Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe linear and nonlinear instability of the Akhmediev breather
127
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $M=N>1$.
rate research
Read More
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.
Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29}, 319324 (2016)). Following this pattern, two integrable equations [eqs.2 and 3] with specific parity-time symmetry are introduced here, under different reduction conditions. For eq. 2, two kinds of periodic solutions are obtained analytically by means of the Hirotas bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the semi-rational states represent RWs built on top of the background of periodic line waves. Similarly, semi-rational solutions consisting of a line RW and line breather are derived. For eq. 3, three kinds of analytical solutions,textit{viz}., breathers, lumps and semi-rational solutions, representing lumps, periodic line waves and breathers are obtained, using the Hirota method. Their dynamics are analyzed and demonstrated by means of three-dimensional plots. It is also worthy to note that eq. 2 can reduce to a (1+1)-dimensional textquotedblleft reverse-space nonlocal NLS equation by means of a certain transformation, Lastly, main differences between solutions of eqs.2 and 3 are summarized.
The generalized perturbative reduction method is used to find the two-component vector breather solution of the nonlinear Klein-Gordon equation. It is shown that the nonlinear pulse oscillates with the sum and difference of frequencies and wave numbers in the region of the carrier wave frequency and wave number. Explicit analytical expressions for the profile and parameters of the nonlinear pulse are obtained. In the particular case, the vector breather coincides with the vector $0pi$ pulse of self-induced transparency.
Solitons and breathers are localized solutions of integrable systems that can be viewed as particles of complex statistical objects called soliton and breather gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media these integrable gases present fundamental interest for nonlinear physics. We develop analytical theory of breather and soliton gases by considering a special, thermodynamic type limit of the wavenumber-frequency relations for multi-phase (finite-gap) solutions of the focusing nonlinear Schrodinger equation. This limit is defined by the locus and the critical scaling of the band spectrum of the associated Zakharov-Shabat operator and yields the nonlinear dispersion relations for a spatially homogeneous breather or soliton gas, depending on the presence or absence of the background Stokes mode. The key quantity of interest is the density of states defining, in principle, all spectral and statistical properties of a soliton (breather) gas. The balance of terms in the nonlinear dispersion relations determines the nature of the gas: from an ideal gas of well separated, non-interacting breathers (solitons) to a special limiting state, which we term breather (soliton) condensate, and whose properties are entirely determined by the pairwise interactions between breathes (solitons). For a non-homogeneous breather gas, we derive a full set of kinetic equations describing slow evolution of the density of states and of its carrier wave counterpart. The kinetic equation for soliton gas is recovered by collapsing the Stokes spectral band. A number of concrete examples of breather and soliton gases are considered, demonstrating efficacy of the developed general theory with broad implications for nonlinear optics, superfluids and oceanography.
We numerically realize breather gas for the focusing nonlinear Schrodinger equation. This is done by building a random ensemble of N $sim$ 50 breathers via the Darboux transform recursive scheme in high precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma and Peregrine breathers as elementary quasi-particles of the respective gases. The interaction properties of the constructed breather gases are investigated by propagating through them a trial generic breather (Tajiri-Watanabe) and comparing the mean propagation velocity with the predictions of the recently developed spectral kinetic theory (El and Tovbis, PRE 2020).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
