Stability properties of standing waves for NLS equations with the \(\delta^\prime\)-interaction. (English) Zbl 1490.35389
Summary: We study the orbital stability of standing waves with discontinuous bump-like profile for the nonlinear Schrödinger model with the repulsive \(\delta^\prime\)-interaction on the line. We consider the model with power non-linearity. In particular, it is shown that such standing waves are unstable in the energy space under some restrictions for parameters. The use of extension theory of symmetric operators by Krein-von Neumann is fundamental for estimating the Morse index of self-adjoint operators associated with our stability study. Moreover, for this purpose we use Sturm oscillation results and analytic perturbation theory. The Perron-Frobenius property for the repulsive \(\delta^\prime \)-interaction is established.
The arguments presented in this investigation have prospects for the study of the stability of stationary waves solutions of other nonlinear evolution equations with point interactions.
The arguments presented in this investigation have prospects for the study of the stability of stationary waves solutions of other nonlinear evolution equations with point interactions.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B20 | Perturbations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
nonlinear Schrödinger equation; orbital stability; bump solutions; self-adjoint extension; deficiency indices; Sturm-Liouville theoryReferences:
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