Dept. Math, Hokkaido Univ. EPrints Server

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);

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Abstract

We 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 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.

Item Type:Preprint
Uncontrolled Keywords:Instability, Collapse, Solitary waves, Nonlinear waves, Dirac delta, Lattice defects
Subjects:65-xx NUMERICAL ANALYSIS
35-xx PARTIAL DIFFERENTIAL EQUATIONS
ID Code:1710