Universal behavior of modulationally unstable media

TL;DR

This paper demonstrates a universal pattern in the evolution of modulationally unstable media across various mathematical models and physical systems, revealing common structural features in their dynamics.

Contribution

It introduces a unified analytical and numerical framework showing that diverse modulationally unstable systems exhibit similar evolving structures.

Findings

All studied systems develop a wedge-shaped oscillation region.

The dynamics consistently produce two quiescent outer sectors.

A heuristic criterion for the oscillation region properties is proposed.

Abstract

Evidence is presented of universal behavior in modulationally unstable media. An ensemble of nonlinear evolution equations, including three partial differential equations, an integro-differential equation, a nonlocal system and a differential-difference equation, is studied. Collectively, these systems arise in a variety of applications in the physical and mathematical sciences, including water waves, optics, acoustics, Bose-Einstein condensation, and more. All these models exhibit modulational instability, namely, the property that a constant background is unstable to long-wavelength perturbations. In this work, each of these systems is studied analytically and numerically for a number of different initial perturbations of the constant background, and it is shown that, for all systems and for all initial conditions considered, the dynamics gives rise to a remarkably similar structure comprised of two outer, quiescent sectors separated by a wedge-shaped central region characterized by modulated periodic oscillations. A heuristic criterion that allows one to compute some of the properties of the central oscillation region is also given.