Domain walls of single-component Bose–Einstein condensates in external potentials

Malomed, Boris A.; Frantzeskakis, D. J.; Bishop, A. R.; Nistazakis, H. E.; Carretero-González, R.

doi:10.1016/j.matcom.2005.01.016

Cited by 12 publications

(14 citation statements)

References 53 publications

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“…Recall that, in the free space, the front solution exists at a single value of the propagation constant, k = 3/4, while an external potential may support a family of front solutions [26] (in Ref. [26], fronts were represented by solutions to the Gross-Pitaevskii equation which included the cubic nonlinearity and a sinusoidal potential). In the present model, it can be shown that front states may also be sustained by a single potential well, in the absence of the second channel.…”

Section: Basic Types Of Spatial Solitons In the Two-channel Systemmentioning

confidence: 99%

Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity

Birnbaum

Malomed

2008

Physica D: Nonlinear Phenomena

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We present eight types of spatial optical solitons which are possible in a model of a planar waveguide that includes a dual-channel trapping structure and competing (cubic-quintic) nonlinearity. Among the families of trapped beams are symmetric and antisymmetric solitons of "broad" and "narrow" types, composite states, built as combinations of broad and narrow beams with identical or opposite signs ("unipolar" and "bipolar" states, respectively), and "single-sided" broad and narrow beams trapped, essentially, in a single channel. The stability of the families is investigated via eigenvalues of small perturbations, and is verified in direct simulations. Three species -narrow symmetric, broad antisymmetric, and unipolar composite states -are unstable to perturbations with real eigenvalues, while the other five families are stable. The unstable states do not decay, but, instead, spontaneously transform themselves into persistent breathers, which, in some cases, demonstrate dynamical symmetry breaking and chaotic internal oscillations. A noteworthy feature is a stability exchange between the broad and narrow antisymmetric states: in the limit when the two channels merge into one, the former species becomes stable, while the latter one loses its stability. Different branches of the stationary states are linked by four bifurcations, which take different forms in the model with the strong and weak inter-channel coupling.

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Section: Basic Types Of Spatial Solitons In the Two-channel Systemmentioning

confidence: 99%

Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity

Birnbaum

Malomed

2008

Physica D: Nonlinear Phenomena

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“…In particular, it has been found that real 1D-version of Eq. (3) ψ xx + (ω − U(x))ψ − σψ 3 = 0 (4) with the model cosine potential U(x) = A cos 2x (5) describes bright and dark gap solitons [7,8,9,10], nonlinear periodic structures (nonlinear Bloch waves) [7,11], domain walls [12], gap waves [13] and so on. Some interesting relations between various nonlinear objects described by Eq.…”

Section: Introductionmentioning

confidence: 99%

Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential

Alfimov

Avramenko

2013

Physica D: Nonlinear Phenomena

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Abstract. We study nonlinear states for NLS-type equation with additional periodic potential U (x) (called the Gross-Pitaevskii equation, GPE in theory of Bose-Einstein Condensate, BEC). We prove that if the nonlinearity is defocusing (repulsive, in BEC context) then under certain conditions there exists a homeomorphism between the set of nonlinear states for GPE (i.e. real bounded solutions of some nonlinear ODE) and the set of bi-infinite sequences of numbers from 1 to N for some integer N . These sequences can be viewed as codes of the nonlinear states. Sufficient conditions for the homeomorphism to exist are given in the form of three hypotheses. For a given U (x), the verification of the hypotheses should be done numerically. We report on numerical results for the case of GPE with cosine potential and describe regions in the plane of parameters where this coding is possible.

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“…The HH bifurcation has also been observed in other NLS-related settings. Several studies have demonstrated numerically the existence of "Krein collisions"-defined in section 5.3 below-in discrete wave equations [10,11,12,13] and in Bose-Einstein condensates (BEC) [14,15,16,17,18]. In these studies, and most others, the bifurcation is discussed only in the context of detecting the instability transition in the linear spectrum, or by performing a small number of numerical solutions to the initial value problem.…”

Section: Introductionmentioning

confidence: 99%

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

Goodman¹

2011

J. Phys. A: Math. Theor.

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We examine the dynamics of solutions to nonlinear Schrödinger/Gross-Pitaevskii equations that arise due to Hamiltonian Hopf (HH) bifurcations-the collision of pairs of eigenvalues on the imaginary axis. To this end, we use inverse scattering to construct localized potentials for this model which lead to HH bifurcations in a predictable manner. We perform a formal reduction from the partial differential equations (PDE) to a small system of ordinary differential equations (ODE). We show numerically that the behavior of the PDE is well-approximated by that of the ODE and that both display Hamiltonian chaos. We analyze the ODE to derive conditions for the HH bifurcation and use averaging to explain certain features of the dynamics that we observe numerically.

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Domain walls of single-component Bose–Einstein condensates in external potentials

Cited by 12 publications

References 53 publications

Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity

Families of spatial solitons in a two-channel waveguide with the cubic-quintic nonlinearity

Coding of nonlinear states for the Gross–Pitaevskii equation with periodic potential

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

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