Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Stefan Le Coz, Reika Fukuizumi, Gadi Fibich, Baruch Ksherim, Yonatan Sivan

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Abstract

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in Hrad1 (R) and unstable in H1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

Original languageEnglish
Pages (from-to)1103-1128
Number of pages26
JournalPhysica D: Nonlinear Phenomena
Volume237
Issue number8
DOIs
StatePublished - 15 Jun 2008
Externally publishedYes

Bibliographical note

Funding Information:
The authors are grateful to Louis Jeanjean for fruitful discussions and helpful advice. Reika Fukuizumi would like to thank Shin-ichi Shirai and Clément Gallo for useful discussions particularly about Section 2 . Stefan Le Coz wishes to thank Mariana Hărăguş for fruitful discussions. The research of Gadi Fibich, Baruch Ksherim and Yonatan Sivan was partially supported by grant 2006-262 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. Reika Fukuizumi was supported by JSPS Postdoctoral Fellowships for Research Abroad.

Funding

The authors are grateful to Louis Jeanjean for fruitful discussions and helpful advice. Reika Fukuizumi would like to thank Shin-ichi Shirai and Clément Gallo for useful discussions particularly about Section 2 . Stefan Le Coz wishes to thank Mariana Hărăguş for fruitful discussions. The research of Gadi Fibich, Baruch Ksherim and Yonatan Sivan was partially supported by grant 2006-262 from the United States–Israel Binational Science Foundation (BSF), Jerusalem, Israel. Reika Fukuizumi was supported by JSPS Postdoctoral Fellowships for Research Abroad.

FundersFunder number
Japan Society for the Promotion of Science
United States-Israel Binational Science Foundation

    Keywords

    • Collapse
    • Dirac delta
    • Instability
    • Lattice defects
    • Nonlinear waves
    • Solitary waves

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