The linear and nonlinear instability of the Akhmediev breather

Abstract

The Akhmediev breather (AB) and its M-breather generalisation, hereafter called AB M , are exact solutions of the focusing NLS equation periodic in space and exponentially localised 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 in nature. It is therefore important to establish the stability properties of these solutions under perturbations. Concerning perturbations of these solutions within the NLS dynamics, there is the following common belief 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 they coincide with this set (M = N), then the AB M solution is neutrally stable. In this paper we argue instead that the AB M solution is always linearly unstable, even in the saturation case M = N, and we prove it in the simplest case M = N = 1, constructing two examples of x-periodic solutions of the linearised theory growing exponentially in time. Then we sketch the proof of completeness of the basis of periodic solutions of the linearised theory. We also investigate the nonlinear instability showing that (i) a perturbed AB initial condition evolves into a recurrence of ABs; (ii) the AB solution is more unstable than the background solution, and its instability increases as T → 0, where T is the AB appearance time. Although the AB solution is linearly and nonlinearly unstable, its instability generates a recurrence of ABs, and this recurrence implies its relevance in the natural phenomena described by the NLS equation, as well as its orbital stability, using a specific definition of orbital stability present in the literature.

The solutions of classical and nonlocal nonlinear Schr\"{o}dinger equations with nonzero backgrounds: Bilinearisation and reduction approach

“…Such a breather is known as the Akhmediev breather [11], which was first studied by Akhmediev in [11] and then bear his name. Stability of the Akhmediev and Kuznetsov-Ma breathers was studied recently [28,31]. The Akhmediev breather is perpendicular to the Kuznetsov-Ma breather, as depicted in Fig.…”

The effect of loss/gain and Hamiltonian perturbations of the Ablowitz—Ladik lattice on the recurrence of periodic anomalous waves

“…The Cauchy problem of the periodic AWs of the focusing NLS equation ( 3) has been recently solved in [23,24], to leading order and in terms of elementary functions, for generic periodic initial perturbations of the unstable background, in the case of a finite number N of unstable modes, using a suitable adaptation of the FG method, showing the relevance of the AB solution and of its N-mode generalization [36] in the description of the AW recurrence. See also [25] for an alternative approach to the study of the AW recurrence in the case of a single unstable mode, based on matched asymptotic expansions; see [26] for a finite-gap model describing the numerical instabilities of the AB and [27] for the analytic study of the linear, nonlinear, and orbital instabilities of the AB within the NLS dynamics; see [28] for the analytic study of the phase resonances in the AW recurrence; see [61] and [16] for the analytic study of the AW recurrence in other NLS type 1 + 1 dimensional partial differential equations: respectively the PT-symmetric NLS equation [4] and the massive Thirring model [48,66], showing the universality of the recurrence properties of periodic AWs, but also the specific differences present in different models. A generalization of the above results to multidimensions has been recently developed in [14,29] on the Davey-Stewartson two equation [17], an integrable generalization of NLS to 2 + 1 dimensions.…”

Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz–Ladik lattices

“…solution of focusing NLS for the arbitrary real parameters ã, ρ, k, X, T, but with different parameters at each appearance [28]. See also [30] for an alternative and effective approach to the study of the AW recurrence in the case of a single unstable mode, based on matched asymptotic expansions; see [33] for a FG model describing the numerical instabilities of the AB and [32] for the analytic study of the linear, nonlinear, and orbital instabilities of the AB within the NLS dynamics; see [29] for the analytic study of the phase resonances in the AW recurrence; see [66] and [19] for the analytic study of the FPUT AW recurrence in other NLS type models: respectively the PT-symmetric NLS equation [4] and the massive Thirring model [52,72],…”