Solitary waves in discrete media with four-wave mixing

Horne, Rudy L.; Kevrekidis, P. G.; Whitaker, N.

doi:10.1103/physreve.73.066601

Cited by 8 publications

(3 citation statements)

References 47 publications

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“…The potential applications of four-wave mixing (FWM) in coupled NLS equations have been discussed in numerous references (e.g. [33–36]). Typical experimentally relevant values, as discussed in [32], are, for example, A = B = D = E = 1 = 2 C = 2 F .…”

Section: Mathematical Formulationmentioning

confidence: 99%

Rogue waves of ultra-high peak amplitude: a mechanism for reaching up to a thousand times the background level

2021

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We unveil a mechanism enabling a fundamental rogue wave, expressed by a rational function of fourth degree, to reach a peak amplitude as high as a thousand times the background level in a system of coupled nonlinear Schrödinger equations involving both incoherent and coherent coupling terms with suitable coefficients. We obtain the exact explicit vector rational solutions using a Darboux-dressing transformation. We show that both components of such coupled equations can reach extremely high amplitudes. The mechanism is confirmed in direct numerical simulations and its robustness is confirmed upon noisy perturbations. Additionally, we showcase the fact that extremely high peak-amplitude vector fundamental rogue waves (of about 80 times the background level) can be excited even within a chaotic background field .

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Section: Mathematical Formulationmentioning

confidence: 99%

Rogue waves of ultra-high peak amplitude: a mechanism for reaching up to a thousand times the background level

2021

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“…Both in this and in the related context of photorefractive crystals, features such as discrete diffraction [3] and its management [4], Talbot revivals [5], PT symmetry and its breaking [6], as well as discrete solitons [3,7] and vortices [8,9] were not only theoretically predicted but also experimentally observed. Variants of the theme of optical waveguide arrays have involved multi-component models bearing multiple polarizations [10,11], waveguides featuring quadratic (so-called χ 2 ) nonlinearities [12,13], and the examination of dark-solitonic states [14,15]. Another theme where extensive related studies have been conducted is the atomic physics realm of Bose-Einstein condensates (BECs) in optical lattices [16,17].…”

Section: Introductionmentioning

confidence: 99%

Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum

Cuevas–Maraver¹,

Aceves²,

Saxena³

2017

J. Phys. A: Math. Theor.

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In the present work, we explore a nonlinear Dirac equation motivated as the continuum limit of a binary waveguide array model. We approach the problem both from a near-continuum perspective as well as from a highly discrete one. Starting from the former, we see that the continuum Dirac solitons can be continued for all values of the discretization (coupling) parameter, down to the uncoupled (so-called anti-continuum) limit where they result in a 9-site configuration. We also consider configurations with 1-or 2-sites at the anti-continuum limit and continue them to large couplings, finding that they also persist. For all the obtained solutions, we examine not only the existence, but also the spectral stability through a linearization analysis and finally consider prototypical examples of the dynamics for a selected number of cases for which the solutions are found to be unstable.

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“…More recently, a number of variants of this theme of optical waveguide arrays have been studied in detail, notable examples being multi-component models involving multiple polarizations [21,22], waveguides featuring quadratic (so-called c 2 ) nonlinearities [23,24], the examination of dark-solitonic states [25,26] and the study of binary waveguide arrays [27][28][29]. Binary waveguide arrays are also used to simulate the behavior of neutrino oscillations in [4].…”

Section: Introductionmentioning

confidence: 99%

Existence, stability and dynamics of discrete solitary waves in a binary waveguide array

Shen¹,

Kevrekidis²,

Srinivasan³

et al. 2016

J. Phys. A: Math. Theor.

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Recent work has explored binary waveguide arrays in the long-wavelength, near-continuum limit, here we examine the opposite limit, namely the vicinity of the so-called anti-continuum limit. We provide a systematic discussion of states involving one, two and three excited waveguides, and provide comparisons that illustrate how the stability of these states differ from the monoatomic limit of a single type of waveguide. We do so by developing a general theory which systematically tracks down the key eigenvalues of the linearized system. When we find the states to be unstable, we explore their dynamical evolution through direct numerical simulations. The latter typically illustrate, for the parameter values considered herein, the persistence of localized dynamics and the emergence for the duration of our simulations of robust quasi-periodic states for two excited sites. As the number of excited nodes increase, the unstable dynamics feature less regular oscillations of the solution's amplitude.

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Solitary waves in discrete media with four-wave mixing

Cited by 8 publications

References 47 publications

Rogue waves of ultra-high peak amplitude: a mechanism for reaching up to a thousand times the background level

Rogue waves of ultra-high peak amplitude: a mechanism for reaching up to a thousand times the background level

Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum

Existence, stability and dynamics of discrete solitary waves in a binary waveguide array

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