On the 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 Schr{\"o}dinger 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 spacediagnostics, 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

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 right-bright (BB) type], or of opposite signs in the two components [the gap-bright (GB) model]. 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

Dipole and quadrupole solitons in optically-induced two-dimensional defocusing photonic lattices

Dipole and quadrupole solitons in a two-dimensional optically induced defocus- ing photonic lattice are theoretically predicted and experimentally observed. It is shown that in-phase nearest-neighbor dipole and out-of-phase next-nearest-neighbor dipoles exist and can be stable in the intermediate intensity regime. The other types of dipoles are always unstable. In-phase nearest-neighbor quadrupoles are also numerically obtained, and may also be linearly stable. Out-of-phase, nearest-neighbor quadrupoles are found to be typically unstable. These numerical results are found to be aligned with the main predictions obtained analytically in the discrete nonlinear Schroedinger model. Finally, experimental results are presented for both dipole and quadrupole structures, indicating that self-trapping of such structures in the defocusing lattice can be realized for the length of the nonlinear crystal (10 mm)

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. © 2007 Elsevier B.V. All rights reserved

Existence and stability of multisite breathers in honeycomb and hexagonal lattices

We study the existence and stability of multisite discrete breathers in two prototypical non-square Klein-Gordon lattices, namely a honeycomb and a hexagonal one. In the honeycomb case we consider six-site configurations and find that for soft potential and positive coupling the out-of-phase breather configuration and the charge-two vortex breather are linearly stable, while the in-phase and charge-one vortex states are unstable. In the hexagonal lattice, we first consider three-site configurations. In the case of soft potential and positive coupling, the in-phase configuration is unstable and the charge-one vortex is linearly stable. The out-of-phase configuration here is found to always be linearly unstable. We then turn to six-site configurations in the hexagonal lattice. The stability results in this case are the same as in the six-site configurations in the honeycomb lattice. For all configurations in both lattices, the stability results are reversed in the setting of either hard potential or negative coupling. The study is complemented by numerical simulations which are in very good agreement with the theoretical predictions. Since neither the form of the on-site potential nor the sign of the coupling parameter involved have been prescribed, this description can accommodate inverse-dispersive systems (e.g. supporting backward waves) such as transverse dust-lattice oscillations in dusty plasma (Debye) crystals or analogous modes in molecular chains. © 2010 IOP Publishing Ltd

Discrete solitons and vortices in hexagonal and honeycomb lattices: Existence, stability, and dynamics

We consider a prototypical dynamical lattice model, namely, the discrete nonlinear Schrödinger equation on nonsquare lattice geometries. We present a systematic classification of the solutions that arise in principal six-lattice-site and three-lattice-site contours in the form of both discrete multipole solitons and discrete vortices. Additionally to identifying the possible states, we analytically track their linear stability both qualitatively and quantitatively. We find that among the six-site configurations, the "hexapole" of alternating phases (0-π), as well as the vortex of topological charge S=2 have intervals of stability; among three-site states, only the vortex of topological charge S=1 may be stable in the case of focusing nonlinearity. These conclusions are confirmed both for hexagonal and for honeycomb lattices by means of detailed numerical bifurcation analysis of the stationary states from the anticontinuum limit, and by direct simulations to monitor the dynamical instabilities, when the latter arise. The dynamics reveal a wealth of nonlinear behavior resulting not only in single-site solitary wave forms, but also in robust multisite breathing structures. © 2008 The American Physical Society

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