Soliton solution

Looking for analytical and stable solutions to the Nonlinear Schroedinger equation, one finds, neglecting attenuation, the so called first order soliton solution:
$$
A(\tau,z)=\frac{2\eta}{\sqrt[3]{\gamma}}\mathrm{sech}\left(j2\eta\frac{\sqrt[6]{\gamma}}{\sqrt{\beta_2}}\tau-2\delta_0-4\zeta\eta\sqrt[3]{\gamma}z \right)\mathrm{exp}\left(2j\sqrt[3]{\gamma}\left(\zeta^2-\eta^2\right)z+2\zeta\frac{\sqrt[6]{\gamma}}{\sqrt{\beta_2}}\tau-j\sigma_0 \right)
$$

This solution is interesting due to its shape being preserved during propagation. That is because the effect of Kerr nonlinearity and chromatic dispersion mutually compensate for this specific impulse.

The pulse duration and the amplitude of a soliton are directly related to each other and can not be choosen independently. The four independent parameters of the soliton above are $\eta$, $\zeta$, $\delta_0$, $\sigma_0$, describing the energy of the signal, its relative velocity, staring position and starting phase.

The stability of solitons can be investigated by creating pulses with a mismatch of amplitude and pulse duration or a non-ideal soltion pulse shape.

Multiple solitons next to each other show interesting phenomena of soliton interaction.