scholarly journals Asymptotic soliton-like solutions to the singularly perturbed Benjamin-Bona-Mahony equation with variable coefficients

2019 ◽  
Vol 60 (1) ◽  
pp. 011501 ◽  
Author(s):  
Valerii Samoilenko ◽  
Yuliia Samoilenko
Keyword(s):  
Variable Coefficients ◽  
Singularly Perturbed ◽  
Asymptotic Soliton

Global Asymptotic Step-Type Solutions to Singularly Perturbed Korteweg-De Vries Equation with Variable Coefficients

2020 ◽  
Vol 52 (9) ◽  
pp. 27-38
Author(s):  
Sergey I. Lyashko ◽  
Valeriy H. Samoilenko ◽  
Yulia I. Samoilenko ◽  
Igor V. Gapyak ◽  
Nataliya I. Lyashko ◽  
...  
Keyword(s):  
Vries Equation ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
De Vries ◽  
Step Type

Asymptotic soliton-like solutions to the Benjamin–Bona–Mahony equation with variable coefficients and a strong singularity

2021 ◽  
Vol 18 (2) ◽  
pp. 226-242
Author(s):  
Valerii Samoilenko ◽  
Yuliia Samoilenko
Keyword(s):  
Asymptotic Solution ◽  
Small Parameter ◽  
Order Term ◽  
Variable Coefficients ◽  
Main Term ◽  
Qualitative Properties ◽  
First Order ◽  
Strong Singularity ◽  
Non Linear ◽  
Asymptotic Soliton

The paper deals with constructing an asymptotic one-phase soliton-like solution to the Benjamin--Bona--Mahony equation with variable coefficients and a strong singularity making use of the non-linear WKB technique. The influence of the small-parameter value on the structure and the qualitative properties of the asymptotic solution, as well as the accuracy with which the solution satisfies the considerable equation, have been analyzed. It was demonstrated that due to the strong singularity, it is possible to write explicitly not only the main term of the asymptotics but at least its first-order term.

Asymptotic analysis of the Korteweg-de Vries equation by the nonlinear WKB technique

2021 ◽  
Vol 8 (3) ◽  
pp. 368-378
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
N. I. Lyashko ◽  
...  
Keyword(s):  
Asymptotic Solution ◽  
Soliton Solution ◽  
Vries Equation ◽  
Wave Process ◽  
Singular Part ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
Main Term ◽  
Phase Variable ◽  
De Vries

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The non-linear WKB technique has been used to construct the asymptotic step-like solution to the equation. Such a solution contains regular and singular parts of the asymptotics. The regular part of the solution describes the background of the wave process, while its singular part reflects specific features associated with soliton properties. The singular part of the searched asymp\-totic solution has the main term that, like the soliton solution, is the quickly decreasing function of the phase variable $\tau$. In contrast, other terms do not possess this property. An algorithm of constructing asymptotic step-like solutions to the singularly perturbed Korteweg--de Vries equation with variable coefficients is presented. In some sense, the constructed asymptotic solution is similar to the soliton solution to the Korteweg-de Vries equation $u_t+uu_x+u_{xxx}=0$. Statement on the accuracy of the main term of the asymptotic solution is proven.

scholarly journals Asymptotic analysis of the singularly perturbed Korteweg-de Vries equation

2019 ◽  
pp. 194-197
Author(s):  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
V. S. Vovk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Vries Equation ◽  
Singular Part ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
De Vries

The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

Sign in / Sign up

Export Citation Format

Share Document