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Hokkaido University Preprint Series in Mathematics >
Instability of bound states of a nonlinear Schrodinger equation with a Dirac potential
| Title: | Instability of bound states of a nonlinear Schrodinger equation with a Dirac potential |
| Authors: | Le Coz, Stefan Browse this author | | Fukuizumi, Reika Browse this author | | Fibich, Gadi Browse this author | | Ksherim, Baruch Browse this author | | Sivan, Yonatan Browse this author |
| Issue Date: | 2007 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 857 |
| Start Page: | 1 |
| End Page: | 43 |
| Abstract: | We study analytically and numerically the stability of the standing waves for a nonlinear Schr¨odinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it 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 H1 rad(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 blow-up 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 non-radial 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. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/69666 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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