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# Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation

Demanet, Laurent and Schlag, Wilhelm (2006) Numerical verification of a gap condition for a linearized nonlinear Schrödinger equation. Nonlinearity, 19 (4). pp. 829-852. ISSN 0951-7715. https://resolver.caltech.edu/CaltechAUTHORS:DEMnonlin06

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## Abstract

We make a detailed numerical study of the spectrum of two Schrödinger operators L± arising from the linearization of the supercritical nonlinear Schrödinger equation (NLS) about the standing wave, in three dimensions. This study was motivated by a recent result of the second author on the conditional asymptotic stability of solitary waves in the case of a cubic nonlinearity. Underlying the validity of this result is a spectral condition on the operators L±, namely that they have no eigenvalues nor resonances in the gap (a region of the positive real axis between zero and the continuous spectrum), which we call the gap property. The present numerical study verifies this spectral condition and shows further that the gap property holds for NLS exponents of the form 2 β + 1, as long as β* < β ≤ 1, where $\begin{equation*}\beta_{\ast} = 0.913\,958\,905 \pm 1e-8.\end{equation*}$ Our strategy consists of rewriting the original eigenvalue problem via the Birman–Schwinger method. From a numerical analysis viewpoint, our main contribution is an efficient quadrature rule for the kernel 1/|x - y| in {\mathbb R}^3 , i.e. proved spectrally accurate. As a result, we are able to give similar accuracy estimates for all our eigenvalue computations. We also propose an improvement in Petviashvili's iteration for the computation of standing wave profiles which automatically chooses the radial solution. All our numerical experiments are reproducible. The Matlab code can be downloaded from http://www.acm.caltech.edu/~demanet/NLS/.

Item Type: Article © Institute of Physics and IOP Publishing Limited 2006. Print publication: Issue 4 (April 2006); Received 9 September 2005, in final form 22 December 2005; Published 28 February 2006 Recommended by F Merle The second author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship. The authors would like to thank Dmitry Pelinovsky and Lexing Ying for interesting discussions as well as the referees for their very careful reading of the paper and many helpful suggestions. 4 CaltechAUTHORS:DEMnonlin06 https://resolver.caltech.edu/CaltechAUTHORS:DEMnonlin06 http://dx.doi.org/10.1088/0951-7715/19/4/004 No commercial reproduction, distribution, display or performance rights in this work are provided. 3398 CaltechAUTHORS Archive Administrator 05 Jun 2006 02 Oct 2019 23:02

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