Bulletin of the
Korean Mathematical Society BKMS

In this paper, the limit behavior of solution for the Schr\"{o}d\-inger equation with random dispersion and time-dependent nonlinear loss/gain: $idu + \frac{1}{\varepsilon} m(\frac{t}{\varepsilon^2})\partial_{xx}udt + |u|^{2\sigma}u dt + i\varepsilon a(t)|u|^{2\sigma_{0}}udt = 0$ is studied. Combining stochastic Strichartz-type estimates with $L^2$ norm estimates, we first derive the global existence for $L^2$ and $H^1$ solution of the stochastic Schr\"{o}dinger equation with white noise dispersion and time-dependent loss/gain:~$idu + \Delta{u} \circ d\beta + |u|^{2\sigma}u dt + ia(t)|u|^{2\sigma_{0}}udt = 0$. Secondly, we prove rigorously the global diffusion-approximation limit of the solution for the former as $\varepsilon\rightarrow0$ in one-dimensional $L^2$ subcritical and critical cases.

Keywords: nonlinear Schr\"{o}dinger equation, random dispersion, time-dependent nonlinear loss/gain, nonlinear fibre optics

MSC numbers: 35Q55, 35R60, 37K40