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## Nonlinear gravity waves in the water flow with inhomogeneous vorticity

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Abrashkin A. A., Pelinovsky E.

Nonlinear Schrodinger equation is derived for weakly modulated nonlinear wave packets in the infinite-depth water flow with inhomogeneous vorticity. Governing 2-D equations are written in Lagrangian variables. Nonlinear Schrodinger equation is obtained in the third order of perturbation theory taking into account weak non-uniform vortex current. Two limiting cases are analyzed. The first one corresponds to the uniform surface flow and is described by the classic nonlinear Schrodinger equation allowed the modulational instability. The second one is the Gerstner’s wave packet. In this limiting case the nonlinear term is absent confirming known fact that nonlinear Gerstner’s wave has the linear dispersion relation.

### In book

Wien : CC Attribution 3.0 License, 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., 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., 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., 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., 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

Chabchoub A., Slunyaev A., Hoffmann N. et al., Frontiers in Physics 2021 Vol. 9 Article 633549

Solitons are coherent structures that describe the nonlinear evolution of wave localizations in hydrodynamics, optics, plasma and Bose-Einstein condensates. While the Peregrine breather is known to amplify a single localized perturbation of a carrier wave of finite amplitude by a factor of three, there is a counterpart solution on zero background known as the degenerate ...

Added: August 25, 2021

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., 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

Parfenyev V., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2022 Vol. 106 No. 2 Article 025102

It is well known that an inverse turbulent cascade in a finite ($2 \pi \times 2 \pi$) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random ...

Added: August 10, 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

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

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

Dymov A. V., Kuksin S., Journal of Statistical Physics 2023 Vol. 190 No. 1 Article 3

In this paper we continue to study small amplitude solutions of the damped cubic NLS equation, driven by a random force [the study was initiated in our previous work Dymov and Kuksin (Commun Math Phys 382:951–1014, 2021) and continued in Dymov et al. (The large-period limit for equations of discrete turbulence 2021, arXiv:2104.11967)]. We write solutions ...

Added: October 31, 2022

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

Didenkulova I., Nikolkina I., Pelinovsky E., JETP Letters 2013 Vol. 97 No. 4 P. 221-225

Properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20m and ...

Added: February 26, 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

Slunyaev A., Степанянц Ю. А., Physics of Fluids 2022 Vol. 34 No. 7 Article 077121

We study the nonlinear modulation property of flexural-gravity waves on a water surface covered by a compressed ice sheet of given thickness and density in a basin of a constant depth. For weakly nonlinear perturbations, we derive the nonlinear Schrodinger equation and investigate the conditions when a quasi-sinusoidal wave becomes unstable with respect to amplitude modulation. The ...

Added: October 13, 2022

Abrashkin A. A., В кн. : Труды X Всероссийской научной конференции «Нелинейные колебания механических систем» (Нижний Новгород, 26–29 сентября 2016 г.). : Н. Новгород : ИД "Наш дом", 2016. С. 7-11.

Предложена модель гипотетического первичного вакуума как континуума, состоящего из частиц с планковскими масштабами длины, времени и массы (эфиронов). Они образуют бозе-конденсат, и их описание сводится к системе уравнений гидродинамики для потенциальных течений сжимаемой жидкости. Уравнение состояния такого планковского конденсата содержит постоянную Планка и является нелокальным. Показано, что в конденсате могут распространяться продольные колебания плотности эфиронов, ...

Added: November 17, 2016

Slunyaev A., Ezersky A., Mouazé D. et al., , in : Nonlinear Waves and Pattern Dynamics. : Switzerland : Springer, 2018. P. 67-76.

Arisingmodulations of surface gravitywaves in a shallow-water resonator under harmonic forcing is discovered in laboratory experiments. Different types of modulations are found. When certain conditions are satisfied (appropriate frequency and sufficient force of excitation), the standing waves become modulated, and the envelopes of standing waves propagate in the channel. Strongly nonlinear numerical simulations of the Euler equations are performed reproducing ...

Added: March 1, 2019

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

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

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., 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

Slunyaev A., , in : Proceedings of the International Conference DAYS on DIFFRACTION 2018. : IEEE, 2018. P. 277-282.

We develop a method for the application of the Inverse Scattering Technique to the analysis of surface
water waves and present here some evidence on its efficiency. The general idea is to interpret nonlinear
wave groups in terms of soliton-type structures - envelope solitons in the framework of the integrable
nonlinear Schrodinger equation. Such analysis can improve understanding of the nonlinear ...

Added: March 1, 2019