Obliquely interacting solitary waves

Abstract

Nonlinear oblique interactions between two slightly dispersive gravity waves (in particular, solitary waves) of dimensionless amplitudes α 1 and α 2 (relative to depth) and relative inclination 2ϕ (between wave normals) are classified as weak if sin 2 ϕ α 1,2 or strong if ϕ 2 = O (α 1,2 ). Weak interactions permit superposition of the individual solutions of the Korteweg-de Vries equation in first approximation; the interaction term, which is O (α 1 α 2 ), 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 |α 2 -α 1 > (2ϕ) 2 but not if |α 2 -α 1 | (2ϕ) 2 (e.g. the reflexion problem, for which the interacting waves are images and α 2 = α 1 ). It also is shown that the interactions are singular, in the sense that regular incoming waves with sech 2 profiles yield singular outgoing waves with - csch 2 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.