The modulation of weakly non-linear dispersive waves near the marginal state of instability

The modulation of a one-dimensional weakly non-linear purely dispersive quasi-monochromatic wave (the carrier) is usually governed by the non-linear Schrodinger (NS) equation. The critical wavenumber for which the carrier is marginally modulationally unstable is determined by the condition that the product of the coefficients of the non-linear and dispersive terms in the NS equation is zero. However, near this marginal state the assumptions that lead to the NS equation are invalid and a modified form of the NS equation that involves higher-order non-linearities is appropriate. This modified NS equation is here derived formally for a general system involving a single dependent variable and a revised form of the instability criterion is obtained. The results are illustrated by considering a particular system described by a generalised Korteweg-de Vries equation.