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Register a new userNumerical instability of the Akhmediev breather and a finite-gap model of it
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P. G. Grinevich
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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.
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The Akhmediev breather (AB) and its M-soliton generalization $AB_M$ are exact solutions of the focusing NLS equation periodic in space and exponentially localized in time over the constant unstable background; they describe the appearance of $M$ unstable nonlinear modes and their interaction, and they are expected to play a relevant role in the theory of periodic anomalous (rogue) waves (AWs) in nature. It is rather well established that they are unstable with respect to small perturbations of the NLS equation. Concerning perturbations of these solutions within the NLS dynamics, there is the following common believe in the literature. Let the NLS background be unstable with respect to the first $N$ modes; then i) if the $M$ unstable modes of the $AB_M$ solution are strictly contained in this set ($M<N$), then the $AB_M$ is unstable; ii) if $M=N$, the so-called saturation of the instability, then the $AB_M$ solution is neutrally stable. We argue instead that the $AB_M$ solution is always unstable, even in the saturation case $M=N$, and we prove it in the simplest case $M=N=1$. We first prove the linear instability, constructing two examples of $x$-periodic solutions of the linearized theory growing exponentially in time. Then we investigate the nonlinear instability using our previous results showing that i) a perturbed AB initial condition evolves into an exact Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of ABs described in terms of elementary functions of the initial data, to leading order; ii) the AB solution is more unstable than the background solution, and its instability increases as $Tto 0$, where $T$ is the AB appearance parameter. Although the AB solution is linearly and nonlinearly unstable, it is relevant in nature, since its instability generates a FPUT recurrence of ABs. These results suitably generalize to the case $M=N>1$.
We describe a numerical procedure to compute the so-called isospectral torus of finite gap sets, that is, the set of Jacobi matrices whose essential spectrum is composed of finitely many intervals. We also study numerically the convergence of specific Jacobi matrices to their isospectral limit. We then extend the analyis to the definition and computation of an isospectral torus for Cantor sets in the family of Iterated Function Systems. This analysis is developed with the ultimate goal of attacking numerically the conjecture that the Jacobi matrices of I.F.S. measures supported on Cantor sets are asymptotically almost-periodic.
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We investigate in detail the qualitative similarities between the pulse localization characteristics observed using sinusoidal phase modulation during linear propagation and those seen during the evolution of Akhmediev breathers during propagation in a system governed by the nonlinear Schr{o}dinger equation. The profiles obtained at the point of maximum focusing indeed present very close temporal and spectral features. If the respective linear and nonlinear longitudinal evolutions of those profiles are similar in the vicinity of the point of maximum focusing, they may diverge significantly for longer propagation distance. Our analysis and numerical simulations are confirmed by experiments performed in optical fiber.
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