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## О СВЯЗИ ДРЕЙФА СТОКСА И ВОЛНЫ ГЕРСТНЕРА

Physics-Uspekhi. 2018. Т. 188. С. 329-334.

Abrashkin A. A., Pelinovsky E.

It is shown in the quadratic approximation that the Gerstner wave vorticity is equal and of different sign to the vorticity of Stokes drift current in deep water. That gives an opportunity to interpret the Stokes wave as a superposition of the Gerstner wave and the Stokes drift and to explain, why the nonlinearity in the nonlinear Shrödinger (NLS) equation for the Gerstner wave is absent. The interpretation of the nonlinearity coefficient in the NLS equation as the Doppler shift of the frequency on the average throughout the vertical drift Stokes current is proposed.

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

Abrashkin A. A., Pelinovsky E., Известия РАН. Физика атмосферы и океана 2018 № 1

The nonlinear Schrödinger (NLS) equation describing the propagation of inhomogeneous vertical wave packets in an infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is shown that the modulation instability criteria of the considered ...

Added: October 16, 2017

Abrashkin A. A., Pelinovsky E., Izvestia, Atmospheric and Oceanic Physic 2018 Vol. 54 No. 1 P. 101-105

A nonlinear Schrцdinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously
vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function
of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is
shown that the modulational instability criteria for the weakly vortical waves and ...

Added: October 3, 2018

Abrashkin A. A., Chaos, Solitons and Fractals 2019 Vol. 118 P. 152-158

We present an analytical description of the class of unsteady vortex surface waves generated by non- uniformly distributed, time-harmonic pressure. The fluid motion is described by an exact solution of the equations of hydrodynamics generalizing the Gerstner solution. The trajectories of the fluid particles are circumferences. The particles on a free surface rotate around circumferences ...

Added: December 17, 2018

Abrashkin A. A., Discrete and Continuous Dynamical Systems 2019 Vol. 39 No. 8 P. 4443-4453

A class of non-stationary surface gravity waves propagating in the
zonal direction in the equatorial region is described in the f -plane approx
imation. These waves are described by exact solutions of the equations of
hydrodynamics in Lagrangian formulation and are generalizations of Gerstner
waves. The wave shape and non-uniform pressure distribution on a free sur
face depend on two ...

Added: June 19, 2019

Abrashkin A. A., Deep-Sea Research Part II: Topical Studies in Oceanography 2019 Vol. 160 P. 3-6

Three Lagrangian invariants are shown to exist for flows in the equatorial region in the β - plane approximation.
They extend the Cauchy invariants to a non-rotating fluid. The relationship between these generalized invariants
and the results following from Kelvin's and Ertel's theorems is ascertained. Explicit expressions of the invariants
for equatorially trapped waves and equatorial Gerstner waves ...

Added: April 2, 2019

A. A. Abrashkin, Monatshefte fur Mathematik 2022 Vol. 199 No. 4 P. 717-732

Propagation of periodic stationaryweakly vortical gravitationalwaves on the freewater surface is considered. Similar wave motion was studied by Gouyon (Ann de la Fac des Sci de l’Université de Toulouse Sér 4(22):1–55, 1958) in linear and quadratic approximations in small parameter of the wave’s steepness ε for the deep water conditions.
In this paper this result is considered for the ...

Added: October 13, 2022

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

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

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

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

Slunyaev A., Pelinovsky E., Water Waves 2020 Vol. 2 No. 1 P. 59-77

The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrödinger ...

Added: April 22, 2020

Abrashkin A. A., Bodunova Y., Fluid Dynamics 2013 Vol. 48 No. 2 P. 223-231

Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the ...

Added: February 25, 2014

Gromov Evgeny, Malomed B., Chaos 2016 Vol. 26 No. 12 P. 123118-1-123118-10

One-parameter families of exact two-component solitary-wave solutions for interacting high-frequency (HF) and low-frequency (LF) waves are found in the framework of Zakharov-type models, which couple the nonlinear Schrödinger equation (NLSE) for intense HF waves to the Boussinesq (Bq) or Korteweg - de Vries (KdV) equation for the LF component through quadratic terms. The systems apply, ...

