Universal scaling laws for pulse propagation in plasma and non-linear media

Pulse propagation in non-linear media is an important topic plasma physics and non-linear optics. Such pulse propagation is subject to scaling laws that go beyond the non-linear extension of the linear dispersion of the carrier wave. These laws govern the relationships (''ideal lines'') between pulse amplitude, spatial width, temporal duration and propagation speed, and the evolution of all these quantities in time. They have ''attractor'' properties: a pulse that is not on an ideal line initially will reshape itself until it is, and then stay on that line. Here we show how the scaling laws and ideal lines that rule non-linear pulse evolution can be derived from non-linear envelope equations. We will do this for non-linear three-wave interaction systems, like those that describe Raman and Brillouin amplification. The scheme also works well for other non-linear equations, such as the non-linear Schrödinger equation, the Korteweg-de Vries equation and the Boussinesq equation. For pancake pulses in multi-dimensional settings, we will show how the scaling laws will cause the pulse to assume a ''horseshoe'' shape, which has already been demonstrated in both parametric pulse amplification and soliton propagation.