On the evolution of a solitary wave for very weak nonlinearity

Abstract

The initial-value problem for a one-dimensional gravity wave of amplitude a and characteristic length l in water of depth d is examined for 0 < a/d [Lt ] d²/l² [Lt ] 1. A preliminary reduction leads to a Korteweg-de Vries (KdV) equation in which the nonlinear term is O(ε) relative to the linear terms, where ε = 3al²/4d³ [Lt ] 1 is a measure of nonlinearity/dispersion. The linear approximation (ε ↓ 0) is found to be valid if and only if $\epsilon\tau^{\frac{1}{3}}\ll 1 $ where $\tau = \frac{1}{2}(d/l)^2(gd)^{\frac{1}{2}}({\rm time})/l $ is the slow time in the KdV equation. The asymptotic solution of the KdV equation is obtained with the aid of inverse-scattering theory and is found to comprise not only a decaying wave train that is qualitatively similar to that predicted by the linear approximation, but also a soliton of amplitude 3V²/4d³ = O(εa) if V > 0, where V is the cross-sectional area of the initial displacement, or of amplitude = O(ε³a) if V = 0 (there is no soliton if V < 0). This soliton is fully evolved, and dominates the solution, only for $\epsilon \tau^{\frac{1}{3}} \gg 1$ if V > 0 or $\epsilon^2\tau^{\frac{1}{3}}\gg 1 $ if V = 0, but nonlinearity already has significant effects for $\epsilon\tau^{\frac{1}{3}} = O(1)$.