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Solutions of Traveling Wave Type for Korteweg-de Vries-Type System with Polynomial Potential
P. 41–41.
This paper deals with the implementation of numerical methods for searching for traveling waves for Korteweg-de Vries-type equations with time delay. Based upon the group approach, the existence of traveling wave solution and its boundedness are shown for some values of parameters. Meanwhile, solutions constructed with the help of the proposed constructive method essentially extend the class of systems, possessing solutions of this type, guaranteed by theory. The proposed method for finding solutions is based on solving a multiparameter extremal problem. Several numerical solutions are demonstrated.
Lerman L., Russian Journal of Nonlinear Dynamics 2025 Vol. 21 No. 1 P. 15 – 31
Added: April 30, 2025
Beklaryan A., Computational Mathematics and Mathematical Physics 2024 Vol. 64 No. 11 P. 2588–2610
This paper is a continuation of the work by L.A. Belkaryan and A.L. Belkaryan published in this journal (64 (7), 1472–1490 (2024)). A theorem is proved formulated as a conjecture in the preceding paper stating the existence and uniqueness of soliton solutions and corresponding solutions of the functional differential equation from a dual pair “function–operator.” For ...
Added: January 27, 2025
Pelinovsky E., Gurbatov S., Chaos, Solitons and Fractals 2025 Vol. 192 Article 116056
The statistical properties of a sequence of spaced solitons and compactons (soliton gas) with random amplitudes
and phases are studied based on the example of solitary waves – solutions of the generalized Korteweg-de Vries
equation with power nonlinearity (including fractional nonlinearity). Such sequences are used to specify initial
conditions in problems of modeling soliton turbulence. It is shown ...
Added: January 25, 2025
Melnikov I., Wave Motion 2024 Vol. 130 Article 103380
Non-reflective wave propagation is of great importance for applications because it allows energy to be transmitted over long distances. The paper discusses the method of reducing the equations of the linear theory of shallow water to a wave equation with a variable coefficient in the form of an inverse hyperbolic sine, the solution of which ...
Added: July 10, 2024
Ioann Melnikov, Pelinovsky E., Physics of Fluids 2024 Vol. 36 No. 7 Article 076609
A method for the transformation of linear shallow water equations based on a generalization of the Carrier–Greenspan transform, well known in the theory of wave rolling on a flat slope, is presented. Thanks to it, the initial equations for waves over arbitrary bathymetry are reduced to a wave equation, from which both the displacement of ...
Added: July 9, 2024
A. E. Rassadin, Agalarov A. M., Ferroelectrics 2021 Vol. 576 No. 1 P. 40–49
Using the Euler-Lagrange dynamics of the Landau-GinzburgDevonshire functional for homogeneous ferroelectric chain in the spatial continuous limit, (1þ 1)D-nonlinear Klein-Gordon-Fock equation has been derived. Elementary four-terminal network of this ferroelectric chain has been considered as a Hamiltonian system corresponding to the point mass in the effective nonlinear potential. Expressing all parameters of automodeling solution of ...
Added: December 8, 2022
Solitary Wave Interactions with an External Periodic Force: The Extended Korteweg-de Vries Framework
Flamarion M. V., Pelinovsky E., Mathematics 2022 Vol. 10 No. 23 Article 4538
In this work we asymptotically and numerically studied the interaction of large amplitude
solitary waves with an external periodic force using the forced extended Korteweg-de Vries equation
(feKdV). Regarding these interactions, we found three types of regimes depending on the amplitude
of the solitary wave and how its speed and the speed of the external force are related. ...
Added: December 1, 2022
Pelinovsky E., Talipova T., Soomere T., Physica D: Nonlinear Phenomena 2021 Vol. 419 No. 5 Article 132785
We analyze the main properties of soliton solutions to the generalized KdV equation u_t +[F (u)]_x+u_xxx
= 0, where the leading term F (u) ∼ qu^α, α > 0, q ∈ R. The far field of such solitons may have
three options. For q > 0 and α > 1 the analysis re-confirmed that all traveling solitons ...
Added: February 13, 2021
Beklaryan L. A., Beklaryan A., Lobachevskii Journal of Mathematics 2020 Vol. 41 No. 11 P. 2136–2142
Solutions of functional differential equation of pointwise type (FDEPT) are in one-to-one correspondence with the traveling-wave type solutions for the canonically induced infinite-dimensional ordinary differential equation and vice versa. In particular, such infinite-dimensional ordinary differential equations are finite difference analogues of equations of mathematical physics. An important class of traveling-wave type solutions is made up of periodic and bounded ...
Added: September 21, 2020
Beklaryan L. A., Beklaryan A., Computational Mathematics and Mathematical Physics 2020 Vol. 60 No. 8 P. 1249–1260
The importance of functional differential equations of pointwise type is determined by the fact that their solutions are used to construct traveling-wave solutions for induced infinite-dimensional ordinary differential equations, and vice versa. Solutions of such equations exhibit bifurcation. A theorem on branching bifurcation is obtained for the solution to a linear homogeneous functional differential equation of pointwise type. ...
