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## Задача Гуревича-Питаевского и ее развитие

Успехи физических наук. 2021. Т. 191. № 1. С. 52-87.

Камчатнов А. М.

We present an introduction to the theory of dispersive shock waves in the framework of the approach proposed by Gurevich and Pitaevskii (Zh. Eksp. Teor. Fiz., Vol. 65, p. 590 (1973) [Sov. Phys. JETP, Vol. 38, p. 291 (1974)]) based on Whitham’s theory of modulation of nonlinear waves. We explain how Whitham equations for a periodic solution can be derived for the Korteweg–de Vries equation and outline some elementary methods to solve them. We illustrate this approach with solutions to the main problems discussed by Gurevich and Pitaevskii. We consider a generalization of the theory to systems with weak dissipation and discuss the theory of dispersive shock waves for the Gross-Pitaevskii equation.

Bienaime T., Isoard M., Fontaine Q. et al., Physical Review Letters 2021 Vol. 126 Article 183901

We report on the formation of a dispersive shock wave in a nonlinear optical medium. We monitor the evolution of the shock by tuning the incoming beam power. The experimental observations for the position and intensity of the solitonic edge of the shock, as well as the location of the nonlinear oscillations are well described ...

Added: October 14, 2021

Kamchatnov A.M., L. F. Calazans de Brito, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2021 Vol. 104 Article 054203

We show that the number of solitons produced from an arbitrary initial pulse of the simple wave type can be calculated analytically if its evolution is governed by a generalized nonlinear Schroedinger equation provided this number is large enough. The final result generalizes the asymptotic formula derived for completely integrable nonlinear wave equations like the standard NLS equation ...

Added: January 17, 2022

Kamchatnov A., Shaykin D. V., Physics of Fluids 2021 Vol. 33 Article 052120

We consider propagation of high-frequency wave packets along a smooth evolving background flow whose evolution is described by a simple-wave type of solutions of hydrodynamic equations. In geometrical optics approximation, the motion of the wave packet obeys the Hamilton equations with the dispersion law playing the role of the Hamiltonian. This Hamiltonian depends also on ...

Added: October 14, 2021

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

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

Savina O. N., Gromov E., Tyutin V. V., Геомагнетизм и аэрономия 2015 Т. 55 № 5 С. 675-679

The possibility of the existence of the solitary internal gravity waves on the Earth's thermosphere heights is discussed. Analytical results are obtained with local approximation weak
nonisothermal atmosphere. For internal gravity waves is derived Korteweg-de Vries equation with the heterogeneity, nonlinearity and dissipation. Theoretical
results are used to interpret the basic parameters of solitary traveling ionospheric disturbances ...

Added: July 2, 2015

Blyakhman L. G., Gromov E., Onosova I. V. et al., Physics Letters A 2017 Vol. 381 No. 17 P. 1490-1492

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger ...

Added: February 3, 2017

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

Wave amplification in the framework of forced nonlinear Schrödinger equation: the rogue wave context

Pelinovsky E., Slunyaev A., Sergeeva A., Physica D: Nonlinear Phenomena 2015 Vol. 303 P. 18-27

Irregular waves which experience the time-limited external forcing within the framework of the nonlinear Schrödinger (NLS) equation are studied numerically. It is shown that the adiabatically slow pumping (the time scale of forcing is much longer than the nonlinear time scale) results in selective enhancement of the solitary part of the wave ensemble. The slow ...

Added: August 4, 2015

Flamarion M. V., Pelinovsky E., Chaos, Solitons and Fractals 2022 Vol. 165 Article 112889

The aim of this work is to study asymptotically and numerically the interaction of solitons with an external
forcing with a variable speed using the forced modified Korteweg–de Vries equation (mKdV). We show that the
asymptotic predictions agree well with numerical solutions for forcings with constant speed and linear variable
speed. Regarding forcing with linear variable speed, we ...

Added: November 19, 2022

Aseeva N., Gromov E., Onosova I. V. et al., JETP Letters 2016 Vol. 103 No. 10 P. 737-741

Evolution of solitons is addressed in the framework of a third-order nonlinear Schrodinger equation (NLSE), including nonlinear dispersion, third-order dispersion and a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term which is well known as a part of the temporal-domain NLSE in optics. In this context, it is induced by the underlying ...

