Weakly nonlinear wavepackets in the Korteweg–de Vries equation: the KdV/NLS connection

If the initial condition for the Korteweg–de Vries (KdV) equation is a weakly nonlinear wavepacket, then its evolution is described by the nonlinear Schrödinger (NLS) equation. This KdV/NLS connection has been known for many years, but its various aspects and implications have been discussed only in asides. In this note, we attempt a more focused and comprehensive discussion including such as issues as the KdV-induced long wave pole in the nonlinear coefficient of the NLS equation, the derivation of NLS from KdV through perturbation theory, resonant effects that give the NLS equation a wide range of applicability, and numerical illustrations. The multiple scales/nonlinear perturbation theory is explicitly extended to two orders beyond that which yields the NLS equation; the wave envelope evolves under a generalized-NLS equation which is third order in space and quintically-nonlinear.