## Analysis & PDE

• Anal. PDE
• Volume 12, Number 5 (2019), 1177-1213.

### Unstable normalized standing waves for the space periodic NLS

#### Abstract

For the stationary nonlinear Schrödinger equation $−Δu+V(x)u−f(u)=λu$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L2$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow us to match the $L2$-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 $L2$-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.

#### Article information

Source
Anal. PDE, Volume 12, Number 5 (2019), 1177-1213.

Dates
Revised: 6 April 2018
Accepted: 12 August 2018
First available in Project Euclid: 5 January 2019

https://projecteuclid.org/euclid.apde/1546657229

Digital Object Identifier
doi:10.2140/apde.2019.12.1177

Mathematical Reviews number (MathSciNet)
MR3892400

Zentralblatt MATH identifier
1405.35191

#### Citation

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

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