Fast algorithms for solving the non-linear Schrödinger equation for digital compensation of signal distortions in fiber-optic communication lines

Initial value problem for the nonlinear Schrodinger equation
$$ i\frac{\partial u}{\partial z} = \frac{\partial^2 u}{\partial t^2} + |u|^2 u, \quad -\infty < t< \infty, \quad z>0, \quad u\bigr\rvert_{z=0} = u_0(t) $$
is the simplest but realistic model for describing signal propagation in a fiber-optic transmission line. When passing through the information transmission line, the signal is completely distorted and requires restoration. The main problem is the need to solve fast this problem. Initial value problem for the linear Schrodinger equation requires only $O(N\ln N)$ operations (complex multiplications). By fast we mean algorithms that require less than $O(N^2)$ actions.