Cauchy Processes, Dissipative Benjamin–Ono Dynamics and Fat-Tail Decaying Solitons

Abstract

In this paper, a dissipative version of the Benjamin–Ono dynamics is shown to faithfully model the collective evolution of swarms of scalar Cauchy stochastic agents obeying a follow-the-leader interaction rule. Due to the Hilbert transform, the swarm dynamic is described by nonlinear and non-local dynamics that can be solved exactly. From the mutual interactions emerges a fat-tail soliton that can be obtained in a closed analytic form. The soliton median evolves nonlinearly with time. This behaviour can be clearly understood from the interaction of mutual agents.

Keywords

• Benjamin–Ono dynamics
• master equations
• jump Markov processes
• interacting Cauchy processes
• swarm dynamics
• fat-tail soliton waves