Weakly Nonlinear Wavepackets in the Korteweg-deVries Equation: The KdV/NLS Connection

If the initial condition for the Korteweg-deVries (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. 1 This work was supported by the National Science Foundation through grant OCE9521133. Dr. Chen is grateful for a three-year graduate fellowship provided by the Department of Education, Taiwan, R. O. C. We also thank Prof. Thiab Taha and the other organizers of the IMACS 1999 conference on “Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory”. Preprint submitted to Elsevier Preprint 18 May 2000