Hydrodynamic Limit of the Gross-Pitaevskii Equation

Robert L. Jerrard; Daniel Spirn

doi:10.1080/03605302.2014.963604

Hydrodynamic Limit of the Gross-Pitaevskii Equation

Robert L. Jerrard, Daniel Spirn

Mathematics

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8 Scopus citations

Abstract

We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i∂_tu = Δu + ϵ⁻²u(1 − |u|²) on ℝ²with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ϵ. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [19].

Original language	English (US)
Pages (from-to)	135-190
Number of pages	56
Journal	Communications in Partial Differential Equations
Volume	40
Issue number	2
DOIs	https://doi.org/10.1080/03605302.2014.963604
State	Published - Feb 1 2015

Bibliographical note

Publisher Copyright:
© 2015, Taylor & Francis Group, LLC.

Keywords

Euler equations
Gross-Pitaevskii
Point vortex method
Vortices

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abstract = "We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i∂tu = Δu + ϵ−2u(1 − |u|2) on ℝ2with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ϵ. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [19].",

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AB - We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i∂tu = Δu + ϵ−2u(1 − |u|2) on ℝ2with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ϵ. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [19].

KW - Euler equations

KW - Gross-Pitaevskii

KW - Point vortex method

KW - Vortices

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Hydrodynamic Limit of the Gross-Pitaevskii Equation

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