Nonlinear Schrödinger equation in optics

Definition

Let a medium have the optical Kerr effect and neglect losses.

1. Pulses in a fibre. Let $A(z, T)$ be the complex envelope of a pulse, normalized so that $\abs{A}^2$ is the power. Here $z$ is the distance along the fibre and $T$ is the time in a frame moving with the group velocity. Let $\beta_2$ be the group-velocity dispersion parameter and $\gamma > 0$ the nonlinear coefficient of the fibre. The nonlinear Schrödinger equation for pulses is $$i \frac{\partial A}{\partial z} - \frac{\beta_2}{2} \frac{\partial^2 A}{\partial T^2} + \gamma \abs{A}^2 A = 0.$$
2. Beams in a bulk medium. Let $A(x, y, z)$ be the envelope of a monochromatic beam travelling along $z$, normalized so that $\abs{A}^2$ is the intensity. Let the medium have linear index $n_0$ and nonlinear index $n_2$. Let $k_0 = 2\pi/\lambda$ be the vacuum wavenumber and $k = n_0 k_0$. The nonlinear Schrödinger equation for beams is $$i \frac{\partial A}{\partial z} + \frac{1}{2k} \left( \frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2} \right) + k_0 n_2 \abs{A}^2 A = 0.$$