Wave instabilities in the presence of non vanishing background in nonlinear Schrödinger systems

Trillo, Stefano; Gongora, Juan Sebastian Totero; Fratalocchi, Andrea

doi:10.1038/srep07285

Cited by 5 publications

(3 citation statements)

References 50 publications

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“…In other words, the numerical results show that, in the supercritical regime, MI prevails over the collapse instability. This finding is compatible with the fact that the presence of the background usually restricts the conditions for the blow-up to occur [51], though further studies (beyond the scope of the present paper) are needed to deepen our understanding of the general interplay between collapse and MI. Conversely, for the DMNLS equation and the saturable nonlinearity model, the weaker nonlinearity results in a narrower oscillation wedge compared to the case of s = 1 and, in the case of the saturable nonlinearity model, a lower peak amplitude.…”

Section: Nonlinear Stage Of Modulational Instabilitysupporting

confidence: 83%

Universal Behavior of Modulationally Unstable Media

Biondini¹,

Li²,

Mantzavinos³

et al. 2018

SIAM Rev.

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Evidence is presented of universal behavior in modulationally unstable media. An ensemble of nonlinear evolution equations, including three partial differential equations, an integro-differential equation, a nonlocal system and a differential-difference equation, is studied analytically and numerically. Collectively, these systems arise in a variety of applications in the physical and mathematical sciences, including water waves, optics, acoustics, Bose-Einstein condensation, and more. All these models exhibit modulational instability, namely, the property that a constant background is unstable to long-wavelength perturbations. In this work, each of these systems is studied analytically and numerically for a number of different initial perturbations of the constant background, and it is shown that, for all systems and for all initial conditions considered, the dynamics gives rise to a remarkably similar structure comprised of two outer, quiescent sectors separated by a wedge-shaped central region characterized by modulated periodic oscillations. A heuristic criterion that allows one to compute some of the properties of the central oscillation region is also given.AMS subject classifications. 35Q55, 37Kxx, 37K40, 74J30 Remarkably, the nonlinear stage of MI in the NLS equation with periodic boundary conditions is described in terms of a homoclinic structure [8,50] characterized by two qualitatively different families of recurrent (or, more strictly, doubly-periodic) solutions, through which the perturbation is cyclically amplified and back-converted to the background. These two families are separated by the so called Akhmediev breather, which represents the separatrix of the homoclinic structure, featuring a single cycle of conversion and back-conversion, with its unstable manifold corresponding to the MI linearized growth.In the more general non-periodic case (i.e., for localized perturbations of the constant background), however, the dynamics of MI is strikingly different. Indeed, using the IST for the focusing NLS equation with non-zero background [13], it was shown in [12] that in this scenario (i.e., constant boundary conditions at infinity, corresponding to localized per- †

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Section: Nonlinear Stage Of Modulational Instabilitysupporting

confidence: 83%

Universal Behavior of Modulationally Unstable Media

Biondini¹,

Li²,

Mantzavinos³

et al. 2018

SIAM Rev.

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“…But because of the limited extinction ratio of high-bandwidth e-o modulators (typically 30 dB) the pulse train is superimposed on a residual continuous background. Even a weak background strongly enhances the extension and the contrast of the temporal oscillations inherent to the dispersive shock [14,15,23].…”

mentioning

confidence: 99%

Spectral broadening of picosecond pulses forming dispersive shock waves in optical fibers

et al. 2017

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We investigate analytically, numerically, and experimentally the spectral broadening of pulses that undergo the formation of dispersive shocks, addressing in particular pulses in the range of tens of ps generated via electro-optic modulation of a continuous-wave laser. We give an analytical estimate of the maximal spectral extension and show that super-Gaussian waveforms favor the generation of flat-topped spectra. We also show that the weak residual background of the modulator produces undesired spectral ripples. Spectral measurements confirm our estimates and agree well with numerical integration of the nonlinear Schrödinger equation.

