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## Existence of Bounded Soliton Solutions in the Problem of Longitudinal Vibrations of an Infinite Elastic Rod in a Field with a Strongly Nonlinear Potential

Computational Mathematics and Mathematical Physics. 2021. Vol. 61. No. 12. P. 1980-1994.

Beklaryan L. A., Beklaryan A.

Translator: I. Ruzanova

The existence of a family of bounded soliton solutions for a finite-difference wave equation with a quadratic potential is established. The proof is based on a formalism establishing a one-to-one correspondence between the soliton solutions of an infinite-dimensional dynamical system and the solutions of a family of functional differential equations of the pointwise type. A key point for the considered class of equations is also the existence of a number of symmetries.

Бекларян Л. А., Beklaryan A., Журнал вычислительной математики и математической физики 2021 Т. 61 № 12 С. 2024-2039

The existence of a family of bounded soliton solutions for a finite difference wave equation with a quadratic potential is established. The proof is carried out within the framework of a formalism that establishes a one-to-one correspondence between soliton solutions of an infinite-dimensional dynamical system and solutions of a family of pointwise functional differential equations. For ...

Added: November 1, 2021

Beklaryan A., Beklaryan L. A., Computational Mathematics and Mathematical Physics 2022 Vol. 62 No. 6 P. 904-919

The existence of a family of bounded soliton solutions for a finite-difference analogue of the wave equation with a general nonlinear potential is proved. The proof is based on a formalism establishing a one-to-one correspondence between soliton solutions of an infinite-dimensional dynamical system and solutions of a family of functional differential equations of the pointwise ...

Added: July 18, 2022

Beklaryan A., Бекларян Л. А., Журнал вычислительной математики и математической физики 2022 Т. 62 № 6 С. 933-950

The existence of a family of bounded soliton solutions for a finite-difference analogue of the wave equation with a general nonlinear potential is proved. The proof is based on a formalism establishing a one-to-one correspondence between soliton solutions of an infinite-dimensional dynamical system and solutions of a family of functional differential equations of the pointwise type. A key ...

Added: May 30, 2022

Trautmann P., Vexler B., Zlotnik A., / Cornell University. Series "Working papers by Cornell University". 2017. No. 1702.00362.

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$
or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level in time bilinear ...

Added: February 2, 2017

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

Trautmann P., Vexler B., Zlotnik A., Mathematical Control and Related Fields 2018 Vol. 8 No. 2 P. 411-449

This work is concerned with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces $\mathcal M_T$ of either measure-valued functions $L^2(I,\mathcal M(\Omega))$ or vector measures $\mathcal M(\Omega,L^2(I))$. The cost functional involves the standard quadratic terms and the regularization term $\alpha\|u\|_{\mathcal M_T}$, $\alpha>0$. We construct and study three-level ...

Added: April 8, 2017

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

Zlotnik A., Kireeva O., / Cornell University. Series arXiv "math". 2020. No. arXiv:2011.14104v2[math.NA].

We consider compact finite-difference schemes of the 4th approximation order for an initial-boundary value problem (IBVP) for the $n$-dimensional non-homogeneous wave equation, $n\geq 1$. Their construction is accomplished by both the classical Numerov approach and alternative technique based on averaging of the equation, together with further necessary improvements of the arising scheme for $n\geq 2$. The ...

Added: December 1, 2020

Beklaryan L., Beklaryan A., , in : Optimization and Applications: 12th International Conference, OPTIMA 2021, Petrovac, Montenegro, September 27 – October 1, 2021, Proceedings. : Switzerland : Springer, 2021. P. 165-175.

Added: November 4, 2021

Romanov I., Шамаев А. С., Известия РАН. Теория и системы управления 2019 № 1 С. 109-116

The boundary controllability of oscillations of a plane membrane is studied. The magnitude
of the control is bounded. The controllability problem of driving the membrane to rest is considered.
The method of proof proposed in this paper can be applied to any dimension but only the two-dimensional case is considered for simplicity. ...

Added: April 30, 2019

Pelinovsky E., Капцов О. В., Доклады Российской академии наук. Физика, технические науки (ранее - Доклады Академии Наук. Физика) 2022 Т. 506 № 2 С. 46-51

Methods to obtain solutions describing traveling waves in strongly inhomogeneous media are discussed
within a linear wave equation with a variable speed (speed of sound). It is shown that there is a wide
range of propagation speed variations that allow for the existence of waves that do not reflect despite the
strong inhomogeneity of the medium. In this ...

