期刊
PHYSICA D-NONLINEAR PHENOMENA
卷 237, 期 8, 页码 1103-1128出版社
ELSEVIER
DOI: 10.1016/j.physd.2007.12.004
关键词
instability; collapse; solitary waves; nonlinear waves; Dirac delta; lattice defects
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 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 H-rad(1)(R) and unstable in H-1(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. (C) 2007 Elsevier B.V. All rights reserved.
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