Numerically induced chaos in the nonlinear Schrödinger equation

Herbst, B. M.; Ablowitz, Mark J.

doi:10.1103/physrevlett.62.2065

Cited by 151 publications

(91 citation statements)

References 17 publications

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“…There exists apparently unassailable evidence both that the outer Solar System is chaotic 1, 2 , and that it is not 3, 4, 5 . The discrepancy is particularly disturbing given that computed chaos is sometimes due to numerical artifacts 6,7 . In this paper we discount the possibility of numerical artifacts, and demonstrate that the discrepancy seen between various investigators is real.…”

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confidence: 99%

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Is the outer Solar System chaotic?

Hayes¹

2007

Nature Phys

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One-sentence summary: Current observational uncertainty in the positions of the Jovian planets precludes deciding whether or not the outer Solar System is chaotic. 100 word technical summary: The existence of chaos in the system of Jovian planets has been in question for the past 15 years. Various investigators have found Lyapunov times ranging from about 5 millions years upwards to infinity, with no clear reason for the discrepancy. In this paper, we resolve the issue. The position of the outer planets is known to only a few parts in 10 million. We show that, within that observational uncertainty, there exist Lyapunov timescales in the full range listed above. Thus, the "true" Lyapunov timescale of the outer Solar System cannot be resolved using current observations. 100 word summary for general public: The orbits of the inner planets (Mercury, Venus, Earth, and Mars) are practically stable in the sense that none of them will collide or be ejected from the Solar System for the next few billion years. However, their orbits are chaotic in the sense that we cannot predict their angular positions within those stable orbits for more than about 20 million years. The picture is less clear for the outer planets (Jupiter, Saturn, Uranus and Neptune). Again their orbits are practically stable, but it is not known for how long we can accurately predict their positions within those orbits.The existence of chaos among the Jovian planets is a contested issue. There exists apparently unassailable evidence both that the outer Solar System is chaotic 1, 2 , and that it is not 3, 4, 5 . The discrepancy is particularly disturbing given that computed chaos is sometimes due to numerical artifacts 6,7 . In this paper we discount the possibility of numerical artifacts, and demonstrate that the discrepancy seen between various investigators is real. It is caused by observational uncertainty in the orbital positions of the Jovian planets, which is currently a few parts in 10 million. Within that observational uncertainty, there exist clearly chaotic 1

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“…One one hand, chaos can be a numerical artifact 6 , even in an n-body integration 7,17 . Furthermore, numerous careful integrations of the outer Solar System have been performed specifically to ensure the accuracy and convergence of the numerical results 4,5 , and these give a clear indication of no chaos.…”

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confidence: 99%

Is the outer Solar System chaotic?

Hayes¹

2007

Nature Phys

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“…It is believed (and proved for a very large class of equations) that integrable equations admit integrable discretizations which preserve the unique features of these equations (infinite number of conservation laws, solitons, transformations generating explicit solutions, etc.) [2,4]. It would be interesting to compare standard integrable discretizations of KdV and modKdV equations (see, e.g., [14]) with discretizations constructed in order to simulate in the best way one soliton solutions ( [1]).…”

Section: Discussionmentioning

confidence: 99%

On simulations of the classical harmonic oscillator equation by difference equations

Cieśliński

Ratkiewicz

2006

Adv. Differ Equ.

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We discuss the discretizations of the second-order linear ordinary diffrential equations with constant coefficients. Special attention is given to the exact discretization because there exists a difference equation whose solutions exactly coincide with solutions of the corresponding differential equation evaluated at a discrete sequence of points. Such exact discretization can be found for an arbitrary lattice spacing.

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“…Recent developments in the study of the DNLS equation are reviewed in Refs [13,14]. However, the standard DNLS equation is nonintegrable [15,16] and does not exhibit exact soliton solutions, though it can be derived as a discretization of the integrable continuous NLS equation. Hence, numerical methods are generally used to investigate nonlinear lattice dynamics on the basis of the DNLS equation.…”

Section: Introductionmentioning

confidence: 99%

Perturbation-induced radiation by the Ablowitz-Ladik soliton

Doktorov

Matsuka²,

Rothos³

2003

Phys. Rev. E

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An efficient formalism is elaborated to analytically describe dynamics of the Ablowitz-Ladik soliton in the presence of perturbations. This formalism is based on using the Riemann-Hilbert problem and provides the means of calculating evolution of the discrete soliton parameters, as well as shape distortion and perturbation-induced radiation effects. As an example, soliton characteristics are calculated for linear damping and quintic perturbations.

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Numerically induced chaos in the nonlinear Schrödinger equation

Cited by 151 publications

References 17 publications

Is the outer Solar System chaotic?

Is the outer Solar System chaotic?

On simulations of the classical harmonic oscillator equation by difference equations

Perturbation-induced radiation by the Ablowitz-Ladik soliton

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