#### 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 $-\Delta u+V\left(x\right)u-f\left(u\right)=\lambda u$ 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.

##### 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