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Alexey Samokhin

Publications and source records attributed to Alexey Samokhin.

9 recordsLinked to original sources
arXiv2026

Shock waves of spherical/cylindrical KdV-B: Asymptotic, stability, superposition

Spherical and cylindrical KdV-B equations have few known exact solutions, yet these solutions are hard to be interpreted physically. But these equations do have a family of diverging shock waves. Their properties such as asymptotic modes, stability, rules of their interactions/superposition are the subject of this paper. It gives a detailed asymptotic description of the one-parameter families of shock wave solutions and proves their stability using a conservation law. Based on these results, effective rules of superposition are obtained. Moreover these rules are applicable to a wide class of shock waves, in particular discontinuous. Typical examples are illustrated by graphs.

nlin.PS↗
arXiv2026

Superposition of shock waves of the generalized BBM equation

The generalized BBM studied in this paper contains an additional dissipative term. Thus instead of solitons for the classic BBM there exists a lot of travelling shock wave solutions. The rules of their interactions or superposition is of high importance. The paper gives a detailed description of the two-parameter family of travelling wave solutions and proves their stability using a conservation law. Based on these results, effective rules of superposition are obtained. Moreover these rules are applicable not exclusively to the travelling wave solutions of BBM, but also to a wider class of shock waves, in particular discontinuous. Characteristic examples are illustrated by numerically worked out graphs.

nlin.PS↗
arXiv2023

On perturbations retaining conservation laws of difftrential equations

The paper deals with perturbations of the equation that have a number of conservation laws. When a small term is added to the equation its conserved quantities usually decay at individual rates, a phenomenon known as a selective decay. These rates are described by the simple law using the conservation laws' generating functions and the added term. Yet some perturbation may retain a specific quantity(s), such as energy, momentum and other physically important characteristics of solutions. We introduce a procedure for finding such perturbations and demonstrate it by examples including the KdV-Burgers equation and a system from magnetodynamics. Some interesting properties of solutions of such perturbed equations are revealed and discussed.

nlin.PS↗
arXiv2020

On periodic boundary solutions for cylindrical and spherical KdV-Burgers equations

For the KdV-Burgers equations for cylindrical and spherical waves the development of a regular profile starting from an equilibrium under a periodic perturbation at the boundary is studied. The equation describes a medium which is both dissipative and dispersive. For an appropriate combination of dispersion and dissipation the asymptotic profile looks like a periodical chain of shock fronts with a decreasing amplitude (sawtooth waves). The development of such a profile is preceded by a head shock of a constant height and equal velocity which depends on spatial dimension as well as on integral characteristics of boundary condition; an explicit asymptotic for this head shock is found.

nlin.PS↗
arXiv2020

The KdV soliton crosses a dissipative and dispersive border

We demonstrate the behavior of the soliton which, while moving in non-dissipative and dispersion-constant medium encounters a finite-width barrier with varying dissipation and/or dispersion; beyond the layer dispersion is constant (but not necessarily of the same value) and dissipation is null. The passed wave either retains the form of a soliton (though of different parameters) or becomes a bi-soliton. And a reflection wave may be negligible or absent. This models a situation similar to a light passing from a humid air to a dry one through the vapour saturation/condensation area. Some rough estimations for a prediction of an output are given using relative decay of the KdV conserved quantities are given. Keywords: KdV- Burgers, non-homogeneous layered media, soliton, bi-soliton, reflection, refraction.

nlin.SI↗
arXiv2019

Soliton transmutations in KdV--Burgers layered media

We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite dissipative layer as well as a wave passing from a dissipative layer into a non-dissipative one and vice versa. New effects are presented in the case of numerically finite barrier on the soliton path: first, if the form of dissipation distribution has a form of a frozen soliton, the wave that leaves the dissipative barrier becomes a bi-soliton and a reflection wave arises as a comparatively small and quasi-harmonic oscillation. Second, if the dissipation is negative (the wave, instead of loosing energy, is pumped with it) the passed wave is a soliton of a greater amplitude and velocity. Third, when the travelling wave solution of the KdV-Burgers (it is a shock wave in a dissipative region) enters a non-dissipative layer this shock transforms into a quasi-harmonic oscillation known for the KdV.

nlin.PS↗
arXiv2017

Modelling Solutions to the Kdv-Burgers Equation in the Case of Non-homogeneous Dissipative Media

We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with finite dissipation. The modelling included the case of a finite dissipative layer similar to a wave passing through the air-glass-air as well as a wave passing from a non-dissipative layer into a dissipative one (similar to the passage of light from air to water). The dissipation predictably results in reducing the soliton amplitude/velocity, but some new effects occur in the case of finite barrier on the soliton path: after the wave leaves the dissipative barrier it retains a soliton form, yet a reflection wave arises as small and quasi-harmonic oscillations (a breather). The breather spreads faster than the soliton as moves through the barrier.

nlin.PS↗
arXiv2016

On nonlinear superposition of shock waves for the KdV-Burgers equation

Superposition of explicit (analytic) monotone non-increasing shock waves for the KdV-Burgers equation is studied, modelled numerically and graphically presented. Initial profile chosen as a sum of two such shock waves gradually transforms into a single shock wave of a somewhat complex yet predictable structure. This transformation is demonstrated in detail.

nlin.PS↗