Nonlinear instability of linearly unstable standing waves for nonlinear Schr\"{o}dinger equations

TL;DR

This paper demonstrates that for nonlinear Schrödinger equations, linear instability of standing waves leads to their orbital instability across all dimensions, supported by a new Strichartz estimate.

Contribution

It establishes a general link between linear and orbital instability for standing waves in nonlinear Schrödinger equations, introducing a novel Strichartz estimate for the linearized propagator.

Findings

Linear instability implies orbital instability in any dimension.

Established a Strichartz type estimate for the linearized operator.

Provided a general framework under broad nonlinearity assumptions.

Abstract

We study the instability of standing waves for nonlinear Schr\"{o}dinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.