Nonlinear Schrödinger equations with exceptional potentials
type: Item
instance of: scholarly article
title: Nonlinear Schrödinger equations with exceptional potentials
zbMATH Open document ID: 1427.35257
published in: Journal of Differential Equations
publication date: 2018-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1707.00503
review text: The author considers an initial boundary value problem, associated to a cubic nonlinear Schrödinger equation in which appears a real-valued given function V, as an exceptional potential. The nonlinear Schrödinger equation is related to the Ginzburg-Landau equation of superconductivity. The results from this paper are extensions of the results from paper [\textit{I. P. Naumkin}, J. Math. Phys. 57, No. 5, 051501, 31 p. (2016; Zbl 1343.81206)], of the same author. In previous paper the author consider a cubic nonlinear Schrödinger equation with a generic potential and proved the existence of unique continuous solutions which decay for long times as solutions to a linear free Schrödinger equation. Also, it is obtained a large time asymptotic representation for these solutions, which have a modified character and it is proved that as in the case of the free NLS equation, the cubic nonlinearity, in the case of generic potentials is critical from the point of view of large time asymptotic behavior. In present paper the author obtained also existence of unique continuous solutions, but by imposing the parity condition on the exceptional potential and on the initial data in order to control these functions The difference between the generic and exceptional cases becomes relevant when the small-energy behavior of the scattering coefficients and of the wave functions is considered. It is important to note that the author's considerations are based on a spectral theorem for the perturbed linear Schrödinger operator and a factorization technique, that allows the control on the resonant nonlinear term.
reviewed by: Marin I. Marin
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q56
Mathematics Subject Classification ID: 35P25
zbMATH DE Number: 6933423
zbMATH Keywords: cubic nonlinear Schrödinger equation
zbMATH Keywords: exceptional potential
zbMATH Keywords: perturbed linear Schrödinger operator
zbMATH Keywords: factorization technique
zbMATH Keywords: wave functions
author: I. P. Naumkin
MaRDI profile type: Publication
OpenAlex ID: W2963921499
arXiv ID: 1707.00503
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DOI: 10.1016/J.JDE.2018.06.016
description: scientific article