^{1}, G. Theocharis

^{2}, P. G. Kevrekidis

^{3}, N. I. Karachalios

^{4}, F. K. Diakonos

^{1}and D. J. Frantzeskakis

^{1,a)}

### Abstract

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.

G.T. acknowledges support from the Alexander S. Onassis Foundation. P.G.K. gratefully acknowledges support from NSF-DMS-0349023, NSF-DMS-0806762, NSF-CMMI-1000337, and from Alexander von Humboldt and Alexander S. Onassis Foundations. The work of F.K.D. and D.J.F. was partially supported by the Special Account for Research Grants of the University of Athens.

I. INTRODUCTION

II. MODEL AND METHODS

A. Physical motivation and the model

B. Stability analysis approach

III. THE LINEAR PROBLEM

IV. EXISTENCE AND BIFURCATIONS OF SOLUTIONS IN THE FULLY NONLINEAR PROBLEM

A. Continuation from the linear to the nonlinear regime

B. Continuation from the anti-continuum limit

V. STABILITY OF THE NONLINEAR STATES

A. The first-excited state

B. The second-excited state

VI. CONCLUSIONS

### Key Topics

- Dark solitons
- 33.0
- Bifurcations
- 25.0
- Eigenvalues
- 23.0
- Wave functions
- 18.0
- Chemical potential
- 17.0

## Figures

The ground-state energy *E* _{0} for the linear quantum harmonic oscillator as a function of the lattice spacing α, for two values of the trap frequency, Ω = 0.1 and Ω = 0.05. Solid and dotted lines show the energy spectrum as found by solving the QHO eigenvalue problem, while (red) circles and (blue) crosses show the respective solutions obtained from the Mathieu equation (20). The dashed and dashed-dotted lines show the analytical result of Eq. (28). The inset shows the wavefunction profile for α = 10.

The ground-state energy *E* _{0} for the linear quantum harmonic oscillator as a function of the lattice spacing α, for two values of the trap frequency, Ω = 0.1 and Ω = 0.05. Solid and dotted lines show the energy spectrum as found by solving the QHO eigenvalue problem, while (red) circles and (blue) crosses show the respective solutions obtained from the Mathieu equation (20). The dashed and dashed-dotted lines show the analytical result of Eq. (28). The inset shows the wavefunction profile for α = 10.

Top right panel: Energy of the lowest four excited states as functions of the lattice spacing α, for a trap strength Ω = 0.1. Solid lines corresponding to the 1st-, 2nd-, 3rd, and 4th-excited states indicate the energy obtained by solving the QHO eigenvalue problem, dashed lines show the respective solutions obtained from the Mathieu equation (20), and circles show the respective analytical results of Eq. (25). Top left panels: spatial profiles of the 1st-, 2nd-, 3d-, and 4th-excited states for α = 10 (corresponding to the strongly discrete limit); solid lines depict the 1st- (top) and 3d- (bottom) excited states, while dotted lines depict the 2nd- (top) and 4th- (bottom) excited states, respectively. Bottom panels (from left to right): spatial profiles of the 1st-, 2nd-, 3d-, and 4th-excited states for α = 1 (corresponding to the discrete regime).

Top right panel: Energy of the lowest four excited states as functions of the lattice spacing α, for a trap strength Ω = 0.1. Solid lines corresponding to the 1st-, 2nd-, 3rd, and 4th-excited states indicate the energy obtained by solving the QHO eigenvalue problem, dashed lines show the respective solutions obtained from the Mathieu equation (20), and circles show the respective analytical results of Eq. (25). Top left panels: spatial profiles of the 1st-, 2nd-, 3d-, and 4th-excited states for α = 10 (corresponding to the strongly discrete limit); solid lines depict the 1st- (top) and 3d- (bottom) excited states, while dotted lines depict the 2nd- (top) and 4th- (bottom) excited states, respectively. Bottom panels (from left to right): spatial profiles of the 1st-, 2nd-, 3d-, and 4th-excited states for α = 1 (corresponding to the discrete regime).

The number of atoms *N* as a function of the chemical potential μ (for α = 0.8 and Ω = 0.1) for the three lowest states: the ground state (solid line), the first excited state (dashed line), and the second excited state (dotted line). Each branch begins from the linear limit (*N* = 0), where μ equals the energy of the corresponding linear state. The insets show the profiles of these nonlinear states for μ = 1.2.

The number of atoms *N* as a function of the chemical potential μ (for α = 0.8 and Ω = 0.1) for the three lowest states: the ground state (solid line), the first excited state (dashed line), and the second excited state (dotted line). Each branch begins from the linear limit (*N* = 0), where μ equals the energy of the corresponding linear state. The insets show the profiles of these nonlinear states for μ = 1.2.

