For the propagation of the optical pulses described by the Nonlinear Schroedinger Equation, there is in general no closed-form solution. Therefore one has to use numerical methods.

For the calculation of the signal propagation, the Split Step Fourier Method is used in this webdemo.

The basic idea is to consider the effects described by the NLSE (chromatic dispersion, Kerr nonlinearity, attenuation) independently of each other. Then, for each of these cases, an analytical solution is available. Solving the equation is done iteratively for small step sizes $\Delta z$: $$A(\tau,z_0+\Delta z)=\mathcal{F}^{-1} \left(\underbrace{\mathcal{F}\left(\underbrace{\underbrace{A(\tau,z_0)\mathrm{exp}\left(-j\gamma |A(\tau,z_0)|^2 \Delta z \right)}_{\mathrm{Kerr\space effect\space (time \space domain)}} \mathrm{exp}\left(-\frac{\alpha}{2} \Delta z \right)}_{\mathrm{+\space attenuation \space (time \space domain)}} \right) \mathrm{exp}\left(-j \frac{\beta_2 \omega^2}{2} \Delta z \right)}_{\mathrm{+\space dispersion \space(frequency \space domain)}} \right)$$