Dynamics of localized states in the stochastic discrete nonlinear Schrödinger equation

Abstract: 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.