An instability mechanism for radially symmetric standing waves of a nonlinear Schrödinger equation. (English) Zbl 0656.35108
Author’s summary: A condition is proved for the spectrum of nonlinear Schrödinger equations linearised at a standing wave to have a positive eigenvalue. The standing waves considered are radially symmetric ones in higher space dimensions. The instability result is applied to show that if there are multiple asymptotically positive, nondegenerate waves with a fixed number of zeroes, then there is an unstable one. The techniques used are dynamical systems arguments and involve a shooting argument in the space of Lagrangian planes.
Reviewer: N.Jacob
MSC:
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
| 35Q99 | Partial differential equations of mathematical physics and other areas of application |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
Keywords:
spectrum; nonlinear Schrödinger equations; standing wave; positive eigenvalue; multiple asymptotically positive; nondegenerate waves; dynamical systems; argumentReferences:
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