Soliton dynamics in a potential

Bronski, Jared C.; Jerrard, Robert L.

doi:10.4310/mrl.2000.v7.n3.a7

Cited by 82 publications

(130 citation statements)

References 13 publications

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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: 99%

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: 99%

Dark solitons in external potentials

Pelinovsky

Kevrekidis

2007

Z. angew. Math. Phys.

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“…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 [8], and Bronski and Jerrard [4]. Roughly speaking, the soliton evolves according to the classical motion of a particle in the external potential.…”

Section: S(λ)mentioning

confidence: 99%

Soliton Splitting by External Delta Potentials

2007

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Abstract. We show that a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons.We also show that the total transmitted mass, that is the square of the L 2 norm of the solution restricted on the transmitted side of the delta potential is in good agreement with the quantum transmission rate of the delta potential.

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“…Later Bronski and Jerrard [2000] proved a similar theorem in the case of a power nonlinearity, and then more general nonlinearities were treated by Fröhlich, Tsai, and Yau [2002] and by Fröhlich, Gustafson, Jonsson, and Sigal [2004]. More recently Jonsson, Fröhlich, Gustafson, and Sigal [2006] have extended the validity of the effective dynamics to longer time in the case of a confining potential V , and Abou Salem [2008] has treated the case of a potential V that is permitted to vary in time.…”

Section: Then Formentioning

confidence: 98%

Solitary waves for the Hartree equation with a slowly varying potential

Datchev¹,

Ventura²

2010

Pacific J. Math.

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We study the Hartree equation with a slowly varying smooth potential, V (x) = W (hx), and with an initial condition that is ε ≤ √ h away in H 1 from a soliton. We show that up to time |log h|/ h and errors of size ε + h 2 in H 1 , the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian. This result is based on methods of Holmer and Zworski, who prove a similar theorem for the Gross-Pitaevskii equation, and on spectral estimates for the linearized Hartree operator recently obtained by Lenzmann. We also provide an extension of the result of Holmer and Zworski to more general initial conditions.

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Soliton dynamics in a potential

Cited by 82 publications

References 13 publications

Dark solitons in external potentials

Dark solitons in external potentials

Soliton Splitting by External Delta Potentials

Solitary waves for the Hartree equation with a slowly varying potential

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