Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes

Goodman, Roy H.

doi:10.1088/1751-8113/44/42/425101

Cited by 20 publications

(23 citation statements)

References 52 publications

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“…This type of collision gives rise to a complex eigenvalue quartet and a different (weak) oscillatory dynamical instability, or a so-called Hamiltonian Hopf bifurcation; see, e.g., the discussion of Ref. [58]. The latter scenario leads to small instability bubbles, as the quartet may form, but subsequently the eigenvalues may return to the imaginary axis, splitting anew into two imaginary pairs.…”

Section: B Adding the Perturbation Potentialmentioning

confidence: 99%

Stabilization of ring dark solitons in Bose-Einstein condensates

et al. 2015

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Earlier work has shown that ring dark solitons in two-dimensional Bose-Einstein condensates are generically unstable. In this work, we propose a way of stabilizing the ring dark soliton via a radial Gaussian external potential. We investigate the existence and stability of the ring dark soliton upon variations of the chemical potential and also of the strength of the radial potential. Numerical results show that the ring dark soliton can be stabilized in a suitable interval of external potential strengths and chemical potentials. We also explore different proposed particle pictures considering the ring as a moving particle and find, where appropriate, results in very good qualitative and also reasonable quantitative agreement with the numerical findings.

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Section: B Adding the Perturbation Potentialmentioning

confidence: 99%

Stabilization of ring dark solitons in Bose-Einstein condensates

et al. 2015

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“…Such configurations have been considered earlier in optical applications theoretically [31,32] and even experimentally [33] in the absence of gain/loss. We examine this case with both linear and nonlinear gain/loss profiles.…”

Section: Analysis Of Stationary Solutions For the Nonlinear Pt -Symmementioning

confidence: 99%

Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis

Duanmu

Horne

et al. 2013

Phil. Trans. R. Soc. A.

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In the present work, we focus on the cases of twosite (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time (PT ) symmetry, i.e. with a spatially odd gain-loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear PT -symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear PT -symmetric trimers, while both of them are absent in linear PT -symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear PT -symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of PT -symmetric quadrimers.

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“…These we will term "nonlinear dressed states" (NDS). Additionally, this new class of states is found to be interconnected as a function of the linear coupling strength via a series of Hamiltonian saddle-node, as well as Hopf bifurcations [20]. A key observation is that the resulting rich bifurcation diagram connects the NDS with two previously studied limits.…”

Section: Fig 1: (Color Online)mentioning

confidence: 83%

“…4) is associated with a Hamiltonian Hopf bifurcation (see e.g. [20] for a recent discussion), whereby quartets of eigenfrequencies emerge and destabilize the branch. Both unstable parts of the branch connect in the limit of Ω = 0 to a train of darkbright solitons but with different even multiplicity.…”

Section: Fig 1: (Color Online)mentioning

confidence: 99%

Nonlinear dressed states at the miscibility-immiscibility threshold

et al. 2015

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The dynamical evolution of spatial patterns in a complex system can reveal the underlying structure and stability of stationary states. As a model system we employ a two-component rubidium Bose-Einstein condensate at the transition from miscible to immiscible with the additional control of linear interconversion. Excellent agreement is found between the detailed experimental time evolution and the corresponding numerical mean-field computations. Analysing the dynamics of the system, we find clear indications of stationary states that we term nonlinear dressed states. A steady state bifurcation analysis reveals a smooth connection of these states with dark-bright soliton solutions of the integrable two-component Manakov model.Bose-Einstein condensates have been established over the past two decades as a prototypical testbed for exciting developments ranging from nonlinear dynamics and wave phenomena to superfluid features and quantum phase transitions [1][2][3][4][5]. Especially two-component ultracold gases are ideal for the study of the connection of topological solutions of integrable systems and their variants in the presence of different types of perturbations.The properties of multi-component Bose Einstein condensates have been studied in numerous contexts. In particular, early experimental efforts produced binary mixtures of two different hyperfine states of 23 Na [6] and of 87 Rb [7]. The progressively improving experimental control has enabled detailed observations of phase separation phenomena and associated multi-component dynamics [8][9][10][11][12][13]. More recently, the mixing-demixing dynamics has been controlled both in pseudo-spinor (twocomponent) [14] and spinor systems [15] via external coupling fields. As a result, formation of domain walls has been observed. In these systems additional topological excitations such as dark-bright solitons do exist. These have been experimentally realized building on dynamical instabilities present in the regime of two counterflowing superfluids [16]. The ability of phase imprinting offers a controlled path for the generation of individual such topological states [17]. All these observations are adequately captured by the mean-field description. Thus, the well established integrable Manakov model [18], i.e. two nonlinearly interacting classical fields in one dimension at the miscibility-immiscibility threshold, forms a basis for understanding the corresponding characteristics. This model is also examined in other physical systems such as nonlinear polarization optics where multiple dark-bright and dark-dark soliton solutions can be systematically constructed [19]. * nlds@matterwave.de As Ω increases further, the amplitude of the oscillatory dynamics decreases and in the large coupling limit the system is well approximated by stationary dressed states.Here, we study the nonlinear dynamics of a twocomponent Bose gas at the transition from miscible to immiscible, arising through linear interconversion between the two components. In particular, we utilize a Rabi c...

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Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes

Cited by 20 publications

References 52 publications

Stabilization of ring dark solitons in Bose-Einstein condensates

Stabilization of ring dark solitons in Bose-Einstein condensates

Linear and nonlinear parity-time-symmetric oligomers: a dynamical systems analysis

Nonlinear dressed states at the miscibility-immiscibility threshold

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