Solitary internal waves with oscillatory tails

Abstract

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.