Obliquely interacting solitary waves

Abstract

Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α1 and α2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin2ϕ α1,2 or strong if ϕ2 = O(α1,2). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O(α1α2), then is determined from these basic solutions.Strong interactions are intrinsically nonlinear. It is shown that these interactions are phase-conserving (the sum of the phases of the incoming waves is equal to the sum of the phases of the outgoing waves) if |α2-α1 > (2ϕ)2 but not if |α2-α1| (2ϕ)2 (e.g. the reflexion problem, for which the interacting waves are images and α2 = α1). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech2 profiles yield singular outgoing waves with - csch2 profiles, if \[ \psi_{-}< |\psi| < \psi_{+},\quad{\rm where}\quad\psi_{\pm}={\textstyle\frac{1}{2}}\left|(3\alpha_2)^{\frac{1}{2}}\pm(3\alpha_1)^{\frac{1}{2}}\right|. \]Regular interactions appear to be impossible within this singular regime, and its end points, |ϕ| = ϕ±, are associated with resonant interactions.