Vol. 12, No. 5, 2019

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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
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.

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