The nontrivial states in one-dimensional nonlinear bichromatic superlattices

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.     

Vortex solitons in the semi-infinite gap of optically induced periodic lattices

Vortex solitons with a ring vortex core residing in a single lattice site in the semi-infinite gap of square optical lattices are reported. These solitons are no longer bound states of the Bloch-wave unit (Bloch-wave distribution in one lattice site) at the band edge of the periodic lattice, and consequently they do not bifurcate from the corresponding band edge. For saturable nonlinearity, one family of such solitons is found, and its existing curve forms a closed loop, which is very surprising. For Kerr nonlinearity, two families of such vortex solitons are found.     

The appearance of gap solitons in a nonlinear Schrödinger lattice

We study the appearance of discrete gap solitons in a nonlinear Schrödinger model with a periodic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit. For finite lattices supporting an anti-phase (q=π/2) gap edge phonon as an anharmonic standing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchfork bifurcations from the gap edge. Analytically, modulational instabilities between pairs of bifurcation points on this “nonlinear gap boundary” are found in terms of critical gap widths, turning to zero in the infinite-size limit, which are associated with the birth of the localized soliton as well as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in exciting soliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensates in periodic potentials. For lattices whose gap edge phonon only asymptotically approaches the anti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to the birth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. The instability-induced dynamics of the localized soliton in the gap regime is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsically localized modes (discrete breathers) from the extended out-gap soliton reveals a phase transition of the solution.     

Surface gap solitons

We put forward the existence of surface gap solitons at the interface between uniform media and an optical lattice with defocusing nonlinearity. Such new type of solitons forms when the incident and reflected waves at the interface of the lattice experience Bragg scattering, and feature a combination of the unique properties of both surface waves and gap solitons. We discover that gap surface solitons exist only when the lattice depth exceeds a threshold value, that they can be made completely stable, and that they can form stable bound states.     

Observation of surface gap solitons in semi-infinite waveguide arrays

We report on the observation of surface gap solitons found to exist at the interface between uniform and periodic dielectric media with defocusing nonlinearity. We demonstrate strong self-trapping at the edge of a LiNbO3 waveguide array and the formation of staggered surface solitons with propagation constant inside the first photonic band gap. We study the crossover between linear repulsion and nonlinear attraction at the surface, revealing the mechanism of nonlinearity-mediated stabilization of the surface gap modes.     

Discrete surface solitons in semi-infinite binary waveguide arrays

We analyze discrete surface modes in semi-infinite binary waveguide arrays, which can support simultaneously two types of discrete solitons. We demonstrate that the analysis of linear surface states in such arrays provides important information about the existence of nonlinear surface modes and their properties. We find numerically the families of both discrete surface solitons and nonlinear Tamm (gap) states and study their stability properties.     

SOLITON EXCITATIONS INA DIATOMIC LATTICE WITH NONLINEAR ON-SITE POTENTIAL

We consider the soliton excitations in a diatomic chain with nonlinear on-site potential based on a quell-discreteness approximation, Different from the theoretical approach devel-oped before, our method is valid in the whole Brillouin zone of phonons. The gap solitons observed recently in a nonlinear diatomic pendulum lattice can be well explained qualita-tively. It is also shown that if the wave number of the carrier wave is near the edge of the Brillouin zone, the results coinside with that of Kivshar and Flytzanis about the gap solitons in diatomic lattices when the difference of mass between two kinds of atoms becomes small.     

SOLITON EXCITATIONS IN A DIATOMIC LATTICE WITH NONLINEAR ON-SITE POTENTIAL

We consider the soliton excitations in a diatomic chain with nonlinear on-site potential based on a quell-discreteness approximation, Different from the theoretical approach devel-oped before, our method is valid in the whole Brillouin zone of phonons. The gap solitons observed recently in a nonlinear diatomic pendulum lattice can be well explained qualita-tively. It is also shown that if the wave number of the carrier wave is near the edge of the Brillouin zone, the results coinside with that of Kivshar and Flytzanis about the gap solitons in diatomic lattices when the difference of mass between two kinds of atoms becomes small.     

