Quantifying the Effect of Random Dispersion for Logarithmic Schrödinger Equation

This paper is concerned with the random effect of the noise dispersion for the stochastic logarithmic Schrödinger equation emerged from the optical fibre with dispersion management. The well-posedness of the logarithmic Schrödinger equation with white noise dispersion is established via the regularization energy approximation and a spatial scaling property. For the small noise case, the effect of the noise dispersion is quantified by the proven large deviation principle under additional regularity assumptions on the initial datum. As an application, we show that for the regularized model, the exit from a neighborhood of the attractor of deterministic equation occurs on a sufficiently large time scale. Furthermore, the exit time and exit point in the small noise case, as well as the effect of large noise dispersion, is also discussed for the stochastic logarithmic Schrödinger equation.