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.
|Number of pages||26|
|Journal||Physica D: Nonlinear Phenomena|
|State||Published - 15 Jun 2008|
Bibliographical noteFunding 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.
- Dirac delta
- Lattice defects
- Nonlinear waves
- Solitary waves