Analysis & PDE

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

Unstable normalized standing waves for the space periodic NLS

Nils Ackermann and Tobias Weth

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Abstract

For the stationary nonlinear Schrödinger equation Δu+V(x)uf(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
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
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

Subjects
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

Keywords
nonlinear Schrödinger equation periodic potential standing wave solution orbitally unstable solution multibump construction prescribed norm

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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