Nonlinear Schroedinger Equation

Making some simplifications, the propagation of an electromagnetic wave on an optical fiber can be described by the Nonlinear Schroedinger Equation (NLSE):
$$
\frac{\mathcal{d}A(\tau,z)}{\mathcal{d}z}-j\frac{\beta_2}{2}\frac{\mathcal{d}^2A(\tau,z)}{\mathcal{d}\tau^2}+j\gamma|A(\tau,z)|^2A(\tau,z)+\alpha A(\tau,z)=0
$$

In this equation, $A(\tau,z)$ is the envelope of an optical signal on the fiber, which depends on the normalized time $\tau$ and the propagation distance z.

The group velocity dispersion parameter $\beta_2$ takes into account chromatic dispersion. The reason for this effect is the frequency dependence of the refractive index.

The Kerr Effect, caused by the refractive index depending on the magnitude of the electical fiel, is considered by the nonlinearity coefficient $\gamma$.

The parameter $\alpha$ represents the attenuation of the optical fiber.

All parameters of the NLSE are material constants of the waveguide and related to its refractive index: $$ \beta=\frac{\omega}{c_0}n(\omega,|E|^2)\approx \left(\beta_0+\beta_1\left(\omega-\omega_0\right)+\frac{\beta_2}{2}\left(\omega-\omega_0\right)^2+\gamma|E|^2-j\alpha\right) $$