Extension of weakly nonlinear theory of solitary wave propagation

Abstract

A localized wave propagating horizontally at speed c through a fluid of arbitrary depth H which consists of a lower vertical layer (thickness L) with a weak-depth-dependent density below an upper layer of constant density is considered. The flow is characterized by Long’s equation for the stream function ψ. Exact analytic results are obtained for ψ through O[1/(h−1)] and the wave speed through order O[1/(h−1)2], h=H/L. The terms in ψ of O[1/(h−1)2] are obtained numerically. Using these results the range of validity of such a perturbative approach is examined.