Added: September 21, 2020
Tobisch E., Pelinovsky E., Journal of Physics A: Mathematical and Theoretical 2020 Vol. 53 No. 34 P. 345703
In this paper we study dispersive enhancement of a wave train in systems described by the fractional Korteweg–de Vries-type equations of the form ut + αn un ux + βm(Dm{u})x = 0,Dm{u} = −|k|m u(k) where the operator Dm{u} is written in the Fourier space, αn, βm are arbitrary constants and n,m being rational numbers (positive or negative). Using ...
Added: August 4, 2020
Beklaryan A., Advances in Systems Science and Applications 2020 Vol. 20 No. 2 P. 56–70
The article discusses construction of traveling wave type solutions for the Frenkel-Kontorova model on the propagation of longitudinal waves. For the first time, based on the existence and uniqueness theorem of traveling wave type solutions, as well as the approximation theorem, a complete family of traveling wave type solutions is constructed in the form of ...
Added: June 30, 2020
Tobisch E., Pelinovsky E., Fluids, Basel, Switzerland 2019 Vol. 4 No. 54 P. 1–13
Our present study is devoted to the constructive study of the modulational instability
for the Korteweg-de Vries (KdV)-family of equations ut + sup ux + uxxx (here s = 1 and p > 0 is
an arbitrary integer). For deducing the conditions of the instability, we first computed the nonlinear
corrections to the frequency of the Stokes wave ...
Added: September 17, 2019
Бекларян Л. А., Дифференциальные уравнения 2018 Т. 54 № 10 С. 1299–1312
The work is devoted to periodic solutions of the functional differential equation of point type. In terms of the right-hand side of the original nonlinear functional-differential equation of point type, easily verifiable conditions of existence and uniqueness are formulated, iterative process of constructing such solution is described. In contrast to the scalar linearization, a more complex matrix ...
Added: February 12, 2019
Beklaryan L. A., Beklaryan A., Journal of machine learning and data analysis 2018 Vol. 4 No. 4 P. 220–234
The problem of existence of soliton solutions (solutions of the traveling wave type) for the Korteweg-de Vries equation with a polynomial potential is considered on the basis of the approach within which the presence of a one-to-one correspondence of such solutions with solutions of the induced functional differential equation of pointwise type is demonstrated. On ...
Added: January 11, 2019
Beklaryan L. A., Beklaryan A., Gornov A., , in: Optimization and Applications 9th International Conference, OPTIMA 2018, Petrovac, Montenegro, October 1–5, 2018, Revised Selected Papers.: Springer International Publishing, 2019. P. 291–305.
This paper deals with the implementation of numerical methods for searching for traveling waves for Korteweg–de Vries-type equations with time delay. Based upon the group approach, the existence of traveling wave solution and its boundedness are shown for some values of parameters. Meanwhile, solutions constructed with the help of the proposed constructive method essentially extend ...
Added: January 10, 2019
Blyakhman L. G., Gromov E., Malomed B. A. et al., Chaos, Solitons and Fractals 2018 No. 117 P. 264–268
The dynamics of two-component solitons with a small spatial displacement of the high-frequency (HF) component relative to the low-frequency (LF) one is investigated in the framework of the Zakharov-type system. In this system, the evolution of the HF field is governed by a linear Schrödinger equation with the potential generated by the LF field, while ...
Added: November 12, 2018
Beklaryan A., Beklaryan L., , in: Тезисы докладов 12-й Международной конференции Интеллектуализация обработки информации.: М.: Торус Пресс, 2018. P. 80–81.
For equations of mathematical physics, which are the Euler-Lagrange equation of the corresponding variational problems, an important class of solutions are soliton solutions. The study of soliton solutions is based on the existence of a one-to-one correspondence between soliton solutions for initial systems and solutions of induced functionaldifferential equations of pointwise type (FDEPT). The existence and uniqueness theorem for ...
Added: October 11, 2018
Vera Ignatenko, Discrete and Continuous Dynamical Systems 2018 Vol. 38 No. 7 P. 3637–3661
A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified ...
Added: May 25, 2018
Beklaryan A., Beklaryan L. A., , in: Proceedings of the VIII International Conference on Optimization and Applications (OPTIMA-2017), Petrovac, Montenegro, October 2-7, 2017.: [б.и.], 2017. P. 81–87.
For equations of mathematical physics, which are the Euler-Lagrange equation of the corresponding variational problem, an important class of solutions are traveling wave solutions (soliton solutions). In turn, soliton solutions for finite-difference analogs of the equations of mathematical physics are in one-to-one correspondence with solutions of induced functional differential equations of pointwise type (FDEPT). The ...
Added: November 14, 2017