28 research outputs found
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Localized structures in kagome lattices
We investigate the existence and stability of gap vortices and multipole gap solitons in a kagome lattice with a defocusing nonlinearity both in a discrete case and in a continuum one with periodic external modulation. In particular, predictions are made based on expansion around a simple and analytically tractable anticontinuum (zero-coupling) limit. These predictions are then confirmed for a continuum model of an optically induced kagome lattice in a photorefractive crystal obtained by a continuous transformation of a honeycomb lattice
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Solitons and vortices in honeycomb defocusing photonic lattices
Solitons and necklaces in the first band gap of a two-dimensional optically induced honeycomb defocusing photonic lattice are theoretically considered. It is shown that dipoles, soliton necklaces, and vortex necklaces exist and may possess regions of stable propagation through a photorefractive crystal. Most of the configurations disappear in bifurcations close to the upper edge of the first band. Solutions associated with such bifurcations are also numerically examined, and it is found that they are often asymmetric and more exotic. The dynamics of the relevant unstable structures are also examined through direct numerical simulations revealing either breathing oscillations or, in some cases, destruction of the original wave form
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Stability of quantized vortices in a Bose-Einstein condensate confined in an optical lattice
We investigate the existence and especially the linear stability of single- and multiple-charge quantized vortex states of nonlinear Schrödinger equations in the presence of a periodic and a parabolic potential in two spatial dimensions. The study is motivated by an examination of pancake-shaped Bose-Einstein condensates in the presence of magnetic and optical confinement. A two-parameter space of the condensate’s chemical potential versus the periodic potential’s strength is scanned for both single- and double-quantized vortex states located at a local minimum or a local maximum of the lattice. Triple-charged vortices are also briefly discussed. Single-charged vortices are found to be stable for cosinusoidal potentials and unstable for sinusoidal ones above a critical strength. Higher-charge vortices are more unstable for both types of potentials, and their dynamical evolution leads to a breakup into single-charged vortices
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Vortex solutions of the discrete Gross-Pitaevskii equation starting from the anti-continuum limit
In this paper, we consider the existence, stability and dynamical evolution of dark vortex states in the two-dimensional defocusing discrete nonlinear Schrödinger model, a model of interest both to atomic physics and to nonlinear optics. Our considerations are chiefly based on initializing such vortex configurations at the anti-continuum limit of zero coupling between adjacent sites, and continuing them to finite values of the coupling. Systematic tools are developed for such continuations based on amplitude-phase decompositions and explicit solvability conditions enforcing the vortex phase structure. Regarding the linear stability of such nonlinear waves, we find that in a way reminiscent of their 1d analogs, i.e., of discrete dark solitons, the discrete defocusing vortices become unstable past a critical coupling strength and, subsequently feature a cascade of alternating stabilization–destabilization windows for any finite lattice. Although the results are mainly geared towards the uniform case, we also consider the effect of harmonic trapping potentials often present in experimental atomic physics settings
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Trapping of two-component matter-wave solitons by mismatched optical lattices
We consider a one-dimensional model of a two-component Bose–Einstein condensate in the presence of periodic external potentials of opposite signs, acting on the two species. The interaction between the species is attractive, while intra-species interactions may be attractive too [the system of the bright–bright (BB) type], or of opposite signs in the two components [the gap–bright (GB) type]. We identify the existence and stability domains for soliton complexes of the BB and GB types. The evolution of unstable solitons leads to the establishment of oscillatory states. The increase of the strength of the nonlinear attraction between the species results in symbiotic stabilization of the complexes, despite the fact that one component is centered around a local maximum of the respective periodic potential
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Emergence of unstable modes in an expanding domain for energy-conserving wave equations
Motivated by recent work on instabilities in expanding domains in reaction–diffusion settings, we propose an analog of such mechanisms in energy-conserving wave equations. In particular, we consider a nonlinear Schrödinger equation in a finite domain and show how the expansion or contraction of the domain, under appropriate conditions, can destabilize its originally stable solutions through the modulational instability mechanism. Using both real and Fourier space diagnostics, we monitor and control the crossing of the instability threshold and, hence, the activation of the instability. We also consider how the manifestation of this mechanism is modified in a spatially inhomogeneous setting, namely in the presence of an external parabolic potential, which is relevant to trapped Bose–Einstein condensates
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Localized structures in kagome lattices
We investigate the existence and stability of gap vortices and multipole gap solitons in a kagome lattice with a defocusing nonlinearity both in a discrete case and in a continuum one with periodic external modulation. In particular, predictions are made based on expansion around a simple and analytically tractable anticontinuum (zero-coupling) limit. These predictions are then confirmed for a continuum model of an optically induced kagome lattice in a photorefractive crystal obtained by a continuous transformation of a honeycomb lattice
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Averaging of nonlinearity management with dissipation
Motivated by recent experiments in optics and atomic physics, we derive an averaged nonlinear partial differential equation describing the dynamics of the complex field in a nonlinear Schrödinger model in the presence of a periodic nonlinearity and a periodically varying dissipation coefficient. The incorporation of dissipation in our model is motivated by experimental considerations. We test the numerical behavior of the derived averaged equation by comparing it to the original nonautonomous model in a prototypical case scenario and observe good agreement between the two
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Geometric stabilization of extended S=2 vortices in two-dimensional photonic lattices: Theoretical analysis, numerical computation, and experimental results
In this work, we focus on the subject of nonlinear discrete self-trapping of S=2 (doubly-charged) vortices in two-dimensional photonic lattices, including theoretical analysis, numerical computation, and experimental demonstration. We revisit earlier findings about S=2 vortices with a discrete model and find that S=2 vortices extended over eight lattice sites can indeed be stable (or only weakly unstable) under certain conditions, not only for the cubic nonlinearity previously used, but also for a saturable nonlinearity more relevant to our experiment with a biased photorefractive nonlinear crystal. We then use the discrete analysis as a guide toward numerically identifying stable (and unstable) vortex solutions in a more realistic continuum model with a periodic potential. Finally, we present our experimental observation of such geometrically extended S=2 vortex solitons in optically induced lattices under both self-focusing and self-defocusing nonlinearities and show clearly that the S=2 vortex singularities are preserved during nonlinear propagation
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Stable higher-charge discrete vortices in hexagonal optical lattices
We show that double-charge discrete optical vortices may be completely stable in hexagonal photonic lattices where single-charge vortices always exhibit dynamical instabilities. Even when unstable the double-charge vortices typically have a much weaker instability than the single-charge vortices, and thus their breakup occurs at longer propagation distances