Coupled Korteweg–de Vries Equations

Abstract

When a system supports two distinct long-wave modes with nearly coincident phase speeds, the weakly nonlinear and linear dispersion unfolding generically leads to two coupled Korteweg–de Vries equations. In this paper, we review the derivation of such systems in stratified fluids, extending previous studies by allowing for background shear flows. Coupled Korteweg–de Vries systems have very rich dynamics, including the phenomenon of leapfrogging solitary waves. An asymptotic approach to describe this is also briefly reviewed.