?
Solutions of Traveling Wave Type for Korteweg-de Vries-Type System with Polynomial Potential
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 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.
In book
Springer International Publishing, 2019.
Melnikov I., Nonlinear Dynamics 2025 No. 113 P. 32641–32648
All possible soliton solutions classification in the generalized KdV equation is given. It is proved that a solitonic solution can be one of four possible types: classical soliton (that is, sign-constant, with one extremum and two inflection points), pyramidal soliton (with a large number of inflection points), classical kink (a monotonic solution that goes to ...
Added: September 11, 2025
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
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
Ekaterina Didenkulova, Pelinovsky E., Journal of Marine Science and Engineering 2023 Vol. 11 No. 3 Article 482
Unexpected large waves known as freak or rogue waves are a phenomenon emerging in the World Ocean and are causing significant damage to vessels and coastal structures. These waves are often associated with deep-water waves; however, they can also be dangerously close to the shore. The present study is devoted to the numerical modeling of ...
Added: March 16, 2023
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
Pelinovsky D., Slunyaev A., Kokorina A. et al., Communications in Nonlinear Science and Numerical Simulation 2021 Vol. 101 Article 105855
Compactons are studied in the framework of the Korteweg–de Vries (KdV) equation with the sublinear nonlinearity. Compactons represent localized bell-shaped waves of either polarity which propagate to the same direction as waves of the linear KdV equation. Their amplitude and width are inverse proportional to their speed. The energetic stability of compactons with respect to ...
Added: May 11, 2021
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
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
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
Didenkulova E., Slunyaev A., Pelinovsky E., European Journal of Mechanics - B/Fluids 2019 Vol. 78 P. 21–31
Direct numerical simulations of irregular unidirectional nonlinear wave evolution are performed within the framework of the Korteweg–de Vries equation for bimodal wave spectra model cases. The additional wave system co-existence effect on the evolution of the wave statistical characteristics and spectral shapes, and also on the attained equilibrium state is studied. The concerned problem describes, for example, the interaction ...
Added: June 15, 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 A., Beklaryan L., Gornov A., , in: Book of abstracts of the IX International Conference on Optimization Methods and Applications (OPTIMA-2018), Petrovac, Montenegro, October 1-5, 2018.: M.: [б.и.], 2018. 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 ...
Added: October 9, 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
Бекларян Л. А., Beklaryan A., Belousov F., Вестник Тамбовского университета. Серия: Естественные и технические науки 2015 Т. 20 № 6 С. 1736–1747
We investigate a problem of the existence of traveling-wave-type solutions for the finite-difference analogue of a nonlinear wave equation. In case of an inhomogeneous medium for vanishing traveling-wave-type solutions a natural extension in the form of quasi-traveling-wave-type solutions is given. ...
Added: December 7, 2015
O.N. Savina, E.M. Gromov, V.V. Tyutin, Geomagnetism and Aeronomy 2015 Vol. 55 No. 5 P. 658–662
The possibility of the existence of a solitary internal gravity wave at heights of the Earth’s thermosphere is considered. Analytical results were obtained in local approximation with weak nonisothermality of the atmosphere. For internal gravity waves, the Korteweg–de Vries equation was derived and studied with allowance for inhomogeneity, nonlinearity, and dissipation. The theoretical results were ...
Added: October 23, 2015
Tyugin D. Y., Наумов А. А., Kurkina O. E. et al., Экологические системы и приборы 2014 № 1 С. 20–28
Simulation of abnormally large internal waves generated by the baroclinic tide is now quite important due to the increased number of offshore platforms installed on offshore oil and gas fields. The height of the internal waves in many areas of the oceans can be up to 100 m, and these waves become really dangerous. All ...
Added: November 18, 2013
Filimonov D., Functional Differential Equations 2004 Vol. 11 No. 3-4 P. 333–339
The system considered in this paper consists of two equations $(k=1,2)$ $\dot x(t)=(-1)^{k-1} (0\le t<\infty),\, k(0)=1,\,x(0)=0,\,x(t)\not\in\{0,1\}(-1\le t<0),$ that change mutually in every instant $t$ for which $x(t-\tau)\in\{0,1\}$, where $\tau={\rm const}>0$ is given. In this paper the behavior of the solutions is characterized for every $\tau\in(\frac{4}{3},\frac{3}{2})$, i. e. in case not covered in \cite{ADM}; as it ...
Added: November 14, 2013
Plaksin M. A., Plaksina V. P., , in: Functional Differential Equations and Applications 2012. August 27–31, 2012. Abstracts.: Ariel: Ariel University Center of Samaria, 2012. P. 34–34.
The report discusses the methodology of abstract theory of FDE in terms of the theory of inventive problem solving (TRIZ). ...
Added: December 2, 2012