Obliquely interacting solitary waves

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 α_1,2 or strong if ϕ² = 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(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.