Vector valley Hall edge solitons in the photonic lattice with type-II Dirac cones

Topological edge solitons represent a significant research topic in the nonlinear topological photonics. They maintain their profiles during propagation, due to the joint action of lattice potential and nonlinearity, and at the same time are immune to defects or disorders, thanks to the topological protection. In the past few years topological edge solitons were reported in systems composed of helical waveguide arrays, in which the time-reversal symmetry is effectively broken. Very recently, topological valley Hall edge solitons have been demonstrated in straight waveguide arrays with the time-reversal symmetry preserved. However, these were scalar solitary structures. Here, for the first time, we report vector valley Hall edge solitons in straight waveguide arrays arranged according to the photonic lattice with innate type-II Dirac cones, which is different from the traditional photonic lattices with type-I Dirac cones such as honeycomb lattice. This comes about because the valley Hall edge state can possess both negative and positive dispersions, which allows the mixing of two different edge states into a vector soliton. Our results not only provide a novel avenue for manipulating topological edge states in the nonlinear regime, but also enlighten relevant research based on the lattices with type-II Dirac cones.     

Bright Bose-Einstein gap solitons of atoms with repulsive interaction

We report on the first experimental observation of bright matter wave solitons for 87Rb atoms with repulsive atom-atom interaction. This counterintuitive situation arises inside a weak periodic potential, where anomalous dispersion can be realized at the Brillouin zone boundary. If the coherent atomic wave packet is prepared at the corresponding band edge, a bright soliton is formed inside the gap. The strength of our system is the precise control of preparation and real time manipulation, allowing the systematic investigation of gap solitons.     

Majorana edge states in interacting one-dimensional systems

We show that one-dimensional electron systems in the proximity of a superconductor that support Majorana edge states are extremely susceptible to electron-electron interactions. Strong interactions generically destroy the induced superconducting gap that stabilizes the Majorana edge states. For weak interactions, the renormalization of the gap is nonuniversal and allows for a regime in which the Majorana edge states persist. We present strategies of how this regime can be reached.     

Controlled generation and steering of spatial gap solitons

We demonstrate the first fully controlled generation of immobile and slow spatial gap solitons in nonlinear periodic systems with band-gap spectra, and observe the key features of gap solitons that distinguish them from discrete solitons, including a dynamical transformation of gap solitons due to nonlinear interband coupling. We also describe theoretically and confirm experimentally the effect of the anomalous steering of gap solitons in optically induced photonic lattices.     

Discrete solitons and nonlinear surface modes in semi-infinite waveguide arrays

We discuss the formation of self-trapped localized states near the edge of a semi-infinite array of nonlinear optical waveguides. We study a crossover from nonlinear surface states to discrete solitons by analyzing the families of odd and even modes centered at finite distances from the surface and reveal the physical mechanism of the nonlinearity-induced stabilization of surface modes.     

Nonlinear light propagation in the defect band of one-dimensional defective structures

In this paper we numerically investigate the nonlinear propagation of defect state in one-dimensional structures with defects. We investigate the nonlinear transmission spectra and the bistable response for defective structures with different index gradients. The results show that positive Kerr nonlinearity can suppress the Wannier-Stark localization. And the nonlinear response of defect states band exhibits an optical switch behavior, which may be applied to all-optical devices. And the gap solitons from these defect states are presented.     

Surface defect gap solitons in superlattices with self-defocusing nonlinearity

We put forward the existence and stability of defect surface gap solitons at the interface between uniform media and an superlattice with self-defocusing nonlinearity. We reveal that the defect plays the significant role in controlling the region of solitons existing. Various solitons are found to be existed in different gaps for different defects. For positive defects, fundamental solitons can exist stably in the semi-infinite gap, and dipole solitons can exist stably in the first gap but they are unstable in the second gap. For zero or negative defects, fundamental and dipole solitons can exist stably in the first gap and the second gap, respectively.     

Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications

We present an overview of nonlinear phenomena related to optical quadratic solitons—intrinsically multi-component localized states of light, which can exist in media without inversion symmetry at the molecular level. Starting with presentation of a few derivation schemes of basic equations describing three-wave parametric wave mixing in diffractive and/or dispersive quadratic media, we discuss their continuous wave solutions and modulational instability phenomena, and then move to the classification and stability analysis of the parametric solitary waves. Not limiting ourselves to the simplest spatial and temporal quadratic solitons we also overview results related to the spatio-temporal solitons (light bullets), higher order quadratic solitons, solitons due to competing nonlinearities, dark solitons, gap solitons, cavity solitons and vortices. Special attention is paid to a comprehensive discussion of the recent experimental demonstrations of the parametric solitons including their interactions and switching. We also discuss connections of quadratic solitons with other types of solitons in optics and their interdisciplinary significance.     

The contrast between defect solitons in parityben time symmetric superlattice and simple-lattice complex potentials

The existence and stability of defect superlattice solitons in parity-time (PT) symmetric superlattice and simple-lattice complex potentials are reported. Compared with defect simple-lattice solitons in similar potentials, the defect soliton in superlattice has a wider stable range than that in simple-lattice. The solitons' power increases with increasing propagation constant. For the positive defect, the solitons are stable in the whole region where solitons exist in the semi-infinite gap. For the zero defect, the solitons are unstable at the edge of the band. For the negative defect, the solitons propagate with the shape of Y at low propagation constant and propagate stably at the large one.     

Higher-Order Topological Spin Hall Effect of Sound

We theoretically propose a reconfigurable two-dimensional(2 D) hexagonal sonic crystal with higher-order topology protected by the six-fold,C_6,rotation symmetry.The acoustic band gap and band topology can be controlled by rotating the triangular scatterers in each unit cell.In the nontrivial phase,the sonic crystal realizes the topological spin Hall effect in a higher-order fashion:(i) the edge states emerging in the bulk band gap exhibit partial spin-momentum correlation and are gapped due to the reduced spatial symmetry at the edges.(ii) The gapped edge states,on the other hand,stabilize the topological corner states emerging in the edge band gap.The partial spin-momentum correlation is manifested as pseudo-spin-polarization of edge states away from the time-reversal invariant momenta,where the pseudospin is emulated by the acoustic orbital angular momentum.We reveal the underlying topological mechanism using a corner topological index based on the symmetry representation of the acoustic Bloch bands.     

Gap solitons of spin-orbit-coupled Bose-Einstein condensates in $\mathcal{PT}$ periodic potential

We numerically investigate the gap solitons in Bose-Einstein condensates (BECs) with spin-orbit coupling (SOC) in the parity-time ($\mathcal{PT}$)-symmetric periodic potential. We find that the depths and periods of the imaginary lattice have an important influence on the shape and stability of these single-peak gap solitons and double-peak gap solitons in the first band gap. The dynamics of these gap solitons are checked by the split-time-step Crank-Nicolson method. It is proved that the depths of the imaginary part of the $\mathcal{PT}$-symmetric periodic potential gradually increase, and the gap solitons become unstable. But the different periods of imaginary part hardly affect the stability of the gap solitons in the corresponding parameter interval.     

Surface defect gap solitons in one-dimensional dual-frequency lattices

We study the surface defect gap solitons in an interface between a defect of one-dimensional dual-frequency lattices and the uniform media. Some unique properties are revealed that such lattices can broaden the region of semi-finite gap, and the semi-finite gap exists not only in the positive and zero defects but also in the negative defect; unlike in the regular lattices, the semi-finite gap exists in the positive and zero defects but does not exist in the negative defect. In particular, stable solitons exist almost in the whole semi-finite gap for the positive and zero defects. These properties are different from other lattices with defects. In addition, it is found that the existence of surface dual-frequency lattice solitons does not need a threshold power.