Added: May 4, 2016

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

Ostrovsky L., E. Pelinovsky, Shrira V. et al., Chaos 2024 Vol. 34 No. 6 Article 062101

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV ...

Added: June 14, 2024

Aseeva N., Gromov E., Onosova I. V. et al., JETP Letters 2016 Vol. 103 No. 10 P. 653-657

Evolution of solitons is addressed in the framework of a third-order nonlinear Schrödinger equation (NLSE), including nonlinear dispersion, third-order dispersion and a pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a spatial-domain counterpart of the SRS term, which is well known as a part of the temporal domain NLSE in optics. In this context, it is induced by the ...

Added: June 28, 2016

Blyakhman L. G., Morozov V. P., Tyutin V. V., Труды НГТУ им. Р.Е. Алексеева 2016 № 3 (114) С. 9-15

Цель работы: Исследована динамика двухкомпонентных (векторных) солитонов Давыдова-Скотта(ДС) при пространственно-скоростном рассогласовании высокочастотных (ВЧ) и низкочастотных (НЧ) компонент. Рассмотрение проведено в рамках Захаровского типа системы двух связанных уравнений для ВЧ и НЧ поля. В этой системе ВЧ поле описывается линейным уравнением Шредингера с переменным во времени и пространстве потенциалом, вызванным НЧ компонентой. НЧ компонента в этой ...

Added: October 5, 2016

Aseeva N., Blyakhman L. G., Gromov E. et al., Труды НГТУ им. Р.Е. Алексеева 2016 № 2 С. 9-16

Цель работы: Рассмотрена динамика солитонов в рамках расширенного нелинейного уравнения Шредингера, учитывающего псевдо-рамановское рассеяние (рассеяние на затухание низкочастотных волнах), нелинейную дисперсию и пространственную неоднородность дисперсии второго порядка.
Результат: Показано, что при наличии нелинейной дисперсии сдвиг вниз спектра волновых чисел солитона, вызванный псевдо-рамановским рассеянием, может быть компенсирован сдвигом спектра вверх, вызванным пространственным уменьшением дисперсии второго порядка. ...

Added: June 24, 2016

Kachulin D., Dyachenko A., Gelash A., Fluids 2019 Vol. 4 No. 2 Article 83

We numerically investigate pairwise collisions of solitary wave structures on the surface of deep water—breathers. These breathers are spatially localised coherent groups of surface gravity waves which propagate so that their envelopes are stable and demonstrate weak oscillations. We perform numerical simulations of breather mutual collisions by using fully nonlinear equations for the potential flow ...

Added: October 22, 2022

Gromov E.M., Malomed B. A., Communications in Nonlinear Science and Numerical Simulation 2020 Vol. 85 Article 105220

We introduce a nonlinear Schrödinger equation (NLSE) which combines the pseudo-stimulated-Raman-scattering (pseudo-SRS) term, i.e., a non-conservative cubic one with the first spatial derivative, and an external potential, which helps to stabilize solitons against the pseudo-SRS effect. Dynamics of solitons is addressed by means of analytical and numerical methods. The quasi-particle approximation (QPA) for the solitons ...

Added: February 5, 2020

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

Pelinovsky E., E.G. Shurgalina, Sergeeva A. et al., Physical Letters A 2013 Vol. 377 No. 1 P. 272-275

Two-soliton interactions play a definitive role in the formation of the structure of soliton turbulence in integrable systems. To quantify the contribution of these interactions to the dynamical and statistical characteristics of the nonlinear wave field of soliton turbulence we study properties of the spatial moments of the two-soliton solution of the Korteweg–de Vries (KdV) ...

Added: January 18, 2013

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

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

Gromov E.M., Malomed B. A., Physica Scripta 2015 Vol. 90 No. 6 P. 068021-1-068021-6

Dynamics of Langmuir solitons is considered in plasmas with spatially inhomogeneous electron temperature. An underlying Zakharov-type system of two unidirectional equations for the Langmuir and ion-sound fields is reduced to an inhomogeneous nonlinear Schrödinger equation (NLSE) with spatial variation of the second-order dispersion (SOD) and self-phase modulation (SPM) coefficients, induced by the spatially inhomogeneous profile ...

Added: April 25, 2015