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## Theory of quasi-simple dispersive shock waves and number of solitons evolved from a nonlinear pulse

Chaos. 2020. Vol. 30. Article 123148.

Kamchatnov A.M.

The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow, which interacts with the edge wave packets or the edge solitons. A conjecture about the existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In the case of localized simple-wave pulses propagating through a quiescent medium, this theory provided a new approach to derivation of an asymptotic formula for the number of solitons eventually produced from such a pulse.

Kamchatnov A.M., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2019 Vol. 99 No. 1 Article 012203

We suggest a method for calculation of parameters of dispersive shock waves in the framework of Whitham modulation theory applied to nonintegrable wave equations with a wide class of initial conditions corresponding to propagation of a pulse into a medium at rest. The method is based on universal applicability of Whitham’s “number of waves conservation ...

Added: February 4, 2021

Ivanov S., Kamchatnov Anatoly M., Physics of Fluids 2020 Vol. 32 Article 126115

The nonlinear dynamics of pulses in a two-temperature collisionless plasma with the formation of dispersion shock waves is studied. An analytical description is given for an arbitrary form of an initial disturbance with a smooth enough density profile on a uniform density background. For large time after the wave breaking moment, dispersive shock waves are ...

Added: February 4, 2021

Ivanov S. K., Kamchatnov A.M., Physics of Fluids 2019 Vol. 31 Article 057102

We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated, and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage ...

Added: February 4, 2021

Kamchatnov A.M., Chaos 2019 Vol. 29 Article 023106

We discuss the problem of breaking of a nonlinear wave in the process of its propagation into a medium at rest. It is supposed that the profile of the wave is described at the breaking moment by the function (−x) 1/n (x < 0, positive pulse) or −x 1/n (x > 0, negative pulse) of ...

Added: February 4, 2021

Камчатнов А. М., Иванов С. К., Nonlinear Dynamics 2022 Vol. 108 P. 2505-2512

We consider the modulationally stable version of the Kaup–Boussinesq system which models propagation of onlinear waves in various physical situations. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich–Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a ...

Added: October 20, 2022

Камчатнов А. М., Шайкин Д. В., EPL 2021 Vol. 136 Article 40001

We study the evolution of pulses propagating through focusing nonlinear media. A small disturbance of a smooth initial non-uniform pulse amplitude leads to formation of a region of strong nonlinear oscillations. We develop here an asymptotic method for finding the law of motion of the front of this region. The method is based on the conjecture that instability ...

Added: October 20, 2022

Камчатнов А. М., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2022 Vol. 105 Article 044204

Equations for contour dynamics of trough-shaped dark solitons are obtained for the general form of the nonlinearity function. Their self-similar solution which describes the nonlinear stage of the bending instability of dark solitons is studied in detail. ...

Added: October 20, 2022

L. F. Calazans de Brito, A. M. Kamchatnov, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2024 Vol. 109 No. 1 Article 015102

We consider nonlinear wave structures described by the modified Korteweg–de Vries equation, taking into account a small Burgers viscosity for the case of steplike initial conditions. The Whitham modulation equations are derived, which include the small viscosity as a perturbation. It is shown that for a long enough time of evolution, this small perturbation leads to the ...

Added: February 1, 2024

Pelinovsky E., Dutykh D., Physical Letters A 2014 Vol. 378 No. 42 P. 3102-3110

The collective behaviour of soliton ensembles (i.e. the solitonic gas) is studied using the methods of the direct numerical simulation. Traditionally this problem was addressed in the context of integrable models such as the celebrated KdV equation. We extend this analysis to non-integrable KdV–BBM type models. Some high resolution numerical results are presented in both ...

Added: November 19, 2014

Slunyaev A., Кокорина А. В., Journal of Ocean Engineering and Marine Energy 2017 Vol. 3 P. 395-408

The results of the probabilistic analysis of the direct numerical simulations of irregular unidirectional deep
water waves are discussed. It is shown that an occurrence of large-amplitude soliton-like groups represents an extraordinary case, which is able to increase noticeably the probability of high waves even in moderately rough sea conditions. The ensemble of wave realizations should be large enough to take these ...

Added: March 1, 2019

Диденкулова (Шургалина) Е. Г., Кокорина А. В., Slunyaev A., Вычислительные технологии 2019 Т. 24 № 2 С. 52-66

The details of the numerical scheme and the method of specifying the initial conditions for the simulation of the irregular dynamics of soliton ensembles within the framework of equations of the Korteweg – de Vries type are given using the example of the modified Korteweg – de Vries equation with a focusing type of nonlinearity. ...

