Unstable normalized standing waves for the space periodic NLS

Ackermann, Nils; Weth, Tobias

doi:10.2140/apde.2019.12.1177

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Unstable normalized standing waves for the space periodic NLS

Nils Ackermann and Tobias Weth

Vol. 12 (2019), No. 5, 1177–1213

DOI: 10.2140/apde.2019.12.1177

Abstract

For the stationary nonlinear Schrödinger equation $- Δ u + V (x) u - f (u) = λ u$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^{2}$ -norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow us to match the $L^{2}$ -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 $L^{2}$ -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.

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Keywords

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

Mathematical Subject Classification 2010

Primary: 35J91, 35Q55

Secondary: 35J20

Milestones

Received: 11 July 2017

Revised: 6 April 2018

Accepted: 12 August 2018

Published: 15 December 2018

Authors

Nils Ackermann
Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad de México Mexico

Tobias Weth
Institut für Mathematik Goethe-Universität Frankfurt am Main Germany

Vol. 12, No. 5, 2019