Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

Gomez, Christophe

doi:10.1007/s00220-013-1711-4

Cited by 7 publications

(20 citation statements)

References 35 publications

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“…10 Proof of estimate (21) We use here the notation of sections 1 and 3. The core of the proof is the following lemma:…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…It is not always the case in practice, as is pointed out in [12,23,31] for geophysical problems, wave propagation in turbulent atmosphere, or medical imaging. This has then stimulated recent mathematical works on wave propagation in random media with long-range dependence [2,18,20,21,22,25,26]. It is shown there that the wave dynamics in such media can be in great contrast with that of waves in media with rapidly decaying correlations.…”

Section: Introductionmentioning

confidence: 99%

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Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

Gomez

Pinaud

2017

Arch Rational Mech Anal

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This work is devoted to the asymptotic analysis of high frequency wave propagation in random media with long-range dependence. We are interested in two asymptotic regimes, that we investigate simultaneously: the paraxial approximation, where the wave is collimated and propagates along a privileged direction of propagation, and the whitenoise limit, where random fluctuations in the background are well approximated in a statistical sense by a fractional white noise. The fractional nature of the fluctuations is reminiscent of the long-range correlations in the underlying random medium. A typical physical setting is laser beam propagation in turbulent atmosphere. Starting from the high frequency wave equation with fast non-Gaussian random oscillations in the velocity field, we derive the fractional Itô-Schrödinger equation, that is a Schrödinger equation with potential equal to a fractional white noise. The proof involves a fine analysis of the backscattering and of the coupling between the propagating and evanescent modes. Because of the long-range dependence, classical diffusion-approximation theorems for equations with random coefficients do not apply, and we therefore use moment techniques to study the convergence.where c 0 is the background velocity (constant for simplicity), and the random field V (z, x), with a stationary covariance, models fluctuations around c 0 in the slab (0, L z ) × R 2 . The parameters σ and l c represent the amplitude and the correlation length of the fluctuations. The main assumption on V is that it satisfies the long-range property in the z-direction, which is translated mathematically into a bounded non-integrable autocorrelation function which decreases at infinity only as Preliminaries and main resultsThroughout this work, we will use the following conventions for the Fourier transform:f denotes the Fourier transform w.r.t. the variable t as in (6), andf that w.r.t. t and x, f (ω, κ) = 1 (2π) 3 f (t, x)e i(ωt+κ·x) dtdx with f (t, x) = f (ω, κ)e −i(ωt+κ·x) dωdκ.

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“…10 Proof of estimate (21) We use here the notation of sections 1 and 3. The core of the proof is the following lemma:…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

Gomez

Pinaud

2017

Arch Rational Mech Anal

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“…In order to be efficient, imaging and communication algorithms require insight about how the wave is affected by the rough medium fluctuations. In view of its potential for applications, mathematical description of wave propagation in multiscale random media with long-range correlations has attracted a lot of interest over the last decade [3,11,14,15,17,18,21].…”

Section: Introductionmentioning

confidence: 99%

“…Multiscale random medium with long-range correlations produce stochastic effects on the waves which are very different from the ones produced by perturbations varying on a well defined microscale and with mixing properties [9]. Wave propagation in random media with long-range correlations has already been considered in one-dimensional propagation media [11,18] or open media under the paraxial approximation [3,7,8,14,15]. In these contexts, it has been observed that stochastic effects appear at different propagation scales, all the stochastic effects do not appear at the same time, which is in contrast with perturbations with mixing properties.…”

Section: Introductionmentioning

confidence: 99%

“…Perturbations with long-range correlations induce first a phase modulation on the waves, a modularion that is driven by a single standard fractional Brownian motion, which does not depend on the frequency band [3,18]. For larger propagation distances, the random phase modulation starts to oscillate very fast to produce anomalous diffusion phenomena [11,14,15]. Here, we follow this line of research by considering the full wave equation in a waveguide with a continuous multiscale medium modeled through a stochastic process with long-range correlations.…”

Section: Introductionmentioning

confidence: 99%

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Wave propagation in random waveguides with long-range correlations

Gomez

Sølna

2018

Communications in Mathematical Sciences

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In this paper we consider an acoustic waveguide with random fluctuations of its sound speed profile. These random perturbations are assumed to have long-range correlation properties. In waveguides a monochromatic wave can be decomposed in propagating modes and evanescent modes, and the random perturbation couples all these modes. Using asymptotic analysis of the modecoupling mechanism we give a description of the effects that the random fluctuations with long-rang statistical properties have on the pulse propagation.

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Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations

Gomez¹,

Pinaud²

2014

ESAIM: M2AN

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This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial condition, and the statistical properties of the random medium. We show that the splitting scheme captures these regimes in a statistical sense for a time stepsize independent of the frequency. Résumé. Nous nous intéressons au comportement asymptotique d'un schéma de time-splitting pour l'équation de Schrödinger avec potential aléatoire de faible amplitude, oscillant rapidement en temps et en espace, et présentant des corrélations longue portée. Cetteéquation décrit par exemple la propagation d'une onde dans un milieu aléatoire dans le cadre de l'approximation paraxiale. La limite haute-fréquence de l'équation de Schrödinger mèneà différents régimes selon la distance de propagation, la fréquence d'oscillation de la condition initiale, et la statistique du milieu aléatoire. Nous montrons que le schéma de splitting capture statistiquement ces régimes asymptotiques pour un pas de discrétisation en temps indépendant de la fréquence d'oscillation.

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Wave Decoherence for the Random Schrödinger Equation with Long-Range Correlations

Cited by 7 publications

References 35 publications

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

Fractional White-Noise Limit and Paraxial Approximation for Waves in Random Media

Wave propagation in random waveguides with long-range correlations

Asymptotics of a Time-Splitting Scheme for the Random Schrödinger Equation with Long-Range Correlations

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