model

Qiu, Yunli; Malomed, Boris A.; Mihalache, Dumitru; Zhu, Xing; Zhang, Li; He, Yan

doi:10.1016/j.chaos.2019.109471

Cited by 89 publications

(39 citation statements)

References 49 publications

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“…( 22) is tantamount to a result recently reported for the cubic nonlinearity with α = 1 in Ref. [72]]. These results may be likened to the well-known VA prediction for the norm of the Townes' solitons in the 2D NLSE with the cubic selffocusing [97]:…”

Section: B Scaling Relations For Soliton Familiessupporting

confidence: 74%

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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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Section: B Scaling Relations For Soliton Familiessupporting

confidence: 74%

“…In the context of a different problem, it was recently demonstrated that VA can be used to look for localized solutions of the fractal NLSE [72,95]. To this end, we note that Eq.…”

Section: B Scaling Relations For Soliton Familiesmentioning

confidence: 95%

Quadratic fractional solitons

Zeng,

Zhu,

Malomed

et al. 2021

Preprint

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“…In particular, the 1D Schrödinger equation with the cubic term subject to singular spatial modulation may emulate fractional dimension 0 < D < 1 [81]. Recent works have demonstrated that the fractional NLSE supports a variety of fractional spatial-soliton solutions [82]: "accessible solitons" [83]- [85], double-hump and fundamental solitons in PT -symmetric potentials [86,87], bulk and surface gap solitons in PT -symmetric photonic lattices [88]- [90], vortex solitons in PT -symmetric azimuthal potentials [91], 2D self-trapped modes [92], spontaneous symmetry breaking in a dual-core system [93], and dissipative solitons in the fractional complex-Ginzburg-Landau equation [94]. Further, the composition relation between nonlinear Bloch waves and gap solitons supported by lattices [95,96], solitons under the action of nonlinearity subject to spatially-periodic modulation [97], dissipative surface solitons [98], as well as discrete solitons [99], have also been addressed.…”

Section: Introductionmentioning

confidence: 96%

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

Malomed

Mihalache

2020

Chaos, Solitons & Fractals

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“…(12) can be solved directly by dint of conjugate gradient iterations [81,82], which yields symmetric, antisymmetric, and asymmetric soliton solutions. In fact, even in the case of α < 1, when solitons, supported by the self-focusing cubic nonlinearity in the free space, are unstable, because of the occurrence of the supercritical collapse [74], the trapping potential may stabilize the solitons and uphold the familiar SSB scenario, as shown in Fig. 3(a) for α = 0.8.…”

Section: B the Numerical Methodsmentioning

confidence: 99%

Symmetry breaking of spatial Kerr solitons in fractional dimension

Malomed

Mihalache

2020

Chaos, Solitons & Fractals

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We study symmetry breaking of solitons in the framework of a nonlinear fractional Schrödinger equation (NLFSE), characterized by its Lévy index, with cubic nonlinearity and a symmetric double-well potential. Asymmetric, symmetric, and antisymmetric soliton solutions are found, with stable asymmetric soliton solutions emerging from unstable symmetric and antisymmetric ones by way of symmetry-breaking bifurcations. Two different bifurcation scenarios are possible. First, symmetric soliton solutions undergo a symmetry-breaking bifurcation of the pitchfork type, which gives rise to a branch of asymmetric solitons, under the action of the self-focusing nonlinearity.Second, a family of asymmetric solutions branches off from antisymmetric states in the case of selfdefocusing nonlinearity through a bifurcation of an inverted-pitchfork type. Systematic numerical analysis demonstrates that increase of the Lévy index leads to shrinkage or expansion of the symmetry-breaking region, depending on parameters of the double-well potential. Stability of the soliton solutions is explored following the variation of the Lévy index, and the results are confirmed by direct numerical simulations. * Recently, the investigation of SSB of solitons has been expanded into non-Hermitian optical systems, which are modeled by NLSEs with parity-time (PT ) symmetric complex potentials [32, 33], where families of stable asymmetric one-dimensional (1D) solitons induced by the SSB have been found for specially designed complex potentials [34-37], as well as under the action of 2D potentials [38]. An interesting generalization of the Schrödinger equation, corresponding to fractionaldimensional Hamiltonians, was proposed in Refs. [39]-[41]. It has been introduced, in the context of the quantum theory, via Feynman path integrals over Lévy-flight trajectories, which leads to the fractional Schrödinger equation (FSE). Although implications of such models are still a matter of debate [42, 43], some experimental schemes have been proposed for emulating them in condensed-matter settings and optical cavities [44, 45]. A number of intriguing dynamical properties have been predicted in the framework of FSEs, including 1D zigzag propagation [46], diffraction-free beams [47-51], beam splitting [52], periodic oscillations of Gaussian beams [53], beam-propagation management [54], optical Bloch oscillations and Zener tunneling [55], resonant mode conversion and Rabi oscillations [56], localization and Anderson delocalization [57], and SSB of PT -symmetric modes in a linear FSE [58].Naturally, FSEs may be applied to nonlinear-optical settings [59,60]. Recent works demonstrate that a variety of fractional optical solitons can be produced by nonlinear FSEs

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Soliton dynamics in a fractional complex Ginzburg-Landau model

Cited by 89 publications

References 49 publications

Quadratic fractional solitons

Quadratic fractional solitons

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

Symmetry breaking of spatial Kerr solitons in fractional dimension

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