Instability of bound states of a nonlinear Schrodinger equation with a Dirac potential
Preprint Series # 857 Le Coz, Stefan and Fukuizumi, Reika and Fibich, Gadi and Ksherim, Baruch and Sivan, Yonatan Instability of bound states of a nonlinear Schrodinger equation with a Dirac potential. (2007); AbstractWe study analytically and numerically the stability of the standing waves
for a nonlinear Schrodinger 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 H^1_rad
and unstable in H^1 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 finitewidth 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 finitewidth instability or a
blowup instability.
