Dynamics of localised states in the stochastic discrete nonlinear Schrödinger equation

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