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## The Hopf equation with certain modular nonlinearities

Physics Letters A. 2024. Vol. 507. Article 129489.

Pelinovsky E., Talipova T., Didenkulova E.

The dynamics of waves of sign-variable shape has been studied within the framework of the Hopf equation (*ut +Fux = 0*) with a non-analytic propagation velocity containing the modulus of the function at the zero crossing (*F ~ |u|α*). It is shown that Riemann waves exist only for a certain smoothness of the function *F[u(x)]* at the initial moment of time. Otherwise, the wave immediately overturns (gradient catastrophe). The modular Hopf equation, the dispersionless Schamel equation, and the dispersionless logarithmic Korteweg-de Vries equation are considered as examples.

Flamarion M., Pelinovsky E., Didenkulova E., Chaos, Solitons and Fractals 2024 Vol. 180 Article 114495

Soliton gas or soliton turbulence is a subject of intense studies due to its great importance to optics, hydrodynamics, electricity, chemistry, biology and plasma physics. Usually, this term is used for integrable models where solitons interact elastically. However, soliton turbulence can also be a part of non-integrable dynamics, where long-lasting solutions in the form of ...

Added: March 15, 2024

Flamarion M., Pelinovsky E., Didenkulova E., Chaos, Solitons and Fractals 2023 Vol. 174 Article 113870

This article presents a numerical investigation of overtaking collisions between two solitary waves in the
context of the Schamel equation. Our study reveals different regimes characterized by the behavior of the wave
interactions. In certain regimes, the collisions maintain two well-separated crests consistently over time, while
in other regimes, the number of local maxima undergoes variations following the ...

Added: March 15, 2024

Kachulin D., Dyachenko A., Zakharov V., Fluids 2020 Vol. 5 No. 2 Article 67

We investigate and compare soliton turbulence appearing as a result of modulational
instability of the homogeneous wave train in three nonlinear models for surface gravity waves:
the nonlinear Schrödinger equation, the super compact Zakharov equation, and the fully nonlinear
equations written in conformal variables. We show that even at a low level of energy and average
wave steepness, the ...

Added: October 22, 2022

Didenkulova E., Pelinovsky E., Flamarion M., Mathematics 2023 Vol. 11 Article 4649

Pair soliton interactions play a significant role in the dynamics of soliton turbulence. The interaction of solitons with different polarities is particularly crucial in the context of abnormally large wave formation, often referred to as freak or rogue waves, as these interactions result in an increase in the maximum wave field. In this article, we ...

Added: March 15, 2024

Ekaterina Didenkulova, Pelinovsky E., Flamarion M., Mathematics 2023 Vol. 11 No. 22 Article 4649

Pair soliton interactions play a significant role in the dynamics of soliton turbulence. The interaction of solitons with different polarities is particularly crucial in the context of abnormally large wave formation, often referred to as freak or rogue waves, as these interactions result in an increase in the maximum wave field. In this article, we ...

Added: December 11, 2023

Slunyaev A., Кокорина А. В., Water Waves, Springer 2019 P. 1-20

The issue of accounting of the wave breaking phenomenon in direct numerical simulations of oceanic waves is discussed. It is emphasized that this problem is crucial for the deterministic description of waves, and also for the dynamical calculation of extreme wave statistical characteristics, such as rogue wave height probability, asymmetry, etc. The conditions for accurate ...

Added: October 13, 2019

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

O. Kurkina, T. Talipova, E. Pelinovsky et al., Journal of Coastal Research 2011 No. SI64 P. 2042-2047

The geographical and seasonal distributions of kinematic and nonlinear parametersof long internal waves obtained on a base of GDEM climatology in the Baltic Sea region are examined. The considered parameters (phase speed of long internal wave, dispersion, quadratic and cubicnonlinearity parameters) of the weakly-nonlinear Korteweg-de Vries-type models (in particular, Gardner model), can be used for ...

Added: October 11, 2012

Flamarion M., Pelinovsky E., Didenkulova E., Chaos, Solitons and Fractals 2023 Vol. 174 Article 113870

This article presents a numerical investigation of overtaking collisions between two solitary waves in the context of the Schamel equation. Our study reveals different regimes characterized by the behavior of the wave interactions. In certain regimes, the collisions maintain two well-separated crests consistently over time, while in other regimes, the number of local maxima undergoes ...

Added: August 24, 2023

М. : Энциклопедия, Инфра-М, 2017

The New Russian Encyclopedia is a fundamental reference publication in 18 volumes that characterizes nature, population, economy, history, science, art, technology and other important aspects. Contains about 60,000 articles, about 30,000 biographies, about 15,000 color illustrations, maps, charts, diagrams, tables. Leaves since 2003. ...

Added: October 29, 2018

Parkhomenko E., Malova H., Grigorenko E. et al., Physics of Plasmas 2019 Vol. 26 No. 4 P. 1-9

This work is devoted to the investigation of particle acceleration during magnetospheric dipolarizations. A numerical model is presented
taking into account the four scenarios of plasma acceleration that can be realized: (A) total dipolarization with characteristic time scales of
3 min; (B) single peak value of the normal magnetic component Bz occurring on the time scale of ...

Added: April 16, 2019

Barskov K., Chechin D., Drozd I. et al., Boundary-Layer Meteorology 2023 Vol. 187 P. 311-338

We present results from the multi-level mast eddy-covariance measurements conducted over flat grassland and over complex urban terrain composed of trees and small buildings. Relationships between the stability parameter and turbulent statistical properties were analysed. Over flat grassland, non-dimensional turbulent moments up to the third-order obey Monin–Obukhov similarity theory. Third-order moments increase for increasing instability, ...

