Correspondence between the NLS equation for optical fibers and a class of integrable NLS equations

The propagation of the optical field complex envelope in a single-mode fiber is governed by a one-dimensional cubic nonlinear Schr\"odinger equation with a loss term. We present a result about $L^2$-closeness of the solutions of the above-mentioned equation and of a one-dimensional nonlinear Schr\"odinger equation that is Painlev\'e integrable.