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## Evolution of wave pulses in fully nonlinear shallow-water theory

Physics of Fluids. 2019. Vol. 31. Article 057102.

Ivanov S. K., Kamchatnov A.M.

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 of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.

Kamchatnov A.M., Chaos 2020 Vol. 30 Article 123148

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 ...

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

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

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

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., 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

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

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

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

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

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

Abrashkin A. A., Pelinovsky E., Physics-Uspekhi 2018 Vol. 61 P. 307-312

We discuss the properties of two-dimensional, non
linear, potential, and vortex waves on the surface of an ideal
liquid of infinite depth. It is shown that in the quadratic order in
the amplitude, the vorticity of the Gerstner wave is equal in
magnitude to and different in sign from that of the Stokes drift
current in a surface layer. This ...

Added: October 3, 2018

Oshmarina O. E., Новый университет. Серия: Вопросы естественных наук 2012 Т. 3 № 6 С. 17-23

В настоящей работе рассмотрены и проанализированы вопросы, связанные с гидрогеологией Горьковского водохранилища. Для учета способности потока переносить различный донный материал исследованы поля течений в бассейне Горьковского водохранилища. На основе полученных результатов представлены прогнозные характеристики эрозии береговой линии Горьковского водохранилища. ...

Added: October 26, 2012

Пелиновский Д. Е., Rouvinskaya E., Kurkina O. E. et al., Теоретическая и математическая физика 2014 Т. 179 № 1 С. 78-89

We prove that line solitons of the two-dimensional hyperbolic nonlinear Schr¨odinger equation are unstable
under transverse perturbations of arbitrarily small periods, i.e., short waves. The analysis is based on
the construction of Jost functions for the continuous spectrum of Schr¨odinger operators, the Sommerfeld
radiation conditions, and the Lyapunov–Schmidt decomposition. We derive precise asymptotic expressions
for the instability growth rate ...

Added: May 13, 2014

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

Li C., Antonina A. Arkhipova, Kartashov Y. et al., ACS Photonics 2023 Vol. 10 No. 6 P. 1976-1982

We study both experimentally and theoretically the propagation of light in the fs-laser written rotating square waveguide arrays and present the first experimental evidence of light localization induced by the rotation of periodic structure in the direction of light propagation. Such linear light localization occurs either in the corners of truncated square array, where it results from the interplay between ...

Added: June 27, 2023

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

Abrashkin A. A., Oshmarina O. E., Physics Letters A 2014 Vol. 378 P. 2866-2871

A vortical model of breather overturning on deep water is proposed. The action of wind is simulated by nonuniform pressure on the free surface. The fluid motion is described by an exact solution of 2D hydrodynamic equations for an inviscid fluid in Lagrangian variables. Fluid particles rotate in circles of different radii. Formation of contraflexure ...

Added: September 16, 2014

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

S. S. Gavrilov, Physical Review B: Condensed Matter and Materials Physics 2020 Vol. 102 Article 104307

It is generally accepted that quantized vortices formed in coherent bosonic fluids are “excitations” and as such do not arise in a one-mode condensate at zero temperature. To excite them, one needs either inhomogeneities (impurities, rotation, etc.) or essentially finite fluctuations. Here, we predict a perfectly spontaneous formation of vortices even at zero temperature, which ...

Added: March 11, 2021

Abrashkin A. A., Bodunova Y., Fluid Dynamics 2012 Vol. 47 No. 6 P. 725-734

Standing surface waves in a viscous infinite-depth fluid are studied. The solution of the problem is obtained in the linear and quadratic approximations. The case of long, as compared with the boundary layer thickness, waves is analyzed in detail. The trajectories of fluid particles are determined and an expression for the vorticity is derived. ...

Added: February 25, 2014

Abrashkin A. A., Pelinovsky E., Journal of Physics A: Mathematical and Theoretical 2021 Vol. 54 No. 39 Article 395701

To study stationary periodic weakly vortical waves on water (the Gouyon waves), the method of the modified Lagrangian variables is suggested. The wave vorticity Ω is specified as a series in the small steepness parameter ε in
the form: Ω =\sum( ε^n · Ω_n (b)), where Ω_n are arbitrary functions of the vertical Lagrangian coordinate b. Earlier Gouyon (1958) studied ...

Added: October 5, 2021

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

A. V. Slunyaev, T. V. Tarasova, Chaos 2022 Vol. 32 Article 101102

Synchronous collisions between a large number of solitons are considered in the context of a statistical description. It is shown that, during the interaction of solitons of the same signs, the wave field is effectively smoothed out. When the number of solitons increases and the sequence of their amplitudes decay slower, the focused wave becomes even smoother ...

Added: October 14, 2022