# Unstable normalized standing waves for the space periodic NLS

## TL;DR

This paper investigates the existence and instability of multibump solutions to the nonlinear Schrödinger equation with periodic potential, introducing new techniques to handle $L^2$-constraints and analyzing their stability properties.

## Contribution

It introduces a new nondegeneracy condition and superposition methods to establish infinitely many solutions with prescribed $L^2$-norm for the periodic NLS.

## Key findings

- Existence of infinitely many multibump solutions.
- All solutions are orbitally unstable under the Schrödinger flow.
- Results hold in both mass-subcritical and supercritical regimes.

## Abstract

For the stationary nonlinear Schr\"odinger equation $-\Delta u+ V(x)u- f(u) = \lambda u$ 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 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 Schr\"odinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.