arXiv subjects

Y. A. Stepanyants

Publications and source records attributed to Y. A. Stepanyants.

9 recordsLinked to original sources

arXiv2026

Soliton dynamics in the Ostrovsky equation with anomalous dispersion

We investigate the formation and interaction of solitons in the non-integrable Ostrovsky equation characterized by anomalous (positive) dispersion. This equation is relevant for describing wave phenomena in various media, including plasma, solids, and optical fibers. Our findings indicate that certain Ostrovsky solitons, which possess zero total ''mass'' and exhibit non-monotonic asymptotic behavior, can arise from initial perturbations of a pulse-like nature. These solitons may organize into regular trains, where they are arranged according to their amplitude, or they may form irregular, nonstationary configurations of bound interacting solitons, or even multi-soliton structures. Furthermore, we demonstrate that the interactions between solitons in the Ostrovsky equation are inelastic, resulting in the emergence of a dominant soliton, or ''soliton-champion,'' within closed systems, such as those with periodic boundary conditions. In such systems, the soliton with the highest amplitude acts as a ''terminator,'' annihilating smaller amplitude solitons and absorbing a portion of their energy. We also analyze the recurrence phenomenon and determine that, although it resembles that of the Korteweg-de Vries equation, it exhibits distinct features.

nlin.PS↗

arXiv2024

Frequency downshifting in decaying wavetrains on the ocean surface covered by ice floes

We study analytically and numerically a frequency downshifting due to power-type frequency-dependent decay of surface waves in the ocean covered by ice floes. The downshifting is obtained both within the linear model and within the nonlinear Schrödinger (NLS) equation augmented by viscous terms for the initial condition in the form of an NLS envelope soliton. It is shown that the frequency-dependent dissipation produces a more substantial downshifting when the spectrum is relatively wide. As a result, the nonlinear adiabatic scenario of wavetrain evolution provides a downshifting remarkably smaller in magnitude than in the linear regime. Meanwhile, interactions between nonlinear wavegroups lead to spectral broadening and thus, result in fast substantial frequency downshifts. Analytic estimates are obtained for an arbitrary power $n$ of the dependence of a dissipation rate on frequency $\sim ω^n$. The developed theory is validated by the numerical modelling of the generalized NLS equation with dissipative terms. Estimates of frequency downshift are given for oceanic waves of realistic parameters.

physics.geo-ph↗

arXiv2021

Transformation of envelope solitons on a bottom step

In this paper we study the transformation of surface envelope solitons travelling over a bottom step in water of a finite depth. Using the transformation coefficients earlier derived in the linear approximation, we find the parameters of transmitted pulses and subsequent evolution of the pulses in the course of propagation. Relying on the weakly nonlinear theory, the analytic formulae are derived which describe the maximum attainable wave amplitude in the neighbourhood of the step and in the far zone. Solitary waves may be greatly amplified (within the weakly nonlinear theory formally even without a limit) when propagating from relatively shallow water to the deeper domain due to the constructive interference between the newly emerging envelope solitons and the residual quasi-linear waves. The theoretical results are in a good agreement with the data of direct numerical modelling of soliton transformation. In particular, more than double wave amplification is demonstrated in the performed simulations.

nlin.PS↗

arXiv2019

Evolution of non-stationary pulses in a cold magnetized quark-gluon plasma

We study weakly nonlinear wave perturbations propagating in a cold nonrelativistic and magnetized ideal quark-gluon plasma. We show that such perturbations can be described by the Ostrovsky equation. The derivation of this equation is presented for the baryon density perturbations. Then we show that the generalized nonlinear Schr{ö}dinger (NLS) equation can be derived from the Ostrovsky equation for the description of quasi-harmonic wave trains. This equation is modulationally stable for the wave number $k < k_m$ and unstable for $k > k_m$, where $k_m$ is the wave number where the group velocity has a maximum. We study numerically the dynamics of initial wave packets with the different carrier wave numbers and demonstrate that depending on the initial parameters they can evolve either into the NLS envelope solitons or into dispersive wave trains.

hep-ph↗

arXiv2019

Interaction of internal solitary waves with long waves within the rotation modified Benjamin-Ono equation

Interaction of a solitary wave with a long background wave is studied within the framework of rotation modified Benjamin-Ono equation describing internal waves in a deep fluid. With the help of asymptotic method, we find stationary and nonstationary solutions for a Benjamin-Ono soliton trapped within a long sinusoidal wave. We show that the radiation losses experienced by a soliton and caused by the Earth rotation can be compensated by the energy pumping from the background wave. However, the back influence of the soliton on the background wave eventually leads to the destruction of the coherent structure and energy dispersion in a quasi-random wave field.

nlin.PS↗

arXiv2018

The effects of interplay between the rotation and shoaling for a solitary wave on variable topography

This paper presents specific features of solitary wave dynamics within the framework of the Ostrovsky equation with variable coefficients in relation to surface and internal waves in a rotating ocean with a variable bottom topography. For solitary waves moving toward the beach, the terminal decay caused by the rotation effect can be suppressed by the shoaling effect. Two basic examples of a bottom profile are analyzed in detail and supported by direct numerical modelling. One of them is a constant-slope bottom and the other is a specific bottom profile providing a constant amplitude solitary wave. Estimates with real oceanic parameters show that the predicted effects of stable soliton dynamics in a coastal zone can occur, in particular, for internal waves.

nlin.PS↗

arXiv2018

Soliton solutions to the fifth-order Korteweg - de Vries equation and their applications to surface and internal water waves

We study solitary wave solutions of the fifth-order Korteweg - de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived and the dependence of its amplitude, width and speed on the parameters of the governing equation are studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.).

physics.flu-dyn↗

arXiv2018

Soliton spectra of random water waves in shallow basins

Interpretation of random wave field on a shallow water in terms of Fourier spectra is not adequate, when wave amplitudes are not infinitesimally small. A nonlinearity of wave fields leads to the harmonic interactions and random variation of Fourier spectra. As has been shown by Osborne and his co-authors, a more adequate analysis can be performed in terms of nonlinear modes representing cnoidal waves; a spectrum of such modes remains unchanged even in the process of nonlinear mode interactions. Here we show that there is an alternative and more simple analysis of random wave fields on shallow water, which can be presented in terms of interacting Korteweg - de Vries solitons. The data processing of random wave field is developed on the basis of inverse scattering method. The soliton component obscured in a random wave field is determined and a corresponding distribution function of number of solitons on their amplitudes is constructed. The approach developed is illustrated by means of artificially generated quasi-random wave field and applied to the real data interpretation of wind waves generated in the laboratory wind tank.

physics.flu-dyn↗

arXiv2007

Picone identities for half-linear differential equations of fourth order

Picone-type identities are established for half-linear ODEs of fourth order (one-dimensional p-biLaplacian). It is shown that in the linear case they reduce to the known identities for fourth order linear ODEs. Picone-type identity known for two half-linear second-order equations is also generalised to set of equations greater than two.

math.CA↗