Soliton Synchronization with Randomness: Rogue Waves and Universality

TL;DR

This paper studies the collision behavior of multiple solitons in the nonlinear Schrödinger equation, revealing universal sinc-shaped collision peaks and deriving CLTs for profile fluctuations under randomness.

Contribution

It establishes conditions for soliton synchronization, demonstrates universality of the collision profile with random amplitudes, and derives CLTs for fluctuation analysis.

Findings

Collision profile converges to sinc(x) function under randomness.

Universality of the collision peak independent of amplitude distribution.

Central Limit Theorems describe fluctuation behavior near and far from collision.

Abstract

We consider an $N$-soliton solution of the focusing nonlinear Schr\"{o}dinger equations. We give conditions for the synchronous collision of these $N$ solitons. When the solitons velocities are well separated and the solitons have equal amplitude, we show that the local wave profile at the collision point scales as the $\operatorname{sinc}(x)$ function. We show that this behaviour persists when the amplitudes of the solitons are i.i.d. sub-exponential random variables. Namely the central collision peak exhibits universality: its spatial profile converges to the $\operatorname{sinc}(x)$ function, independently of the distribution. We derive Central Limit Theorems for the fluctuations of the profile in the near-field regime (near the collision point) and in the far-regime.