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In this work, we study a model of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators. We focus on periodic and localized traveling waves of the model, in the presence of loss and an external drive. Employing a Melnikov analysis we study the existence and persistence of such traveling waves, and study their linear stability. We show that, under certain conditions, the presence of dissipation and/or driving may stabilize or destabilize the solutions. Our analytical results are found to be in good agreement with direct numerical computations. [ABSTRACT FROM AUTHOR]

In this paper, we revisit one of the prototypical PT-symmetric oligomers, namely the trimer. We find all the relevant branches of 'regular' solutions and analyze the bifurcations and instabilities thereof. Our work generalizes the formulation that was recently proposed in the case of dimers for the so-called 'ghost states' of trimers, which we also identify and connect to symmetrybreaking bifurcations from the regular states. We also examine the dynamics of unstable trimers, as well as those of the ghost states in the parametric regime where the latter are found to exist. Finally, we present the current state-of-theart for optical experiments in PT -symmetric trimers, as well as experimental results in a gain-loss-gain three channel waveguide structure. [ABSTRACT FROM AUTHOR]

We study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive Bose-Einstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that - in the discrete regime - all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is stable, while the excited states feature a chain of stability/instability bands. We illustrate the mechanisms under which discreteness destabilizes the dark-soliton configurations, which become stable only in the continuum regime. Continuation from the anti-continuum limit is also considered, and a rich bifurcation structure is revealed. [ABSTRACT FROM AUTHOR]

We study the existence and stability of solutions of the two-dimensional nonlinear Schrödinger equation in the combined presence of a parabolic and a periodic potential. The motivating physical example consists of Bose–Einstein condensates confined in an harmonic (e.g., magnetic) trap and an optical lattice. By connecting the nonlinear problem with the underlying linear spectrum, we examine the bifurcation of nonlinear modes out of the linear ones for both focusing and defocusing nonlinearities. In particular, we find real-valued solutions (such as multipoles) and complex-valued ones (such as vortices). A primary motivation of the present work is to develop “rules of thumb” about what waveforms to expect emerging in the nonlinear problem and about the stability of those modes. As a case example of the latter, we find that among the real-valued solutions, the one with larger norm for a fixed value of the chemical potential is expected to be unstable. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligorić et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations. [ABSTRACT FROM AUTHOR]

In the present work, we consider the problem of a system of few vortices N≤5 as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion for an axially symmetric trapped condensate, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. We use a Monte Carlo method parametrizing the vortex particles by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for N=2,...,5 and different vortex particle angular momenta. We then complement this picture with a dynamical system analysis of the possible rigidly rotating states. The latter reveals a supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise, exposing the full wealth of the problem even for such low-dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte Carlo approach selects for different values of the angular momentum that is used as a bifurcation parameter. [ABSTRACT FROM AUTHOR]