Solitary Wave Dynamics in an External Potential

Fröhlich, Jürg; Gustafson, Stephen J.; Jonsson, B. L. G.; Sigal, Israel Michael

doi:10.1007/s00220-004-1128-1

Cited by 148 publications

(236 citation statements)

References 43 publications

(72 reference statements)

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“…In the end, we mention that the rigorous derivation of the Newton's particle equation for slow dynamics of a bright soliton in an external potential has been reported recently in [6,13,16]. Derivation of its counterpart (1.7) for slow dynamics of a dark soliton is an open problem of analysis.…”

Section: Numerical Simulations Of the Gp Equationmentioning

confidence: 89%

“…Let γ = Γ(ǫ) be an eigenvalue of the generalized eigenvalue problem (3.22) for sufficiently small ǫ, such that w ∈ L 2 (R), ImΓ ≤ 0, and lim ǫ→0 Γ(ǫ) = 0. Then, the spectral problem (3.19) for sufficiently small ǫ admits two eigenvalues λ = ±Λ(ǫ) = ± −Γ(ǫ) with (u, w) ∈ L 2 (R, C 2 ), ReΛImΛ ≥ 0, and lim (ii) (u, w) is infinitely smooth with respect to ǫ 1/2 and 13) up to an arbitrary multiplicative factor.…”

Section: Definition 45mentioning

confidence: 99%

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Dark solitons in external potentials

Pelinovsky

Kevrekidis

2007

Z. angew. Math. Phys.

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We consider the persistence and stability of dark solitons in the Gross-Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation with cubic nonlinearity. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton's particle law for its position.

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Section: Numerical Simulations Of the Gp Equationmentioning

confidence: 89%

Section: Definition 45mentioning

confidence: 99%

Dark solitons in external potentials

Pelinovsky

Kevrekidis

2007

Z. angew. Math. Phys.

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“…If φ ∈ S(R) and h(x, t) is the solution to (A.1) with initial data h(x, 0) = φ(x), then h(x, t) H 2 x ≤ c (A. 5) xh(x, t) L 2 x ≤ c t (A.6) x∂ x h(x, t) L 2 ≤ c t 3/2 (A.7)…”

Section: Appendix : Properties Of Free Nonlinear Evolutionmentioning

confidence: 99%

Fast Soliton Scattering by Delta Impurities

Holmer

Marzuola

Zworski

2007

Commun. Math. Phys.

139

214

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Abstract. We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L 2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

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“…The heuristic insight that at high velocities "linear scattering" by the external potential should dominate the partition of mass is certainly present there. In the mathematical literature the dynamics of solitons in the presence of external potentials has been studied in high velocity or semiclassical limits following the work of Floer and Weinstein [5], and Bronski and Jerrard [1] -see [6] for recent results and a review of the subject. Roughly speaking, the soliton evolves according to the classical motion of a particle in the external potential.…”

Section: Theorem 2 Under the Hypothesis Of Theorem 1 And Formentioning

confidence: 99%

“…The possible novelty in (6) and (7) lies in seeing quantum effects of the external potential strongly affecting soliton dynamics. As shown in Fig.…”

Section: Theorem 2 Under the Hypothesis Of Theorem 1 And Formentioning

confidence: 99%