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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.

, 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

, 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

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

, 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

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

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

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

Physics of Fluids 2011 Vol. 23 No. 11 P. 116602-1-13-116602-13

We address a specific but possible situation in natural water bodies when the three-layer stratification has a symmetric nature, with equal depths of the uppermost and the lowermost layers. In such case, the coefficients at the leading nonlinear terms of the modified Korteweg-de Vries (mKdV) equation vanish simultaneously. It is shown that in such cases ...

Added: November 6, 2012

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

Труды НГТУ им. Р.Е. Алексеева 2016 № 3 (114) С. 9-15

Цель работы: Исследована динамика двухкомпонентных (векторных) солитонов Давыдова-Скотта(ДС) при пространственно-скоростном рассогласовании высокочастотных (ВЧ) и низкочастотных (НЧ) компонент. Рассмотрение проведено в рамках Захаровского типа системы двух связанных уравнений для ВЧ и НЧ поля. В этой системе ВЧ поле описывается линейным уравнением Шредингера с переменным во времени и пространстве потенциалом, вызванным НЧ компонентой. НЧ компонента в этой ...

Added: October 5, 2016

Physics Letters A 2017 Vol. 381 No. 17 P. 1490-1492

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger ...

Added: February 3, 2017

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

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

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

Journal of Coastal Research 2011 No. SI64 P. 2042-2047

The geographical and seasonal distributions of kinematic and nonlinear parametersof long internal waves obtained on a base of GDEM climatology in the Baltic Sea region are examined. The considered parameters (phase speed of long internal wave, dispersion, quadratic and cubicnonlinearity parameters) of the weakly-nonlinear Korteweg-de Vries-type models (in particular, Gardner model), can be used for ...

Added: October 11, 2012

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

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

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

, 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

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

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