Derivation of the equation for an ultrashort pulse in a fibre

Pulses propagating in a non-linear dispersive (glass) fibre can be described by the non-linear Schrödinger equation il the pulse is longer than a picosecond; for shorter pulses, this equation must be extended. In this paper we systematically derive this extended equation using the method of multiple scales. By using an inherent freedom in the method of multiple scales, a technique is developed such that perturbation terms are greatly simplified. The limits of validity of the derived equation are discussed. It is shown to be valid for pulses longer than 30 fs