ГЛОБАЛЬНІ АСИМПТОТИЧНІ РОЗВ’ЯЗКИ СХІДЧАСТОГО ТИПУ ДЛЯ СИНГУЛЯРНО ЗБУРЕНОГО РІВНЯННЯ КОРТЕВЕГА-ДЕ ФРІЗА ЗІ ЗМІННИМИ КОЕФІЦІЄНТАМИ

• Date: 2020
• Main Authors: Lyashko, S.I., Samoilenko, V.H., Samoilenko, Yu.I., Gapyak, I.V., Lyashko, N.I., Orlova , M.S.
• Format: Article
• Language: English
• Published: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Subjects: математичні моделі; рівняння Кортевега-де Фріза зі змінними коефіцієнтами; асимптотичні; солітоноподібні розв’язки; малий параметр; сингулярні збурення
• Online Access: https://jais.net.ua/index.php/files/article/view/488
• Journal Title: Problems of Control and Informatics

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. Similar equations arise while mathematical simulation of wave processes in inhomogeneous media with variable characteristics and a small dispersion. A specific feature of the problem is the presence of nonlinear terms and variable coefficients that does not allow us to obtain its exact solution in explicit form. By means of the nonlinear WKB technique its asymptotic soliton like solutions are constructed, that in general case approximately describe the deformations of soliton solutions of the corresponding equations with constant coefficients. The searched solution is given in the form of the sum of the regular and singular parts of the asymptotics. The equations for terms of the asymptotic expansions are obtained as well as the procedure of constructing their solutions is presented. There is considered the case of zero background when the regular part of the asymptotics is trivial. The first approximation is obtained for an asymptotic step like solution that is defined for all values of independent variables. It means that the solution is global. The graphs of the constructed solutions for various values of a small parameter are presented. Their analysis allows us to conclude that for an adequate description of the asymptotic step like solution it is sufficient to construct its first asymptotic approximation. The last property is consistent with the theoretical results obtained by the authors previously.