Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

Sivan, Yonatan; Fibich, Gadi; Ilan, Boaz; Weinstein, Michael I.

doi:10.1103/physreve.78.046602

Cited by 76 publications

(106 citation statements)

References 88 publications

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“…In the case of the bright solitons of the form u(x,t) = u s (x)e i t , the Vakhitov-Kolokolov criterion states that if the corresponding energy Hessian has only one negative eigenvalue, then the soliton is stable if dN/d > 0 and unstable otherwise [21][22][23]. Here N = |u| 2 dx; depending on the physical context, N is referred to as the number of particles contained in the soliton or the total power of the optical beam.…”

Section: U + δU = R[u(xt); Xt]mentioning

confidence: 99%

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

et al. 2011

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We investigate the dynamics of traveling oscillating solitons of the cubic nonlinear Schrödinger (NLS) equation under an external spatiotemporal forcing of the form f (x,t) = a exp[iK(t)x]. For the case of time-independent forcing, a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where K(t) is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" p(v), where p(t) and v(t) are the normalized momentum and the velocity of the soliton, has a section with a negative slope. In the case of a constant K and zero damping, we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.

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Section: U + δU = R[u(xt); Xt]mentioning

confidence: 99%

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

et al. 2011

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“…The bifurcation mechanism analyzed herein (bifurcation from simple eigenvalues) is one of the main types for generation of nonlinear states, as it has been highlighted in Ref. 52 (see Section II 52 , pg. 046602-2).…”

Section: Discussionmentioning

confidence: 99%

“…The change of slope manifests instability according to the slope criterion, as suggested by the general stability criteria summarized in Section III of Ref. 52 . It should be noted that instability occurs when either the slope criterion (well known also as a Vakhitov-Kolokolov (VK) criterion 43 ), or the spectral condition 39,40 fails.…”

Section: B Continuation From the Anti-continuum Limitmentioning

confidence: 99%

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap

Achilleos

Theocharis

Kevrekidis

et al. 2011

Journal of Mathematical Physics

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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 inside the continuum regime. Continuation from the anti-continuum limit is also considered, and a rich bifurcation structure is revealed.

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“…In the case of the bright solitons of the form u(x,t) = u s (x)e i t , the Vakhitov-Kolokolov criterion states that if the corresponding energy Hessian has only one negative eigenvalue, then the soliton is stable if the variation of the norm dN/d > 0 and unstable otherwise [1][2][3]. Here, N = |u| 2 dx is denoted as the norm, or as the number of particles contained in the soliton.…”

Section: Introductionmentioning

confidence: 99%

Soliton stability criterion for generalized nonlinear Schrödinger equations

2015

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A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p (v) < 0 is a sufficient condition for instability, while p (v) > 0 is a necessary condition for stability; here, v is the soliton velocity and p = P /N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate p(v) exactly for several classes and cases of the generalized NLSE: a soliton moving in a real potential, in particular a time-dependent ramp potential, and a time-dependent confining quadratic potential, where the nonlinearity in the NLSE also has a time-dependent coefficient. Moreover, we investigate a logarithmic and a cubic NLSE with a time-independent quadratic potential well. In the latter case, there is a bisoliton solution that consists of two solitons with asymmetric shapes, forming a bound state in which the shapes and the separation distance oscillate. Finally, we consider a cubic NLSE with parametric driving. In all cases, the p(v) curve is calculated either analytically or numerically, and the stability criterion is confirmed.

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Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

Cited by 76 publications

References 88 publications

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap

Soliton stability criterion for generalized nonlinear Schrödinger equations

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