Added: October 5, 2023

Kachulin D., Dyachenko A., Dremov S., Fluids 2020 Vol. 5 No. 2 Article 65

The paper presents the long-time dynamics with multiple collisions of breathers in the super compact Zakharov equation for unidirectional deep water waves. Solutions in the form of breathers were found numerically by the Petviashvili method. In the terms of envelope and the assumption of the narrow spectral width the super compact equation turns into the ...

Added: October 22, 2022

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

Садовничий В. А., Панасюк М. И., Липунов В. М. et al., Космические исследования 2019 Т. 57 № 1 С. 46-56

The article discusses the results of experiments on the Lomonosov satellite to observe natural and man-made space threats, including electromagnetic transients, space debris. A new space project of the Moscow State University named after M.V. Lomonosov Univesat-SOCRAT on the creation of a constellation of satellites for real-time monitoring in near-earth space: radiation situation; potentially dangerous ...

Added: October 15, 2021

Egor Stadnichuk, Свечникова Е., Atmospheric Research 2022 Vol. 277 Article 106329

Relativistic runaway electron avalanches (RREA) occurring in thunderstorm large-scale electric fields are one of the sources of atmospheric gamma radiation. Relativistic feedback is the generation of RREAs by positrons or backscattered gamma rays of the previously produced RREAs. In strong electric fields, relativistic feedback can make RREAs self-sustainable that hypothetically can cause a terrestrial gamma-ray flash (TGF). This paper introduces ...

Added: May 24, 2023

A. A. Abrashkin, Constantin A., Journal of Differential Equations 2023 Vol. 374 P. 632-641

We investigate steady flow moving purely in the azimuthal direction on rotating sphere and having a meridionally localized jet structure. An exact solution for a stratified invicsid fluid, which admits a depth-dependent velocity profile below the surface, is constructed in spherical coordinates. This soluton is relevant to the modelling of the Antarctic Circumpolar Current. We ...

Added: September 14, 2023

Kolokolov I., Lebedev V., Gamba A. et al., Physical Review Letters 2007 Vol. 99 P. 158101-1-158101-4

Eukaryotic cells possess a sensible chemical compass allowing them to orient toward sources of soluble chemicals. The extracellular chemical signal triggers separation of the cell membrane into two domains populated by different phospholipid molecules and oriented along the signal anisotropy. We propose a theory of this polarization process, which is articulated into subsequent stages of ...

Added: February 10, 2017

Gavryushkin P., Belonoshko Anatoly B., Sagatov N. et al., Crystal Growth & Design 2021 Vol. 21 No. 1 P. 65-74

Using molecular dynamics simulation and evolutionary metadynamic calculations, a series of structures were revealed that possessed enthalpies and Gibbs energies lower than those of aragonite but higher than those of calcite. The structures are polytypes of calcite, differing in the stacking sequence of close-packed (cp) Ca layers. The two- and six-layered polytypes have hexagonal symmetry P6322 ...

Added: January 21, 2021

Могилевский М. М., Чугунин Д. В., Chernyshov A. et al., Письма в Журнал экспериментальной и теоретической физики 2021 Т. 114 № 1 С. 18-23

According to measurements of the electric component of the electromagnetic field in the frequency range from 2 kHz to 10 MHz by the Japanese ERG satellite, generation regions of kilometric continuum radiation and new hectometric continuum radiation have been determined. It is demonstrated that the kilometric continuum is mainly observed on the dayside of the ...

Added: January 21, 2022

Dyachenko A., Kachulin D., Zakharov V., Journal of Ocean Engineering and Marine Energy 2017 Vol. 3 No. 4 P. 409-415

Water waves have long been a subject of attention of both mathematicians and physicists. The formulation of the problem is simple enough to be considered fundamental, but as of yet many questions still remain unanswered and many phenomena associated with wind-driven turbulence remain puzzling. We consider a “unidirectional” motion of weakly nonlinear gravity waves, i.e., ...

Added: October 22, 2022

Didenkulova E., Pelinovsky E., Touboul ., Wave Motion 2021 Vol. 100 Article 102668

The role of various long-wave approximations in the description of the wave field and bottom pressure caused by surface waves, and their relation to evolution equations are being considered. In the framework of the linear theory, these approximations are being tested on the well-known exact solution for the wave spectral amplitudes and pressure variations. The ...

Added: January 26, 2021

Klein M., Clauss G., Slunyaev A., Physics of Fluids 2017 Vol. 29 P. 047103-1-047103-17

The investigation of dynamics of intense solitary wave groups of collinear surface waves is performed by means of numerical simulations of the Euler equations and laboratory experiments. The processes of solitary wave generation, reflection from a wall and collisions are considered. Steep solitary wave groups with characteristic steepness up to kAcr ~ 0.3 (where k is the ...

Added: February 14, 2019

Стадничук Е. М., Хамитов Т. М., Нозик А. А. et al., EPL 2020 Vol. 132 No. 3 Article 35001

The connection between thunderstorms and relativistic runaway electron avalanches is an important topic that has attracted the attention of many researchers. Among other things, there are a lot of various simulations of the dynamics of electron avalanches. This article was written mostly in response to the article The critical avalanche of runaway electrons by Evgeny ...

Added: May 24, 2023