Added: April 17, 2019

Slunyaev A., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2020 Vol. 101 Article 062214

A method of windowed spatiotemporal spectral filtering is proposed to segregate different nonlinear wave components and to calculate the surface of free waves. The dynamic kurtosis (i.e., produced by the free wave component) is shown able to contribute essentially to the abnormally large values of the surface displacement kurtosis, according to the direct numerical simulations of realistic sea ...

Added: June 28, 2020

Isoard M., Kamchatnov A.M., Pavloff N., EPL 2020 Vol. 129 Article 64003

We consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson equation which is valid for any kind of nonlinearity. The approach is exact for monoatomic classical gas and agrees very well ...

Added: February 4, 2021

Switzerland : Springer, 2018

The present book gathers chapters from colleagues of A. Ezersky from Russia, especially
those from Nizhny Novgorod Institute of Applied Physics of the Russian Academy
of Science and from France, with whom he has been collaborating on experimental
and theoretical developments.
The book is subdivided into two parts. Part I contains eight chapters related to
nonlinear water waves and Part II ...

Added: October 21, 2018

Pelinovsky E., Ezersky A., Abha N., La Houille Blanche 2016 No. 1 P. 58-65

We have prepared two sets of experiments in a wave flume to model effects occurring in nature and to demonstrate resonance phenomena in laboratory conditions. The first set was performed to investigate non‑linear wave run‑up on the beach caused by harmonic wave maker located at some distance from the shore line. It is revealed that ...

Added: October 29, 2016

Slunyaev A., Studies in Applied Mathematics 2019 Vol. 142 P. 385-413

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. ...

Added: March 11, 2019

Slunyaev A., Известия высших учебных заведений. Радиофизика 2018 Т. 61 № 1 С. 1-23

We propose a method for the analysis of groups of unidirectional waves on the surface of deep water, which is based on spectral data of the scattering problem in the approximation of a nonlinear Schrodinger equation. The main attention is paid to the robustness and accuracy of the numerically obtained spectral data. Various methods of choosing the wave ...

Added: March 1, 2019

Pelinovsky E., Touboil J., European J Mechanics B Fluids (Elsivier) 2014 Vol. 48 P. 13-18

The bottom pressure distribution under solitonic waves, travelling or fully reflected at a wall is analysed here. Results given by two kind of numerical models are compared. One of the models is based on the Green–Naghdi equations, while the other one is based on the fully nonlinear potential equations. The two models differ through the ...

Added: November 19, 2014

Boiti M., Pempinelli F., Pogrebkov A., Journal of Mathematical Physics 2011 Vol. 52 No. 083506 P. 1-21

Properties of Jost and dual Jost solutions of the heat equation, F (x,k)
and Y(x,k), in the case of a pure solitonic potential are studied in
detail.We describe their analytical properties on the spectral parameter k
and their asymptotic behavior on the x-plane and we show that the values
of e(−qx)F (x, k) and the residues of exp(qx ...

Added: February 16, 2013

Kurkina O., Rouvinskaya E., Talipova T. et al., Physica D: Nonlinear Phenomena 2016 Vol. 333 P. 222-234

Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg-de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear ...

Added: March 3, 2016

Chabchoub A., Hoffmann N., Onorato M. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2012 Vol. 86 No. 5 P. 156601-1-156601-6

We present experimental observations of the hierarchy of rational breather solutions of the nonlinear Schrodinger equation (NLS) generated in a water wave tank. First, five breathers of the infinite hierarchy have been successfully generated, thus confirming the theoretical predictions of their existence. Breathers of orders higher than five appeared to be unstable relative to the ...

Added: January 18, 2013

Pelinovsky D., Slunyaev A., Kokorina A. et al., Communications in Nonlinear Science and Numerical Simulation 2021 Vol. 101 Article 105855

Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to ...

Added: May 11, 2021

O.E. Kurkina, A.A. Kurkin, T. Soomere et al., Physics of Fluids 2011 Vol. 23 No. 11 P. 116602-1-13-116602-13

We address a specific but possible situation in natural water bodies when the three-layer stratification has a symmetric nature, with equal depths of the uppermost and the lowermost layers. In such case, the coefficients at the leading nonlinear terms of the modified Korteweg-de Vries (mKdV) equation vanish simultaneously. It is shown that in such cases ...

Added: November 6, 2012

Kartashova E., Talipova T., Pelinovsky E., Nonlinear Processes in Geophysics 2013 Vol. 20 No. 4 P. 571-580

The nonlinear deformation of long internal waves in the ocean is studied using the dispersionless Gardner equation. The process of nonlinear wave deformation is determined by the signs of the coefficients of the quadratic and cubic nonlinear terms; the breaking time depends only on their absolute values. The explicit formula for the Fourier spectrum of ...

Added: October 15, 2013