[Paper Review] Numerical instability of the Akhmediev breather and a finite-gap model of it

P. G. Grinevich, P. M. Santini|arXiv (Cornell University)|Jul 30, 2017

Nonlinear Waves and Solitons38 references4 citations

TL;DR

This paper investigates the numerical instability of the Akhmediev breather in the focusing Nonlinear Schrödinger equation using the split-step Fourier method (SSFM). It shows that when round-off errors are negligible, the numerical solution is well-approximated by a genus 2 finite-gap solution, exhibiting exact recurrence of rogue waves. A key finding is that the SSFM induces a vertical unstable gap proportional to the inverse square of the number of time steps, explaining recurrence time and phase stability in high-precision simulations.

ABSTRACT

In this paper we study the numerical instabilities of the NLS Akhmediev breather, the simplest space periodic, one-mode perturbation of the unstable background, limiting our considerations to the simplest case of one unstable mode. In agreement with recent theoretical findings of the authors, in the situation in which the round-off errors are negligible with respect to the perturbations due to the discrete scheme used in the numerical experiments, the split-step Fourier method (SSFM), the numerical output is well-described by a suitable genus 2 finite-gap solution of NLS. This solution can be written in terms of different elementary functions in different time regions and, ultimately, it shows an exact recurrence of rogue waves described, at each appearance, by the Akhmediev breather. We discover a remarkable empirical formula connecting the recurrence time with the number of time steps used in the SSFM and, via our recent theoretical findings, we establish that the SSFM opens up a vertical unstable gap whose length can be computed with high accuracy, and is proportional to the inverse of the square of the number of time steps used in the SSFM. This neat picture essentially changes when the round-off error is sufficiently large. Indeed experiments in standard double precision show serious instabilities in both the periods and phases of the recurrence. In contrast with it, as predicted by the theory, replacing the exact Akhmediev Cauchy datum by its first harmonic approximation, we only slightly modify the numerical output. Let us also remark, that the first rogue wave appearance is completely stable in all experiments and is in perfect agreement with the Akhmediev formula and with the theoretical prediction in terms of the Cauchy data.

Motivation & Objective

• To analyze the numerical instability of the Akhmediev breather under the split-step Fourier method (SSFM) in the focusing Nonlinear Schrödinger equation.
• To investigate how discrete numerical schemes, particularly SSFM, affect the recurrence of rogue waves in periodic, one-mode perturbations of the unstable background.
• To determine the role of round-off errors in disrupting recurrence periods and phases in standard double-precision simulations.
• To test the robustness of the Akhmediev breather's first-peak characteristics under numerical perturbations and harmonic approximations.
• To validate theoretical predictions from finite-gap theory by comparing numerical outputs with genus 2 solutions and deriving empirical scaling laws.

Proposed method

• Numerical simulations of the focusing NLS equation using the split-step Fourier method (SSFM) with varying numbers of time steps.
• Comparison of numerical outputs using the exact Akhmediev breather as initial condition versus its first harmonic approximation.
• High-precision (quadruple) arithmetic used to suppress round-off errors and isolate instabilities due to the discrete scheme.
• Application of finite-gap theory to model the numerical instability as a genus 2 solution, with different elementary functions describing behavior in distinct time regions.
• Empirical derivation of formulas linking recurrence time and gap opening in the SSFM to the number of time steps.
• Spatial grid sensitivity analysis by reducing the number of spatial points to assess discretization effects on recurrence and phase.

Experimental results

Research questions

• RQ1 How does the split-step Fourier method (SSFM) numerically approximate the Akhmediev breather when round-off errors are negligible?
• RQ2 What is the relationship between the number of time steps in the SSFM and the recurrence time and phase stability of numerically generated rogue waves?
• RQ3 Why do standard double-precision simulations exhibit severe instabilities in recurrence periods and phases despite the Akhmediev breather being analytically stable?
• RQ4 To what extent does replacing the exact Akhmediev initial condition with its first harmonic approximation alter the numerical output?
• RQ5 How does the finite-gap theory, particularly genus 2 solutions, describe the observed numerical instability in the SSFM?

Key findings

• When round-off errors are negligible, the numerical output of the SSFM for the Akhmediev breather is well-described by a genus 2 finite-gap solution, which exhibits exact recurrence of rogue waves.
• A remarkable empirical formula links the recurrence time and the gap opening in the SSFM to the inverse square of the number of time steps, with the gap length proportional to $ N_t^{-2} $, where $ N_t $ is the number of time steps.
• The SSFM introduces a vertical unstable gap in the spectral plane ($ \mathrm{Re}\,E_1 = \mathrm{Re}\,E_2 = 0 $), explaining the observed recurrence without phase shifts.
• In standard double-precision simulations, round-off errors dominate and cause significant instabilities in recurrence periods and phases, which increase with more time steps.
• Replacing the exact Akhmediev initial condition with its first harmonic approximation causes only a negligible change in the numerical output, confirming the robustness of the first-peak dynamics.
• The first rogue wave appearance—its time, position, and amplitude—is perfectly stable and matches the Akhmediev formula and theoretical predictions across all simulation settings.