Top left panel: The normalized number of atoms *N*/α as a function of the chemical potential μ, for α = 10 (i.e., in the vicinity of the anti-continuum limit) and Ω = 0.1. The black square indicates the region where this panel is magnified, as shown in top right panel. The letters A, B, …, H denote certain points in the diagram for which corresponding wavefunction profiles are shown in the middle and bottom panels. Stable (unstable) branches and respective states are depicted by solid (dashed or dotted) lines.

Top left panel: The normalized number of atoms *N*/α as a function of the chemical potential μ, for α = 10 (i.e., in the vicinity of the anti-continuum limit) and Ω = 0.1. The black square indicates the region where this panel is magnified, as shown in top right panel. The letters A, B, …, H denote certain points in the diagram for which corresponding wavefunction profiles are shown in the middle and bottom panels. Stable (unstable) branches and respective states are depicted by solid (dashed or dotted) lines.

The pitchfork bifurcation, relevant to the (green) branches D, F (and its parity-symmetric one), and G, as viewed by the difference ψ_{1} − ψ_{−1} of the two outer sites as a function of the normalized atom number *N*/α.

The pitchfork bifurcation, relevant to the (green) branches D, F (and its parity-symmetric one), and G, as viewed by the difference ψ_{1} − ψ_{−1} of the two outer sites as a function of the normalized atom number *N*/α.

The linear stability analysis for the first-excited state, corresponding to a single discrete dark soliton. The three top panels show the real part of the lowest-order eigenvalues and the two left bottom panels show the maximum of the imaginary part of the eigenvalues, both as functions of the chemical potential μ. The bottom right panel shows the dependence of the critical value of the chemical potential, μ_{ c }, for the onset of the instability as a function of α. Branches shown with circles (in red) in the two top left panels denote dynamically unstable modes, which have emerged upon collision of modes with opposite Krein sign. The parameter values are α = 1.2 (left column), α = 0.6 (middle column), and α = 0.1 (right column); the trap strength is in all cases Ω = 0.1.

The linear stability analysis for the first-excited state, corresponding to a single discrete dark soliton. The three top panels show the real part of the lowest-order eigenvalues and the two left bottom panels show the maximum of the imaginary part of the eigenvalues, both as functions of the chemical potential μ. The bottom right panel shows the dependence of the critical value of the chemical potential, μ_{ c }, for the onset of the instability as a function of α. Branches shown with circles (in red) in the two top left panels denote dynamically unstable modes, which have emerged upon collision of modes with opposite Krein sign. The parameter values are α = 1.2 (left column), α = 0.6 (middle column), and α = 0.1 (right column); the trap strength is in all cases Ω = 0.1.

The top panel shows a spatio-temporal contour plot of the density of the first-excited state (corresponding to a single discrete dark soliton), for parameter values μ = 1, α = 1.2, and Ω = 0.1. The soliton stays at rest, up to *t* ≈ 1500, and then starts to perform oscillations of growing amplitude. The bottom panel shows the initial density profile.

The top panel shows a spatio-temporal contour plot of the density of the first-excited state (corresponding to a single discrete dark soliton), for parameter values μ = 1, α = 1.2, and Ω = 0.1. The soliton stays at rest, up to *t* ≈ 1500, and then starts to perform oscillations of growing amplitude. The bottom panel shows the initial density profile.

The BdG analysis for the second-excited state, corresponding to a discrete dark soliton pair. The top (bottom) panels show the real (imaginary) part of the lowest-order eigenfrequencies as functions of the chemical potential μ. Branches shown with circles (in red or green) in the top panels denote dynamically unstable modes, which have emerged upon collision of modes with opposite Krein sign. The parameter values are α = 1.2 (left column), α = 0.6 (middle column), and α = 0.1 (right column); the trap strength is in all cases Ω = 0.1.

The BdG analysis for the second-excited state, corresponding to a discrete dark soliton pair. The top (bottom) panels show the real (imaginary) part of the lowest-order eigenfrequencies as functions of the chemical potential μ. Branches shown with circles (in red or green) in the top panels denote dynamically unstable modes, which have emerged upon collision of modes with opposite Krein sign. The parameter values are α = 1.2 (left column), α = 0.6 (middle column), and α = 0.1 (right column); the trap strength is in all cases Ω = 0.1.

Same as Fig. 7, but for the second-excited state, corresponding to a discrete dark soliton pair, for parameter values μ = 0.5 (corresponding to the first instability band – see bottom middle panel of Fig. 8), α = 0.6, and Ω = 0.1.

Same as Fig. 7, but for the second-excited state, corresponding to a discrete dark soliton pair, for parameter values μ = 0.5 (corresponding to the first instability band – see bottom middle panel of Fig. 8), α = 0.6, and Ω = 0.1.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content