Added: November 26, 2016

Abrashkin A. A., Pelinovsky E., Nonlinear Processes in Geophysics 2017 Vol. 24 P. 255-264

The nonlinear Schrödinger (NLS) equation describing the propagation of weakly rotational wave packets in an infinitely deep fluid in Lagrangian coordinates has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. The vorticity effects manifest themselves in a shift ...

Added: June 26, 2017

Abrashkin A. A., Oshmarina O. E., Communications in Nonlinear Science and Numerical Simulation 2016 Vol. 34 P. 66-76

The process of rogue wave formation on deep water is considered. A wave of extreme amplitude is born against the background of uniform waves (Gerstner waves) under the action of external pressure on free surface. The pressure distribution has a form of a quasi-stationary “pit”. The fluid motion is supposed to be a vortex one ...

Added: November 3, 2015

Abrashkin A. A., Бодунова Ю. П., Известия РАН. Механика жидкости и газа 2012 № 6 С. 50-59

Изучаются стоячие поверхностные волны в вязкой жидкости бесконечной глубины. Дано решение задачи в линейном и квадратичном приближениях. Подробно проанализирован случай длинных по сравнению с толщиной пограничного слоя волн. Определены траектории жидких частиц и выражение для завихренности. ...

Added: November 19, 2013

Dymov A. V., Kuksin S., Communications in Mathematical Physics 2021 Vol. 382 P. 951-1014

We consider the damped/driven (modified) cubic NLS equation on a large
torus with a properly scaled forcing and dissipation, and decompose its solutions to
formal series in the amplitude. We study the second order truncation of this series and
prove that when the amplitude goes to zero and the torus’ size goes to infinity the energy
spectrum of the ...

Added: June 29, 2021

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

Smirnov A., Matveenko S., Semenova E., Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 2015 Vol. 11

In the article, we describe three-phase finite-gap solutions of the focusing nonlinear Schrödinger equation and Kadomtsev-Petviashvili and Hirota equations that exhibit the behavior of almost-periodic ''freak waves''. We also study the dependency of the solution parameters on the spectral curves. ...

Added: October 15, 2015

Abrashkin A. A., Бодунова Ю. П., Известия РАН. Механика жидкости и газа 2013 № 2 С. 81-89

В рамках лагранжевого подхода разработан метод описания волнового пакета на поверхности бесконечно глубокой вязкой жидкости. Проанализирован случай, когда обратное число Рейнольдса порядка квадрата крутизны волны. Выражения для траекторий жидких частиц определены с точностью до куба крутизны. Указаны условия, при которых эволюция огибающей пакета описывается нелинейным уравнением Шредингера с линейным по амплитуде диссипативным членом. Сформулировано правило, ...

Added: November 19, 2013

Abrashkin A. A., Pelinovsky E., Physics-Uspekhi 2022 Vol. 65 P. 453-467

To mark 220 years since the appearance of Gerstner's paper that proposed an exact solution to the hydrodynamic
equations, an overview of exact solutions for water waves is given, each of which is a generalization of the Gerstner wave. Additional factors are coastal geometry, fluid rotation, varying pressure on the free surface, stratification, fluid compressibility, and background flows. Waves on ...

Added: October 13, 2022

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

Dymov A. V., Куксин С. Б., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2020 Т. 491 № 1 С. 29-37

Авторы обсуждают ряд строгих результатов в стохастической модели волновой турбулентности Захарова–Львова. А именно, рассматривают уравнение Шрёдингера с (модифицированной) кубической нелинейностью и вязкостью на торе большого периода, возмущенное случайной силой, и раскладывают его решение в формальный ряд по амплитуде. Авторы показывают, что в пределе, когда
амплитуда стремится к нулю, а период тора – к бесконечности, спектр энергии ...

Added: June 29, 2021