Chaotic scattering in solitary wave interactions: A singular iterated-map description

TL;DR

This paper develops a singular iterated map framework to analyze chaotic solitary wave interactions, accounting for pre-collision excitation of internal modes, and connects it to broader physical systems.

Contribution

It introduces a generalized singular iterated map derived via Melnikov integrals, capturing complex wave interactions and exotic periodic orbits, extending previous models.

Findings

Map exhibits singular behavior with infinite winding regions

Allows analysis of interactions with pre-collision internal mode excitation

Connects solitary wave chaos to laser, celestial, and fluid mechanics problems

Abstract

We derive a family of singular iterated maps--closely related to Poincare maps--that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary wave collisions depends on the transfer of energy to a secondary mode of oscillation, often an internal mode of the pulse. Unlike previous analyses, this map allows one to understand the interactions in the case when this mode is excited prior to the first collision. The map is derived using Melnikov integrals and matched asymptotic expansions and generalizes a ``multi-pulse'' Melnikov integral and allows one to find not only multipulse heteroclinic orbits, but exotic periodic orbits. The family of maps derived exhibits singular behavior, including regions of infinite winding. This problem is shown to be a singular version of the conservative Ikeda map from laser physics and connections are made with problems from celestial mechanics and fluid mechanics.