Nonlinear dynamics of a soliton gas: Modified Korteweg–de Vries equation framework

Shurgalina, Ekaterina; Pelinovsky, Efim

doi:10.1016/j.physleta.2016.04.023

Cited by 64 publications

(46 citation statements)

References 23 publications

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“…As the KdV equation possesses positive solitons only, they never produce large waves in the focal point (as shown in Figure 1), according to (46). This conclusion is in qualitative agreement with the results of direct numerical simulations of irregular soliton ensembles, 26,30 which did not observe extreme wave events in the ensembles of unipolar solitons.…”

Section: Discussionsupporting

confidence: 91%

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On the optimal focusing of solitons and breathers in long‐wave models

Slunyaev

2019

Stud Appl Math

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Conditions of optimal (synchronized) collisions of any number of solitons and breathers are studied within the framework of the Gardner equation (GE) with positive cubic nonlinearity, which in the limits of small and large amplitudes tends to other long-wave models, the classic and the modified Korteweg-de Vries equations. The local solution for an isolated soliton or breather within the GE is obtained. The wave amplitude in the focal point is calculated exactly. It exhibits a linear superposition of partial amplitudes of the solitons and breathers. The crucial role of the choice of proper soliton polarities and breather phases on the cumulative wave amplitude in the focal point is demonstrated. Solitons are most synchronized when they have alternating polarities. The straightforward link to the problem of synchronization of envelope solitons and breathers in the focusing nonlinear Schrödinger equation is discussed (then breathers correspond to envelope solitons propagating above a condensate). K E Y W O R D S amplitude in the focal point, Gardner equation, Korteweg-de Vries equation, modified Korteweg-de Vries equation, nonlinear Schrödinger equation, optimal focusing of solitons and breathers Stud Appl Math. 2019;142:385-413.wileyonlinelibrary.com/journal/sapm

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Section: Discussionsupporting

confidence: 91%

“…. Such "absorb-emit" collisions were also considered in Ref 26. as the reason for the generation of abnormally high waves in a soliton gas.The presented scenarios of soliton collisions are similar to the ones depicted in Ref 17.…”

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confidence: 97%

On the optimal focusing of solitons and breathers in long‐wave models

Slunyaev

2019

Stud Appl Math

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“…Hence, the peculiarity of the 'absorb-emit' collision of bipolar solitons yields occurrence of higher waves than could happen in the situation of a unipolar soliton gas. In particular, the occurrence of high waves which are twice higher than the typical soliton height was observed in numerical simulations [6].…”

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confidence: 79%

Role of Multiple Soliton Interactions in the Generation of Rogue Waves: The Modified Korteweg–de Vries Framework

Slunyaev¹,

Pelinovsky²

2016

Phys. Rev. Lett.

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The role of multiple soliton and breather interactions in formation of very high waves is disclosed within the framework of integrable modified Korteweg -de Vries (mKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and thus cannot be taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of focusing mKdV equation, when solitons possess 'frozen' phases (polarities), though the approach works in other integrable systems which admit soliton and breather solutions. Introduction.-The generation of unexpectedly large waves from stochastic fields has attracted much interest in many recent studies thanks to recognition of the rogue wave phenomenon in marine and optical realms (see [1] and many others). The modulational instability is the most recognized physical effect capable of generation of very high waves due to the energy transfer from many waves towards an inoculating perturbation of the wavetrain. Most importantly, the nonlinear dynamics alters essentially the statistical properties of stochastic waves, favouring occurrence of very high waves. The accounting for significant deviations from the quasi Gaussian states breaks down the classic assumptions of the wave turbulence theory. The wave phase averaging becomes inappropriate, thus direct simulations of irregular waves are involved to discover the statistics of high waves.

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“…If two eigenvalues λ n and λ m turn out to be close to each other, the numerical implementation of both the dressing method and the determinant formula may fail due to very small denominators in Eqs. (12), (14) (for λ n = λ m these equations become indeterminate). To avoid such situations, we use threshold δλ = 10 −9 for minimal distance between the eigenvalues |λ n − λ m | > δλ, n, m = 1, ..., N, m = n.…”

Section: A Initial Conditionsmentioning

confidence: 99%

Strongly interacting soliton gas and formation of rogue waves

Gelash

Агафонцев

2018

Phys. Rev. E

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We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional Nonlinear Schrödinger (NLS) equation. In order to model such gas we use N -soliton solutions (N -SS) with N ∼ 100, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wave intensity.We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multi-soliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

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Nonlinear dynamics of a soliton gas: Modified Korteweg–de Vries equation framework

Cited by 64 publications

References 23 publications

On the optimal focusing of solitons and breathers in long‐wave models

On the optimal focusing of solitons and breathers in long‐wave models

Role of Multiple Soliton Interactions in the Generation of Rogue Waves: The Modified Korteweg–de Vries Framework

Strongly interacting soliton gas and formation of rogue waves

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