On the stability of self-similar solutions of 1D cubic Schrödinger equations

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On the stability of self-similar solutions of 1D cubic Schrödinger equations. / Gutierrez, S.; Vega, L.

In: Mathematische Annalen, Vol. 356, No. 1, 05.2013, p. 259-300.

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@article{6975b573cb85469eb13cc4ac8c2017d2,
title = "On the stability of self-similar solutions of 1D cubic Schr{\"o}dinger equations",
abstract = "In this paper we will study the stability properties of self-similar solutions of 1D cubic NLS equations with time-dependent coefficients of the form i u + u + u/2 ({pipe}u{pipe} - A/t) = 0, A ∈ ℝ. The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation iv + v + v/2t({pipe}v{pipe}-A) = 0. As a by-product of our results we prove that Eq. (0. 1) is well-posed in appropriate function spaces when the initial datum is given by u(0, x) = z p. v 1/x for some values of z ∈ ℂ\ {0}, and A is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution. {\textcopyright} 2012 Springer-Verlag.",
author = "S. Gutierrez and L. Vega",
note = "38 pages, 8 figures",
year = "2013",
month = may,
doi = "10.1007/s00208-012-0847-4",
language = "English",
volume = "356",
pages = "259--300",
journal = "Mathematische Annalen",
issn = "0025-5831",
publisher = "Springer",
number = "1",

}

RIS

TY - JOUR

T1 - On the stability of self-similar solutions of 1D cubic Schrödinger equations

AU - Gutierrez, S.

AU - Vega, L.

N1 - 38 pages, 8 figures

PY - 2013/5

Y1 - 2013/5

N2 - In this paper we will study the stability properties of self-similar solutions of 1D cubic NLS equations with time-dependent coefficients of the form i u + u + u/2 ({pipe}u{pipe} - A/t) = 0, A ∈ ℝ. The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation iv + v + v/2t({pipe}v{pipe}-A) = 0. As a by-product of our results we prove that Eq. (0. 1) is well-posed in appropriate function spaces when the initial datum is given by u(0, x) = z p. v 1/x for some values of z ∈ ℂ\ {0}, and A is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution. © 2012 Springer-Verlag.

AB - In this paper we will study the stability properties of self-similar solutions of 1D cubic NLS equations with time-dependent coefficients of the form i u + u + u/2 ({pipe}u{pipe} - A/t) = 0, A ∈ ℝ. The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation iv + v + v/2t({pipe}v{pipe}-A) = 0. As a by-product of our results we prove that Eq. (0. 1) is well-posed in appropriate function spaces when the initial datum is given by u(0, x) = z p. v 1/x for some values of z ∈ ℂ\ {0}, and A is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution. © 2012 Springer-Verlag.

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-84875720183&partnerID=8YFLogxK

U2 - 10.1007/s00208-012-0847-4

DO - 10.1007/s00208-012-0847-4

M3 - Article

VL - 356

SP - 259

EP - 300

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 1

ER -