Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one. (English) Zbl 1421.35299
Summary: The paper discusses nonlinear singular delta-type perturbations of the fractional Schrödinger equation \(\imath\partial_t\psi=(-\Delta)^s\psi\), with \(s\in (\frac{1}{2}, 1]\), in dimension one. In particular, we investigate local and global well-posedness (in a strong sense), conservation laws and the existence of blow-up solutions and standing waves.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35R11 | Fractional partial differential equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 34K37 | Functional-differential equations with fractional derivatives |
| 35B44 | Blow-up in context of PDEs |
| 35B25 | Singular perturbations in context of PDEs |
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