Dynamics of localized states in the stochastic discrete nonlinear Schr\"odinger equation

We revisit aspects of dynamics and stability of localized states in the deterministic and stochastic discrete nonlinear Schr\"odinger equation. By a combination of analytic and numerical techniques, we show that localized initial conditions disperse if the strength of the nonlinear part drops below a threshold and that localized states are unstable in a noisy environment. As expected, the constants of motion in the nonlinear Schr\"odinger equation play a crucial role. An infinite temperature state emerges when multiplicative noise is applied, while additive noise yields unbounded dynamics since conservation of normalization is violated.