Travelling waves for the Gross-Pitaevskii equation in dimension larger than two
type: Item
instance of: scholarly article
title: Travelling waves for the Gross-Pitaevskii equation in dimension larger than two
zbMATH Open document ID: 1054.35091
author: David Chiron
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-08-19T00:00:00Z
review text: The article deals with the existence of travelling vortex ring solutions \(\psi(x,t)=U(x_1-Ct, x_2,\dots,x_N)\) of the Gross-Pitaevskii equation \[ i\;\frac{\partial \psi}{\partial t}+\Delta \psi+(1-|\psi|^2)=0\quad (\psi:\mathbb{R}^N\times \mathbb{R}\to\mathbb{C},\quad N\geq 3). \] The equation on \(U\) reads \(iC\partial U/\partial x_1 =\Delta U + (1- | H|^2)U\) and the result is that there exists \(C_0>0\) such that a nontrivial solution \(U_C\) exists if \(0<C<C_0\) with \[ P(U_C)/2\pi \varepsilon^{1-N}\to |\mathbb{B}^{N-1}|,\quad E(U_C)/\pi\varepsilon^{2-N}|\log\varepsilon|\to\| S^{N-2}\| \] if \(C\to 0\), where \(\varepsilon\) is defined by \(C = (N-2)\varepsilon|\log\varepsilon|\). Here \[ E(\psi)=\tfrac12 \int|\nabla\psi|^2 +\tfrac12(1-|\psi|^2)\,dx,\quad P(\psi)=\text{Im}\int\psi\overline{\nabla\psi}\,dx \] are the energy and the momentum. The functional \(F(U) = E(U) - CP(U)\) with his mountain pass critical point are employed.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 76B47
Mathematics Subject Classification ID: 37K40
zbMATH DE Number: 2092999
zbMATH Keywords: travelling wave
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: vortex ring
zbMATH Keywords: mountain pass critical point
zbMATH Keywords: Ginzburg-Landau functional
reviewed by: Jan Chrastina
MaRDI profile type: Publication
full work available at URL: j.na.2003.10.028
OpenAlex ID: W4299649154
cites work: Variational convergence for functionals of Ginzburg-Landau type
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cites work: Ginzburg-Landau vortices
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cites work: Q4239008
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cites work: The Jacobian and the Ginzburg-Landau energy
DOI: 10.1016/J.NA.2003.10.028
description: scientific article; zbMATH DE number 2092999