Akhmediev breathers and Peregrine solitary waves in a quadratic medium

Abstract

We investigate the formation of optical localized nonlinear structures, evolving upon a non-zero background plane wave, in a dispersive quadratic medium. We show the existence of quadratic Akhmediev breathers and Peregrine solitary waves, in the regime of cascading second-harmonic generation. This finding opens a novel path for the excitation of extreme rogue waves and for the description of modulation instability in quadratic nonlinear optics.In dispersive optical media, with cubic Kerr nonlinearity, the nonlinear Schrödinger equation (NLSE) provides a central description of a variety of nonlinear localization effects [1]. In fact, since the 1970s there has been wide and continued investigation into the properties of analytic soliton solutions of the NLSE [2], both for their intrinsic scientific interest, as well as for their potential to provide new insights into important applications such as optical propagation in nonlinear waveguides [3]. For the case of a self-focussing nonlinearity, the most celebrated solution of this type is very likely the propagation-invariant hyperbolic secant bright soliton, but there is also an extensive literature studying various types of solitons on finite background consisting of a localized nonlinear structure evolving upon a nonzero background plane wave [4][5][6].Solitons on finite background have recently attracted significant interest as their localization dynamics have been proposed as an important mechanism underlying the formation of the infamous extreme amplitude rogue waves on the surface of the ocean [7,8]. Much of this work has also been motivated by a parallel research effort using nonlinear optical fibre systems to implement controlled experiments studying NLSE dynamics and rogue waves in an optical context [9][10][11][12][13][14][15]. Many of the recent studies have focussed on the characteristics of the Akhmediev breather [6], a soliton on finite background solution which is excited from a weak periodic modulation and which is localized in the longitudinal dimension as it undergoes growth and decay. Experiments in optics have demonstrated important links to modulation instability (MI) [9,[12][13][14]: of importance has been the realization that many properties of MI previously described only approximately (via numerical or truncated mode approaches) can in fact be described almost exactly using Akhmedievbreathers. Another significant application of the theory of Akhmediev breathers has been to design experiments generating the rational Peregrine soliton [5], an important and limiting case of a solitons on finite background solution that is localised in both transverse and longitudinal dimensions [10,11].In dispersive optical media, with quadratic nonlinearity, the existence of localized nonlinear structures evolving upon a non-zero background plane wave remained unexplored to date.In this Letter, we investigate the formation of localized nonlinear structures, evolving upon a non-zero background plane wave, in optical systems described by second harmonic generati...

Super chirped rogue waves in optical fibers

“…On the other side, as Eq. (1) has significantly generalized such integrable models as the NLS equation, the CLL-NLS equation, the KN-NLS equation, the GI equation, and the KE equation, we expect that the universal solutions presented here might be used as a platform for exploring the interesting rogue wave dynamics of many complex and non-integrable systems, which, to the first order approximation, are well described by the latter equations [64,65].…”

Rogue Wave Modes for the Coupled Nonlinear Schrödinger System with Three Components: A Computational Study

“…Furthermore, rogue waves from well-studied equations like the NLS equation can be utilized to study rogue waves in less thoroughly studied physical systems. With a suitable physical assumption, optical quadratic solution can be related to the solution of the NLS equation through the second-harmonic asymptotic expansion and the method of repeated substitution [46]. Hence, useful approximations of rogue waves in a quadratic medium can be obtained.…”

Observation of a group of dark rogue waves in a telecommunication optical fiber

“…Although studies on rogue wave solutions of the NLSE are blooming in different areas of physics, moving beyond the standard NLSE model is relevant and needful for describing more general classes of physical systems and applications. In this direction, recent studies have extended the search for rogue waves to coupled wave systems [32][33][34][35][36][37][38]. In fact, a variety of physical phenomena require the modeling of waves with two or more components, in order to account for different spectral peaks, modes, or polar-arXiv:1802.09865v1 [physics.optics] 27 Feb 2018 ization states.…”