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“…Moreover, nonlinear dynamics of extended systems, having infinite degrees of freedom, and the nonlinear dynamics of waves in particular, is one of its most fruitful areas, as historically many of its achievements were later applied to explain miscellaneous physical phenomena that unexpectedly shared the same theoretical basis. Pervasive equations such as the nonlinear Schrödinger equation (central in this work) or the Korteweg-de Vries equation successfully describe nonlinear Introduction effects such as solitons in water waves (Russell 1844), laser beam self-focusing in nonlinear media (Konno and Suzuki 1979), Bose-Einstein condensates (Dodd 1996;Tian et al 2006), Langmuir waves (Kuo 2014), wave breaking and collapse (Silberberg 1990;Trillo, Totero-Góngora, and Fratalocchi 2014) and ocean turbulence (Costa et al 2014) in fields as diverse as optical communications, relativity and hydrodynamics. Solitons, singularities, vortices and other complex structures often emerge from these equations, requiring everincreasing levels of detail and precision in analytical and numerical models that must be faithful to experimental results done at higher energies or finer scales of space and time measurements.…”

Section: Introduction "Uh What?"mentioning

confidence: 99%

Estabilidad e inestabilidad en la propagación no lineal de haces ópticos de Bessel con momento angular orbital

Carvalho¹,

da²

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Nevertheless, after an uncertain first year, nonlinear waves and their vortices finally caught me up. And also my doctoral advisor, Professor MiguelÁngel Porras. It's hard to express how thankful I am for his generosity in sharing with me both his time and his work the way he did. He didn't teach me all the Physics I know, but he definitely taught me how to persevere in discovering the Physics I don't know (which is a lot more!), and that's priceless. Thank you, MiguelÁngel. And as a bonus, I was also very lucky to have a senior Ph.D. candidate colleague that still to this day shows me the way to follow through when I don't know where the end of the tunnel is, Dr. Carlos Ruiz Jiménez. You helped me a lot more than you can imagine, believe me, both with our work and your words. During the last years I learned much from the two of you, and your patience and kindness were always there, during good and bad moments. I will never forget the endless rehearsal for my presentation at the XXXVI Bienal de Física in Santiago de Compostela in 2017, and the adrenaline rush when I finally got it right for the first time. And I truly have the feeling to have collaborated in bringing some new and useful knowledge around, which makes everything worthwhile.Also, I would like to thank the Grupo de Sistemas Complejos, in particular Professor Rosa María Benito, for kindly accepting me in the research group and offering me the opportunity to know first-hand what the world of scientific research is; and Professor Juan Carlos Losada, for always volunteering when help was needed, both for scientific and organizational matters. The research group is fairly large, and it would be out of place to name everybody here, but I'd like to say I'm grateful for the kindness I always received from all of you, and that I'm sorry for the CPU time Matlab R ruthlessly ate up at the servers with our endless numerical simulations. Some special words of gratitude must be dedicated to Marta Antón, who helped me a lot with the front cover during those idle moments at work. I guess my sense of aesthetics needed more inspiration than I expected in first place.And at last, but surely not least, I would like to thank my family for putting up with me during my long isolation periods necessary to do what I had to do these last years. In fact, I wanted to make a point with that by using quotations from people who are closest to me to introduce each chapter, instead of picking more glamorous ones from historically relevant characters. I would like this choice to be seen not as the result of indolence or arrogance, but as a way of thanking people that, although feeling distant from the "weird things" physicist are usually interested in, they should know that they are still part of it all, at least for me. I hope I will now have more spare time to make up for it correctly. Thank you Marisa, Xavier, Ana and Nico for your support. This particular acknowledgement may be short, but it is from the heart.

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Wave instabilities in the presence of non vanishing background in nonlinear Schrödinger systems

Cited by 5 publications

References 50 publications

Universal Behavior of Modulationally Unstable Media

Universal Behavior of Modulationally Unstable Media

Spectral broadening of picosecond pulses forming dispersive shock waves in optical fibers

Estabilidad e inestabilidad en la propagación no lineal de haces ópticos de Bessel con momento angular orbital

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