Maximal intensity higher-order Akhmediev breathers of the nonlinear Schrödinger equation and their systematic generation

Abstract

It is well known that Akhmediev breathers of the nonlinear cubic Schrödinger equation can be superposed nonlinearly via the Darboux transformation to yield breathers of higher order. Surprisingly, we find that the peak height of each Akhmediev breather only adds linearly to form the peak height of the final breather. Using this new peak-height formula, we show that at any given periodicity, there exist a unique high-order breather of maximal intensity. Moreover, these high-order breathers form a continuous hierarchy, growing in intensity with increasing periodicity. For any such higherorder breather, a simple initial wave function can be extracted from the Darboux transformation to dynamically generate that breather from the nonlinear Schrödinger equation.

The height of an n th-order fundamental rogue wave for the nonlinear Schrödinger equation

“…Another work that gave the proof for the same expression for the maximal amplitude is [26]. The problem was also addressed in [27]. The purpose of this paper is to provide a direct proof of the above conjecture, which differs from above two works given in [19,26].…”

“…The NLS equation is a widely applicable integrable system [31] in physics, which is solved by several methods such as the inverse scattering method [31], the Hirota method [32] and the Darboux transformation(DT) [33]. Recently, the height of multi-breather of the NLS has been given in references [27,34], but which can not imply the height of the RWs because of the appearing of an indeterminate form in reference [27] or ν i → 1 in reference [34]. In general, this indeterminate form is unavoidable to construct RWs for many equations by the DT.…”

Talbot carpets by rogue waves of extended nonlinear Schrödinger equations

“…In this sense, ABs seem to be especially relevant [11,12], which can be generalized to the doubly periodic solutions (as well as to the extended NLSEs). They allow for an easy systematic buildup of higher-order breathers that can be regarded as prototype RWs [13,14].…”

Breather structures in a plasma with warm ions and Cairns electrons

“…[52] Mathematically, localized solutions of Equation 28 have a hierarchy of rational solutions that represent a class of special solitons localized in space as well as in time. [57,58] The general first-order breather solution of the NLSE 28 can be written as follows: [29,[59][60][61][62] The first-order Peregrine soliton of this hierarchy is characterized by a maximum amplitude amplification (at = 0 and = 0) of thrice the background wave amplitude.…”