Some Models for the Interaction of Long and Short Waves in Dispersive Media: Part I—Derivation

Abstract

It is universally accepted that the cubic, nonlinear Schrodinger equation (NLS) models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves, while the Korteweg–de Vries equation (KdV) models the propagation of long waves in dispersive media. A system that couples the two equations seems attractive to model the interaction of long and short waves, and such a system has been studied over the last few decades. However, questions about the validity of this system in the study of water waves were raised in a previous work of one of us where the analysis was presented using the fifth-order KdV as the starting point. These questions will now be settled unequivocally in a series of papers. In this first part, we show that the NLS-KdV system (or even the linear Schrodinger–KdV system) cannot be resulted from the full Euler equations formulated in the study of water waves. In the process of so doing, we also propose a few alternative models for describing the interaction of long and short waves.