Modulation instability: The beginning

Proceedings of the National Academy of Sciences, 2021

Significance Modulation instability (MI) is a ubiquitous phenomenon in physics, corresponding to the growth of a weakly modulated continuous wave in a nonlinear medium and leading to the generation of a large-amplitude periodic wave train. In space, it transforms weakly modulated plane waves into spatially periodic patterns. In frequency domain, the MI is the result of energy transfer from a strong single spectral component into sidebands. While linear stability analysis predicts a limited band of unstable frequencies of modulation, recent developments based on a nonlinear theory revealed the existence of MI beyond this limited frequency range. These experimental studies are the first experimental demonstrations of the “extraordinary” MI phenomenon. Achieved both in optics and hydrodynamics, they clearly further highlight the interdisciplinary of this process.

2011

Different from statistical considerations on stochastic wave fields, this paper aims to contribute to the understanding of (some of) the underlying physical phenomena that may give rise to the occurence of extreme, rogue, waves. To that end a specific deterministic wavefield is investigated that develops extreme waves from a uniform background. For this explicitly described nonlinear extension of the Benjamin-Feir instability, the soliton on finite background of the NLS equation, the global down-stream evolving distortions, the time signal of the extreme waves, and the local evolution near the extreme position are investigated. As part of the search for conditions to obtain extreme waves, we show that the extreme wave has a specific optimization property for the physical energy, and comment on the possible validity for more realistic situations.

Physics Letters A, 1993

Modulation (Benjamin-Feir) instability of propagating quasi-harmonic wave in a nonlinear dispersive medium near cut-off frequency is analyzed. It is shown, that the increasing of wave amplitude causes the transition from convective to absolute instability. Physically this phenomena is explained by expansion of instability region to the area of backward waves, that have negative group velocity. The results of numerical simulations that verify the developed theory are presented. Application of the theory to the problem of soliton tunnelling is discussed.

Journal of Applied Mathematics, 2012

Modulation instability is one of the most ubiquitous types of instabilities in nature. As one of the key characteristics of modulation instability, the most unstable condition attracts lots of attention. The most unstable condition is investigated here with two kinds of initial wave systems via a numerical high-order spectral method (HOS) for surface water wave field. Classically, one carrier wave and a pair of sidebands are implied as the first kind of initial wave system: “seeded” wave system. In the second kind of initial wave system: “un-seeded” wave system, only one carrier wave is implied. Two impressive new results are present. One result shows that the grow rates of lower and upper sideband are different within the “seeded” wave system. It means that, for a given wave steepness, the most unstable lower sideband is not in pair with the most unstable upper sideband. Another result shows the fastest growing sidebands are exactly in pair from “unseeded” wave system. And the most...

Journal of the Optical Society of America B, 1997

Technical Physics Letters, 2004

Modulation instability of a quasi-harmonic wave propagating in a cubic nonlinear medium is analyzed in the vicinity of a critical frequency. As the wave amplitude increases, the character of instability changes from convective to absolute. This is explained by expansion of the range of unstable perturbations to the region of counterpropagating waves possessing a negative group velocity. The proposed theory is confirmed by the results of computer simulations. © 2004 MAIK "Nauka/Interperiodica".

Nature Photonics, 2012

Stochastically driven nonlinear processes are responsible for spontaneous pattern formation and instabilities in numerous natural and artificial systems, including well-known examples such as sand ripples, cloud formations, water waves, animal pigmentation and heart rhythms 1-3 . Technologically, a type of such self-amplification drives free-electron lasers 4,5 and optical supercontinuum sources 6,7 whose radiation qualities, however, suffer from the stochastic origins . Through timeresolved observations, we identify intrinsic properties of these fluctuations that are hidden in ensemble measurements. We acquire single-shot spectra of modulation instability produced by laser pulses in glass fibre at megahertz real-time capture rates. The temporally confined nature of the gain physically limits the number of amplified modes, which form an antibunched arrangement as identified from a statistical analysis of the data. These dynamics provide an example of pattern competition and interaction in confined nonlinear systems.

Lecture Notes in Physics, 2016

It is a matter of experience that nonlinear waves in dispersive media, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristicsamplitude, phase, wave number, etc.-slowly vary in large space and time scales. In the 1970's, Whitham developed an asymptotic (WKB) method to study the effects of small "modulations" on nonlinear periodic wave trains. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham's formal theory. We discuss recent advances in the mathematical understanding of the dynamics, in particular, the instability of slowly modulated wave trains for nonlinear dispersive equations of KdV type.

Natural Hazards and Earth System Sciences, 2010

The evolution of modulational instability, or Benjamin-Feir instability is investigated within the framework of the two-dimensional fully nonlinear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles' theory. The introduction of dissipation in the equations is briefly discussed. Evolution of this instability in the presence of damping was considered by Segur et al. (2005a) and Wu et al. (2006). Their results were extended theoretically by Kharif et al. (2010) who considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon.