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## 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

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

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

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

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

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

Savina O. N., Gromov E., Tyutin V. V., 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

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

Pelinovsky D. E., Slunyaev A., Kokorina A. V. 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

Maksimov V. P., Chadov A. L., Известия высших учебных заведений. Математика 2010 № 10 С. 82-86

We consider linear boundary-value problems for systems of functional differential equations when the number of boundary conditions is greater than the dimension of the system. We allow the boundary conditions to be fulfilled approximately. We propose an approach based on theorems whose conditions allow the verification by special reliable computing procedures. ...

Added: November 14, 2012

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

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

Ignatenko V., 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

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

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