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## Pressure induced breather overturning on deep water: Exact solution

Physics Letters A. 2014. Vol. 378. P. 2866-2871.

Abrashkin A. A., Oshmarina O. E.

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 points on the breather profile is studied. The mechanism of wave breaking and the role of flow vorticity are discussed.

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., 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., 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

Abrashkin A. A., Yakubovich E. I., Radiophysics and Quantum Electronics 2016 Vol. 58 No. 11 P. 852-857

We show that the discrete frequency spectrum of a plane hydrodynamic flow of ideal incompressible liquid with localized trajectories of the liquid particles can contain only one, two, or an infinite number of harmonics. ...

Added: November 17, 2016

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

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., Pelinovsky E., Успехи физических наук 2018 Т. 188 С. 329-334

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 ...

Added: October 17, 2017

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., Soloviev A., Fluid Dynamics 2013 Vol. 48 No. 5 P. 679-686

Plane periodic oscillations of an infinitely deep fluid are studied in the case of a nonuniform pressure distribution over its free surface. The fluid flow is governed by an exact solution of the Euler equations in the Lagrangian variables. The dynamics of an oscillating standing soliton are described, together with the scenario of the soliton ...

Added: November 19, 2013

Abrashkin A. A., Journal of Physics A: Mathematical and Theoretical 2022 Vol. 55 No. 41 Article 415701

An analytical description of unsteady edge waves over a uniform slope is proposed. It is assumed that the waves are excited by time-harmonic external pressure with inhomogeneous spatial distribution. The problem is considered in Lagrangian variables. An exact solution of the hydrodynamic equations is obtained. It generalizes the stationary Gerstner–Constantin solution. The proposed model describes the dynamics of coastal splashes ...

Added: October 13, 2022

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

Tamm M., Stadnichuk V., Ilyina A. et al., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2014 Vol. 89 P. 042137

We consider two random walkers starting at the same time t = 0 from different points in space separated by a given distance R. We compute the average volume of the space visited by both walkers up to time t as a function of R and t and dimensionality of space d. For d < 4, this volume, after proper renormalization, is shown to be ...

Added: May 23, 2014

Bedrikovetsky P., Osipov Y., l.Kuzmina et al., Water Resources Research 2019 Vol. 55 No. 2 P. 1011-1039

The article investigates one‐dimensional (1D) suspension‐colloidal transport of size distributed particles with particle attachment. A population balance approach is presented for computing the particle transport and capture by porous media. The occupied area of each attached particle is particle‐size dependent. The main model assumption is the retention‐rate dependency of the overall vacancy concentration for all ...

Added: January 18, 2019

Sinelshchikov D., Garashchuk I., Kudryashov N. A., European Journal of Physics: Web of Conference 2018 Vol. 173 P. 03008-1-03008-4

We consider a generalization of the Rayleigh equation for the description of
the dynamics of a spherical gas bubble oscillating near an elastic or rigid wall. We show
that in the non–dissipative case, i.e. neglecting the liquid viscosity and compressibility, it
is possible to construct the general analytical solution of this equation. The corresponding
general solution is expressed via ...

Added: December 16, 2019

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

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

Sinelshchikov D., Garashchuk I., Kudryashov N. A., Journal of Physics: Conference Series 2017 Vol. 788 P. 012013 -1-012013 -6

Non-linear dynamical systems describe many physical processes. In this work we investigate a three-dimensional Lorenz-like system - the Glukhovsky-Dolzhansky system. We consider analytical properties of the studied system. The problem of existence of meromorphic solution is discussed. We perform the Painlev`e test and find conditions imposed on parameters of the system for which meromorphic solutions ...

Added: December 16, 2019

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

Abrashkin A. A., Якубович Е. И., Известия высших учебных заведений. Радиофизика 2015 Т. 58 № 11 С. 953-959

It is shown that the discrete frequency spectrum of the plane hydrodynamic flow of an ideal incompressible liquid with localized trajectories of liquid particles can contain only one harmonic, two harmonics, or an infinite number of the latter. ...

Added: March 2, 2016

Teplitskaya Y., Степанов Е. О., Paolini E., Advances in Calculus of Variations 2015 Vol. 8 No. 3 P. 267-290

We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, ...

Added: January 31, 2018

Abrashkin A. A., Соловьев А. Г., Известия РАН. Механика жидкости и газа 2013 № 5 С. 125-133

Изучены плоские периодические колебания бесконечно глубокой жидкости при неоднородном распределении давления на ее свободной поверхности. Движение жидкости описывается точным решением уравнения Эйлера в лагранжевых переменных. Исследованы динамика колеблющегося стоячего солитона, а также сценарий эволюции солитона и возникновение волны аномально большой амплитуды на фоне однородного герстнеровского волнения (модель волны-убийцы). Все течения являются неоднородно завихренными. ...

Added: November 19, 2013

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

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Okounkov A., Aganagic M., Moscow Mathematical Journal 2017 Vol. 17 No. 4 P. 565-600

We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties.
This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik Zamolodchikov and dynamical q-difference equations. ...

Added: October 25, 2018