Kink-antikink collisions in the $\phi^4$ equation: The $n$-bounce resonance and the separatrix map

We provide a detailed mathematical explanation of a phenomenon known as the two-bounce resonance observed in collisions between kink and anti-kink traveling waves of the phi-four equations of mathematical physics. This behavior was discovered numerically in the 1980’s by Campbell and his collaborators and subsequently discovered in several other equations supporting traveling waves. We first demonstrate the effect with new high-resolution numerical simulations. A pair of kink-like traveling waves may coalesce into a localized bound state or may reflect off each other. In the two bounce-resonance, they first coalesce, but later escape each others’ embrace, with a very regular pattern governing the behaviors. Studying a finite-dimensional “collective coordinates” model, we use geometric phase-plane based reasoning and matched asymptotics to explain the mechanism underlying the phenomenon, including the origin of several mathematical assumptions needed by previous researchers. We derive a separatrix map for this problem—a simple algebraic recursion formula that explains the complex fractal-like dependence on initial velocity for kink-antikink interactions.

SIAM Journal on Applied Dynamical Systems, 4, 1195–1228