On the infinitely many standing waves of some nonlinear Schrödinger equations. (English) Zbl 0594.35003
Nonlinear systems of partial differential equations in applied mathematics, Proc. SIAM-AMS Summer Semin., Santa Fe/N.M. 1984, Lect. Appl. Math. 23, Pt. 2, 3-21 (1986).
[For the entire collection see Zbl 0579.00008.]
The author develops a geometric (ODE) approach to prove the existence of spherically symmetric standing waves of the nonlinear Schrödinger equation \(iu_ t=\Delta u+| u|^ Su\) with a prescribed number of zeros. The problem of uniqueness of solutions is studied. The author discusses also the stability question and describes a mechanism for forcing an instability.
The author develops a geometric (ODE) approach to prove the existence of spherically symmetric standing waves of the nonlinear Schrödinger equation \(iu_ t=\Delta u+| u|^ Su\) with a prescribed number of zeros. The problem of uniqueness of solutions is studied. The author discusses also the stability question and describes a mechanism for forcing an instability.
Reviewer: Z.Guo
MSC:
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35B35 | Stability in context of PDEs |
| 35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
