Instability of bound states of a nonlinear Schr\\"odinger equation with a\n Dirac potential

Abstract

We study analytically and numerically the stability of the standing waves for\na nonlinear Schr\\"odinger equation with a point defect and a power type\nnonlinearity. A main difficulty is to compute the number of negative\neigenvalues of the linearized operator around the standing waves, and it is\novercome by a perturbation method and continuation arguments. Among others, in\nthe case of a repulsive defect, we show that the standing wave solution is\nstable in $\\hurad$ and unstable in $\\hu$ under subcritical nonlinearity.\nFurther we investigate the nature of instability: under critical or\nsupercritical nonlinear interaction, we prove the instability by blowup in the\nrepulsive case by showing a virial theorem and using a minimization method\ninvolving two constraints. In the subcritical radial case, unstable bound\nstates cannot collapse, but rather narrow down until they reach the stable\nregime (a {\\em finite-width instability}). In the non-radial repulsive case,\nall bound states are unstable, and the instability is manifested by a lateral\ndrift away from the defect, sometimes in combination with a finite-width\ninstability or a blowup instability.\n