Separatrix Map Analysis for Fractal Scatterings in Weak Interactions of Solitary Waves

TL;DR

This paper provides a detailed analysis of a universal separatrix map that describes fractal scatterings in weak solitary wave interactions, deriving scaling laws and an explicit criterion for fractal occurrence, confirmed by numerical simulations.

Contribution

It offers a comprehensive analytical characterization of fractal scatterings in solitary wave interactions using the separatrix map, including new scaling laws and an explicit occurrence criterion.

Findings

Derived analytical scaling laws for fractal scatterings.

Confirmed laws through numerical simulations.

Provided an explicit criterion for fractal scattering occurrence.

Abstract

Previous studies have shown that fractal scatterings in weak interactions of solitary waves in the generalized nonlinear Schr\"odinger equations are described by a universal second-order separatrix map. In this paper, this separatrix map is analyzed in detail, and hence a complete characterization of fractal scatterings in these weak interactions is obtained. In particular, scaling laws of these fractals are derived analytically for different initial conditions, and these laws are confirmed by direct numerical simulations. In addition, an analytical criterion for the occurrence of fractal scatterings is given explicitly.