Semiclassical Propagation of Coherent States for the Hartree Equation

Athanassoulis, Agissilaos; Paul, Thierry; Pezzotti, Federica; Pulvirenti, Mario

doi:10.1007/s00023-011-0115-2

Cited by 9 publications

(25 citation statements)

References 21 publications

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“…2.2. The same ideas can be used to derive the exact, closed nonlinear Wigner equation corresponding to a wide range of nonlinear dispersive equations, including the MMT equation (Majda et al 1997;Cousins and Sapsis 2014), Ginzburg-Landau models (Aranson and Kramer 2002), Hartree equations (Athanassoulis et al 2011), and the Szegö equation (Pocovnicu 2011). In particular, the Whitham pseudodifferential operator (Whitham 1967) can be treated in the place of the Laplacian, so that fully realistic dispersion is used.…”

Section: Observe That W [U(t)](x K) Is Real Valued For Any Complex Vmentioning

confidence: 99%

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Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

Athanassoulis

Sapsis

2017

J. Ocean Eng. Mar. Energy

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In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose's method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin-Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength λ 0 as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to δ spectra, where the standard Benjamin-Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes.B A. G. Athanassoulis

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Section: Observe That W [U(t)](x K) Is Real Valued For Any Complex Vmentioning

confidence: 99%

“…This is due to the pseudodifferential calculus available for the Wigner transform (Athanassoulis et al 2011;Gérard et al 1997). …”

Section: Introductionmentioning

confidence: 99%

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

Athanassoulis

Sapsis

2017

J. Ocean Eng. Mar. Energy

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“…Technically, the reason is two-fold. First, in [3,9], it is assumed that ∇ K (0) = 0, so the first line in (9.4) vanishes. Then, the second line in (9.4) accounts for the presence of two wave packets, and measures some coupling through a phase modulation; it vanishes in the case of a single wave packet.…”

Section: Case α = 1/2mentioning

confidence: 99%

Interaction of Coherent States for Hartree Equations

Carles

2012

Arch Rational Mech Anal

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We consider the Hartree equation with a smooth kernel and an external potential, in the semiclassical regime. We analyze the propagation of two initial wave packets and show different possible effects of the interaction, according to the size of the nonlinearity in terms of the semiclassical parameter. We show three different sorts of nonlinear phenomena. In each case, the structure of the wave as a sum of two coherent states is preserved. However, the envelope and the center (in phase space) of these two wave packets are affected by nonlinear interferences, which are described precisely.

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“…Besides, E has a (kind of) classical analogue: Theorem 2.5 provides an upper bound for the classical quadratic MongeKantorovich distance between the solution of the Vlasov equation and the Husimi function of the solution of the Hartree equation, up to terms of order O( ) which vanish in the semiclassical limit. The case of pure states is treated in [2] for initial data given by coherent states, and the solution of Hartree's equation is computed at leading order in terms of the linearization of the "Vlasov flow". Most likely, the method used in [2] can be extended to any order in the expansion of the Hartree solution in powers of 1/2 .…”

Section: Introductionmentioning

confidence: 99%

“…The case of pure states is treated in [2] for initial data given by coherent states, and the solution of Hartree's equation is computed at leading order in terms of the linearization of the "Vlasov flow". Most likely, the method used in [2] can be extended to any order in the expansion of the Hartree solution in powers of 1/2 . Our first main result in the present paper, Theorem 2.5, bears on a quantitative estimate for the classical limit of Hartree's equation leading to the Vlasov equation, i.e.…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

Golse

Paul

2016

Arch Rational Mech Anal

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113

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Abstract. In this paper, we establish (1) the classical limit of the Hartree equation leading to the Vlasov equation, (2) the classical limit of the N -body linear Schrödinger equation uniformly in N leading to the N -body Liouville equation of classical mechanics and (3) the simultaneous mean-field and classical limit of the N -body linear Schrödinger equation leading to the Vlasov equation. In all these limits, we assume that the gradient of the interaction potential is Lipschitz continuous. All our results are formulated as estimates involving a quantum analogue of the Monge-Kantorovich distance of exponent 2 adapted to the classical limit, reminiscent of, but different from the one defined in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343 (2016), 165-205]. As a by-product, we also provide bounds on the quadratic MongeKantorovich distances between the classical densities and the Husimi functions of the quantum density matrices.

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Semiclassical Propagation of Coherent States for the Hartree Equation

Cited by 9 publications

References 21 publications

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

Localized instabilities of the Wigner equation as a model for the emergence of Rogue Waves

Interaction of Coherent States for Hartree Equations

The Schrödinger Equation in the Mean-Field and Semiclassical Regime

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