Oscillatory dynamics of the classical Nonlinear Schrodinger equation

TL;DR

This paper numerically investigates the oscillatory statistical behavior of solutions to the classical 1D Nonlinear Schrödinger equation during modulation instability, revealing sinusoidal amplitude oscillations, decay laws, and Rayleigh PDF convergence.

Contribution

It provides a detailed numerical analysis of the oscillatory dynamics and statistical properties of the NLS equation during modulation instability, highlighting universal behaviors across different initial noise conditions.

Findings

Amplitude oscillations decay as t^{-3/2}

Oscillation period is π

PDF fluctuations around Rayleigh distribution

Abstract

We study numerically the statistical properties of the modulation instability (MI) developing from condensate solution seeded by weak, statistically homogeneous in space noise, in the framework of the classical (integrable) one-dimensional Nonlinear Schrodinger (NLS) equation. We demonstrate that in the nonlinear stage of the MI the moments of the solutions amplitudes oscillate with time around their asymptotic values very similar to sinusoidal law. The amplitudes of these oscillations decay with time $t$ as $t^{-3/2}$, the phases contain the nonlinear phase shift that decays as $t^{-1/2}$, and the period of the oscillations is equal to $\pi$. The asymptotic values of the moments correspond to Rayleigh probability density function (PDF) of waves amplitudes appearance. We show that such behavior of the moments is governed by oscillatory-like, decaying with time, fluctuations of the PDF around the Rayleigh PDF; the time dependence of the PDF turns out to be very similar to that of the moments. We study how the oscillations that we observe depend on the initial noise properties and demonstrate that they should be visible for a very wide variety of statistical distributions of noise.