Gap solitons and Bloch waves in nonlinear periodic systems

Zhang, Yongping; Liang, Zhaoxin; Wu, Biao

doi:10.1103/physreva.80.063815

Cited by 41 publications

(39 citation statements)

References 52 publications

(82 reference statements)

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“…As discussed in Refs. [16,[28][29][30], gap solitons develop in the linear band gaps and originate from the stable bound states of a single periodic well. So they can be divided different family according to the locations of the band gaps.…”

Section: Gap Solitons and Composition Relationsmentioning

confidence: 99%

The nontrivial states in one-dimensional nonlinear bichromatic superlattices

Liu

Wang

Yin

et al. 2017

Physica E: Low-dimensional Systems and Nanostructures

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We study topological properties of one-dimensional nonlinear bichromatic superlattices and unveil the effect of nonlinearity on topological states. We find the existence of nontrivial edge solitions, which distribute on the boundaries of the lattice with their chemical potential located in the linear gap regime and are sensitive to the phase parameter of the superlattice potential. We further demonstrate that the topological property of the nonlinear Bloch bands can be characterized by topological Chern numbers defined in the extended two-dimensional parameter space. In addition, we discuss that the composition relations between the nolinear Bloch waves and gap solitions for the nonlinear superlattices. The stabilities of edge solitons are also studied.

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Section: Gap Solitons and Composition Relationsmentioning

confidence: 99%

The nontrivial states in one-dimensional nonlinear bichromatic superlattices

Liu

Wang

Yin

et al. 2017

Physica E: Low-dimensional Systems and Nanostructures

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“…However it is known that it can miss the situations of weak oscillatory instabilities caused by quartets of complex eigenvalues with small real parts (see e.g. 39 and references therein, while the SFSs are chiefly unstable in that case, having a small stability region 20 (strictly speaking, FSs in models with linear lattice potentials may also feature a very weak oscillatory instability, having at the same time great lifetime, see 39 ). Therefore, stable DSs supported by the nonlinear pseudopotential, whose shape is very similar to that of the chiefly unstable SFSs in the systems with linear lattice potentials, deserve a detailed consideration, which is given in Sect.…”

Section: The Linear-stability Analysismentioning

confidence: 99%

Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity

Lebedev

Alfimov

Malomed

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, the one which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.Periodic (alias lattice) potentials is a well-known ingredient of diverse physical settings represented by the nonlinear Schrödinger/Gross-Pitaevskii equations. The lattice potentials help to create self-trapped modes (solitons) which do not exist otherwise, or stabilize those solitons which are definitely unstable in free space. In particular, the lattice potentials generate the bandgap spectrum in the linearized version of the equation, and adding local cubic nonlinearity gives rise to a great variety of gap solitons and their bound complexes residing in the spectral gaps. On the other hand, an essential extension of the concept of lattice potentials is the introduction of nonlinear pseudopotentials, which are induced by spatially periodic modulation of the coefficient in front of the cubic term. While single-peak fundamental solitons (FSs) in nonlinear potentials were studied in detail, more sophisticated ones, such as narrow antisymmetric dipole solitons (DSs), which essentially reside in a single cell of the nonlinear lattice, were not previously considered in this setting. Their shape is similar to that of the so-called subfundamental species of gap solitons in linear lattices, which have a small stability region. In this work, we first develop a general classification of a potentially infinite number of different types of soliton complexes supported by the nonlinear lattice. For physical applications, the most significant finding is the existence of two branches of the DS family, one of which is entirely stable. Its stability is readily predicted by the celebrated Vakhitov-Kolokolov criterion, while the shape of a) Electronic

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“…For convenience, dimensionless scaling will be made for the length and the energy [6,7]. Position x is to be scaled in the unit of Λ/(2π); Periodic potential V (x), interaction energy F (ρ), and the chemical potential µ are scaled in the unit of 8E r with E r = 2 π 2 /2mΛ 2 being the recoil energy and m the atom mass.…”

Section: Model Equationmentioning

confidence: 99%

Gap solitons and Bloch waves of interacting bosons in one-dimensional optical lattices: From the weak- to the strong-interaction limits

Guo

Jing

et al. 2011

Phys. Rev. A

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We study the gap solitons and nonlinear Bloch waves of interacting bosons in one-dimensional optical lattices, taking into account the interaction from the weak to the strong limits. It is shown that composition relation between the gap solitons and nonlinear Bloch waves exists for the whole span of the interaction strength. The linear stability analysis indicates that the gap solitons are stable when their energies are near the bottom of the linear Bloch band gap. By increasing the interaction strength, the stable gap solitons can turn into unstable. It is argued that the stable gap solitons can easily be formed in a weakly interacting system with energies near the bottoms of the lower-level linear Bloch band gaps.

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Gap solitons and Bloch waves in nonlinear periodic systems

Cited by 41 publications

References 52 publications

The nontrivial states in one-dimensional nonlinear bichromatic superlattices

The nontrivial states in one-dimensional nonlinear bichromatic superlattices

Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity

Gap solitons and Bloch waves of interacting bosons in one-dimensional optical lattices: From the weak- to the strong-interaction limits

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