KDV Soliton Gas: Interactions and Turbulence

Pelinovsky, Efim; Shurgalina, Ekaterina

doi:10.1007/978-3-319-53673-6_18

Cited by 18 publications

(15 citation statements)

References 19 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: 90%

“…Assume that the seed function for 1 is cosh. Then according to (30) and (31), the soliton is positive with amplitude A 1+ = 2 1 tan 1 . Meanwhile from (33) u(0,0) = -2i 1 tanh i 1 = 2 1 tanh 1 .…”

Section: Focal Point Of the Multisoliton Solutionmentioning

confidence: 99%

“…If N = 2, then 1 < 0 and 2 > 0. If the seed function for 1 is cosh, then due to (30), (31) the first soliton is negative with amplitude A 1-= 2 1 cot 1 < 0. Meanwhile, the first summand in (32) for even N gives -2i 1 1 (0, 0)∕ 1 (0, 0) = -2i 1 coth i 1 = 2 1 cot 1 = A 1-.…”

Section: Focal Point Of the Multisoliton Solutionmentioning

confidence: 99%

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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: 90%

Section: Focal Point Of the Multisoliton Solutionmentioning

confidence: 99%

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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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“…This is relevant for the dynamics of discrete chains after transients have decayed such that the chain contains propagating solitary waves with a wide spectrum of energies. Of notable interest here is the recent observation that rogue waves can form in such circumstances [76][77][78].…”

Section: Discussionmentioning

confidence: 99%

Long-wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes

Anco

Przedborski

2018

Phys. Rev. E

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1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of physics brock university st. catharines, on l2s3a1, canada 3 department of applied mathematics university of waterloo waterloo, on n2l 3g1, canada Abstract. Properties of solitary waves in pre-compressed Hertzian chains of particles are studied in the long wavelength limit using a well-known continuum model. Several main results are obtained by parameterizing the solitary waves in terms of their wave speed and their asymptotic amplitude. First, the asymptotic amplitude is shown to be directly related to the continuum sound speed, and the ratio of asymptotic amplitude to peak amplitude is shown to describe the degree of dynamical nonlinearity in the underlying discrete system. Second, an algebraic relation is derived that determines the dynamical nonlinearity ratio in terms of the ratio of the solitary wave speed to the sound speed. In particular, highly supersonic solitary waves correspond to highly nonlinear propagating pulses in weakly compressed systems, and slightly supersonic solitary waves correspond to weakly nonlinear propagating pulses in strongly compressed systems. Third, explicit formulas for the physical height, width, impulse and energy of the solitary waves are obtained in both the strongly nonlinear regime and the weakly nonlinear regime. Asymptotic expansions are used to show that in the strongly nonlinear regime, solitary waves are well-approximated by Nesterenko's compacton (having the same wave speed), while in the weakly nonlinear regime, solitary waves coincide with solitons of the Korteweg-de Vries (KdV) equation, with the same wave speed. All of these results are illustrated by means of exact solitary wave solutions, including the physically important case that models a chain of spherical particles. email: mp06lj@brocku.ca, sanco@brocku.ca 4

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“…We may explain the sign of the computed skewness applying the weakly nonlinear theory of internal waves based on the Korteweg-de Vries equation (Pelinovsky and Shurgalina, 2017). For given area in the Yellow Sea the sign of quadratic 20 nonlinear term in this equation is negative, because the water stratification (see in Wang and Gao, 2002) is practically twolayer with pycnocline situated above the half-depth (Djorjevich and Redekopp, 1978;Kakutani and Yamasaki, 1978).…”

Section: Probability Density Function In the Yellow Seamentioning

confidence: 96%

Exceedance frequency of appearance of extreme internal waves in the World Ocean

Talipova¹,

Pelinovsky²,

Kurkina³

et al. 2018

Preprint

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Abstract. Statistical estimates of internal wave appearance in different regions of the World Ocean are discussed. It is found that the observed exceedance probability of large-amplitude internal waves in most cases can be described by the Poisson curve, which is one of the typical curves of extreme statistics. Detailed analysis is done for the internal waves in the several areas of the World Ocean: tropical part of the Atlantic Ocean, the North-West shelf of Australia, the Mediterranean Sea near 10 the Egyptian Coast and the Yellow Sea.

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KDV Soliton Gas: Interactions and Turbulence

Cited by 18 publications

References 19 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

Long-wavelength solitary waves in Hertzian chains and their properties in different nonlinearity regimes

Exceedance frequency of appearance of extreme internal waves in the World Ocean

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