Added: January 28, 2023

Petropavlovsky S., Цынков С. В., Журнал вычислительной математики и математической физики 2020 Т. 60 № 4 С. 725-737

A boundary method for simulation of wave propagation in 3D is proposed. The approach relies on the method of difference potentials and Huygens'principle allowing for an update of the solution only on the boundaries of a scatterer/ ...

Added: April 13, 2020

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

Maslov V., Теоретическая и математическая физика 2021 Т. 206 № 3 С. 448-452

We consider the construction of asymptotic solutions of linear equations related to equations of classical mechanics: the Hamilton–Jacobi equation and the transport equation. We show that these methods and also the theory of the mechanics of an infinitely narrow beam as a whole can be applied to some objects in bioenergy if the thin organic ...

Added: July 14, 2021

Zlotnik A., Čiegis R., / Cornell University. Series arXiv "math". 2021. No. ArXiv: 2101.10575v2[math.NA].

We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. They are closely connected to some ...

Added: February 2, 2021

Romanov I., Shamaev A., Journal of Optimization Theory and Applications 2021 Vol. 188 No. 3 P. 925-938

The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the
control is to drive the system to rest in finite time. ...

Added: May 23, 2021

Zlotnik A., Čiegis R., Journal of Scientific Computing 2023 Vol. 95 No. 1 Article 3

We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time and compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. We also show that some ...

Added: January 20, 2023

Деркач М. М., Филимонов А. М., Filimonov D., М. : МИИТ, 2013

В настоящем выпуске помещены методические указания к решению задач по разделу «Метод бегущих волн» дисциплины «Уравнения математической физики». В первой части настоящих методических указаний метод бегущих волн рассматривается применительно к одномерному волновому уравнению. Во второй части рассматриваются гиперболические системы довольно общего вида, поскольку метод бегущих волн можно с успехом распространить и на гиперболические системы уравнений ...

Added: January 9, 2014

Vexler B., Zlotnik A., Trautmann P., Comptes Rendus Mathematique 2018 Vol. 356 No. 5 P. 523-531

The paper deals with the optimal control problems governed by the 1D wave equation with variable coefficients and the control spaces of either measure-valued functions or vector measures. Bilinear finite element discretizations are constructed and their stability and error analysis is accomplished. ...

Added: April 8, 2017

Zlotnik A., Lomonosov T., Applied Numerical Mathematics 2024 Vol. 195 P. 54-74

We study a three-level explicit in time higher-order vector compact scheme, with additional n sought functions approximating second order non-mixed spatial derivatives of the solution, for an initial-boundary value problem for the n-dimensional wave equation and acoustic wave equation, with the variable speed of sound, n⩾1. We also approximate the solution at the first time level in a ...

Added: October 7, 2023

Melnikov I. E., 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

Manita L., Современная математика и ее приложения 2011 Т. 69

Рассматриваются задачи минимизации квадратичного функционала на траекториях волнового уравнения, которые на основе метода Фурье сводятся к задаче управления счетной системой обыкновенных дифференциальных уравнений. В качестве управления выступает плотность внешних сил. Исследуется проблема управляемости. ...

Added: October 7, 2013

Pelinovsky E., Talipova T., Didenkulova E., Fluids 2022 Vol. 7 No. 9 Article 294

Rational solutions of nonlinear evolution equations are considered in the literature as a
mathematical image of rogue waves, which are anomalously large waves that occur for a short time.
In this work, bounded rational solutions of Gardner-type equations (the extended Korteweg-de
Vries equation), when a nonlinear term can be represented as a sum of several terms with arbitrary
powers ...

Added: September 6, 2022

Zlotnik A., Čiegis R., Applied Mathematics and Computation 2022 Vol. 412 Article 126565

We consider an initial-boundary value problem for the $n$-dimensional wave equation, $n\geq 2$, with the variable sound speed with the nonhomogeneous Dirichlet boundary conditions. We construct and study three-level in time and compact in space three-point in each spatial direction alternating direction implicit (ADI) schemes having the approximation orders $\mathcal{O}(h_t^2+|h|^4)$ and $\mathcal{O}(h_t^4+|h|^4)$ on the uniform rectangular mesh.
The study includes stability ...

Added: August 11, 2021