2023-03-20T13:24:41Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/696662022-11-17T02:08:08Zhdl_2115_45007hdl_2115_116Instability of bound states of a nonlinear Schrodinger equation with a Dirac potentialLe Coz, StefanFukuizumi, ReikaFibich, GadiKsherim, BaruchSivan, Yonatan410We 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.Department of Mathematics, Hokkaido UniversityDepartmental Bulletin Paperapplication/pdfhttp://hdl.handle.net/2115/69666info:doi/10.14943/84007https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/69666/1/pre857.pdfHokkaido University Preprint Series in Mathematics8571432007engpublisher