Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

Vladimirov, Vsevolod; ̧czka, Czesław Ma; Sergyeyev, Artur; Skurativskyi, Sergii

doi:10.1016/j.cnsns.2013.10.027

Cited by 11 publications

(19 citation statements)

References 38 publications

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“…Finally, there are solitary waves moving without preserving of their selfsimilar shapes. While the numerical results concerning the stable properties of single solitary wave can be confirmed by analytical treatments [10], the studies of wave collisions require mostly the application of improved numerical schemes. The results presented above can be useful for modelling the behaviour of complex media in vibrational fields, for instance when we deal with the intensification of extracting oil and natural gas [4].…”

Section: Discussionmentioning

confidence: 99%

On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua

Skurativskyi¹,

Danylenko²

2015

Preprint

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The model we deal with is the mathematical model for mutually penetrating continua one of which is the carrying medium obeying the wave equation whereas the other one is the oscillating inclusion described by the equation for oscillators. These equations of motion are closed by the cubic constitutive equation for the carrying medium. Studying the wave solutions we reduce this model to a plane dynamical system of Hamiltonian type. This allows us to derive the relation describing the homoclinic trajectory going through the origin and obtain the solitary wave with infinite support. Moreover, there exist a limiting solitary wave with finite support, i.e. compacton. To model the solitary waves dynamics, we construct the three level finite-difference numerical scheme and study its stability. We are interested in the interaction of the pair of solitary waves. It turns out that the collisions of solitary waves have non-elastic character but the shapes of waves after collisions are preserved.

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Section: Discussionmentioning

confidence: 99%

On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua

Skurativskyi¹,

Danylenko²

2015

Preprint

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“…which is fulfilled providing that −2 < ν < 1. It follows from the results of the test, that the stability of the solitary wave arising in the model considered can take place at moderate values of the non-linearity of the equation of state [2] connecting the pressure with the average density of the medium.…”

Section: Stability Testmentioning

confidence: 99%

“…where γ = β/(ν + 2) > 0, κ = −σ/(ν + 2) > 0. In accordance with the work [2], we denote the soliton solutions of the system (4) by the symbols u = u s (z), ρ = R s (z). Inserting the perturbations of the form u(t, z) = u s (z) + e λ tÛ (z), ρ(t, z) = R s (z) + e λ tρ (z) into (67)-(68) and dropping out the terms of the order O( 2 ), we get the following spectral problem:…”

Section: Multi-symplectic Representation Of the System (3)-(4) And Evmentioning

confidence: 99%

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On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model

Vladimirov

Skurativskyi²

2020

Communications in Nonlinear Science and Numerical Simulation

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A model of nonlinear elastic medium with internal structure is considered. The medium is assumed to contain cavities, microcracks or inclusions consisting of substances that differ sharply in physical properties from the base material. To describe the wave processes in such a medium, the averaged values of physical fields are used. This leads to nonlinear evolutionary PDEs, differing from the classical balance equations. Using some transformations, these equations can be presented in the Hamiltonian form. It is shown that the system under investigation has a family of invariant soliton-like solutions. The present work is devoted to study of the spectral stability of the solitary wave solutions.

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“…e.g. [7,24,27,29,34] and references therein. The system (1) is no exception: it is Hamiltonian (we refer the reader to [24] for details on the Hamiltonian formalism for PDEs), with an operator…”

Section: Introductionmentioning

confidence: 99%

On symmetries and conservation laws of the Majda–Biello system

Vodova-Jahnova

2015

Nonlinear Analysis: Real World Applications

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In 2003, A.J. Majda and J.A. Biello derived and studied the so-called reduced equations for equatorial baroclinic-barotropic waves, to which we refer as to the Majda-Biello system. The equations in question describe the nonlinear interaction of long-wavelength equatorial Rossby waves and barotropic Rossby waves with a significant midlatitude projection in the presence of suitable horizontally and vertically sheared zonal mean flows.Below we present a Hamiltonian structure for Majda-Biello system and describe all generalized symmetries and conservation laws for the latter. It turns out that there are only three symmetries corresponding to x-translations, t-translations and to a scaling of t, x, u and v, and four conservation laws, one of which is associated to the conservation of energy, the second conserved quantity is just the Hamiltonian functional and the other two are Casimir functionals of the Hamiltonian operator admitted by our system. Our result provides inter alia a rigorous proof of the fact that the Majda-Biello system has just the conservation laws mentioned in the paper by Majda and Biello.2010 Mathematics Subject Classification. 37K05, 37K10.

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Stability and dynamical features of solitary wave solutions for a hydrodynamic-type system taking into account nonlocal effects

Cited by 11 publications

References 38 publications

On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua

On the dynamics of solitary wave solutions supported by the model of mutually penetrating continua

On the spectral stability of soliton-like solutions to a non-local hydrodynamic-type model

On symmetries and conservation laws of the Majda–Biello system

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