Analysis & PDE
- Anal. PDE
- Volume 12, Number 5 (2019), 1177-1213.
Unstable normalized standing waves for the space periodic NLS
For the stationary nonlinear Schrödinger equation with periodic potential we study the existence and stability properties of multibump solutions with prescribed -norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow us to match the -constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated -sphere, and we show their orbital instability with respect to the Schrödinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.
Anal. PDE, Volume 12, Number 5 (2019), 1177-1213.
Received: 11 July 2017
Revised: 6 April 2018
Accepted: 12 August 2018
First available in Project Euclid: 5 January 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35J20: Variational methods for second-order elliptic equations
Ackermann, Nils; Weth, Tobias. Unstable normalized standing waves for the space periodic NLS. Anal. PDE 12 (2019), no. 5, 1177--1213. doi:10.2140/apde.2019.12.1177. https://projecteuclid.org/euclid.apde/1546657229