Stepwise Asymptotic Solutions to the Korteweg–De Vries Equation with Variable Coefficients and a Small Parameter at the Higher-Order Derivative

2020 ◽  
Vol 56 (6) ◽  
pp. 934-942
Author(s):  
S. I. Lyashko ◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
N. I. Lyashko
Keyword(s):  
Small Parameter ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Higher Order ◽  
Asymptotic Solutions ◽  
Order Derivative ◽  
De Vries ◽  
Higher Order Derivative

ASYMPTOTIC SOLUTIONS TO THE KORTEWEG-DE VRIES EQUATION WITH VARIABLE COEFFICICNTES AT NON-ZERO BACKGROUND

2019 ◽  
pp. 35-40
Author(s):  
V. Samoilenko ◽  
Yu. Samoilenko ◽  
M. Orlova
Keyword(s):  
Small Parameter ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Asymptotic Solutions ◽  
De Vries

Problem on studying asymptotic one-phase soliton-like solutions to the Korteweg–de Vries equation with variable coefficients and a small parameter of the first degree at the highest derivative is considered for the case of non-zero background. There is given an algorithm of constructing the solutions for general case. The algorithm is demonstrated for the equation with some given variable coefficients.

Asymptotic stepwise solutions of the Korteweg--de Vries equation with a singular perturbation and their accuracy

2021 ◽  
Vol 8 (3) ◽  
pp. 410-421
Author(s):  
S. I. Lyashko ◽  
◽  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
I. V. Gapyak ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Small Parameter ◽  
Asymptotic Approximation ◽  
Vries Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Main Term ◽  
Asymptotic Approximations ◽  
De Vries ◽  
The Given

The paper deals with the Korteweg-de Vries equation with variable coefficients and a small parameter at the highest derivative. The asymptotic step-like solution to the equation is obtained by the non-linear WKB technique. An algorithm of constructing the higher terms of the asymptotic step-like solutions is presented. The theorem on the accuracy of the higher asymptotic approximations is proven. The proposed technique is demonstrated by example of the equation with given variable coefficients. The main term and the first asymptotic approximation of the given example are found, their analysis is done and statement of the approximate solutions accuracy is presented.

scholarly journals Asymptotic solutions of soliton type of the Korteweg–de Vries equation with variable coefficients and singular perturbation

2019 ◽  
Vol 6 (2) ◽  
pp. 374-385 ◽  
Author(s):  
V. H. Samoilenko ◽  
◽  
Yu. I. Samoilenko ◽  
V. O. Limarchenko ◽  
V. S. Vovk ◽  
...  
Keyword(s):  
Singular Perturbation ◽  
Vries Equation ◽  
Variable Coefficients ◽  
Asymptotic Solutions ◽  
De Vries

scholarly journals Whitham equations and phase shifts for the Korteweg–de Vries equation

2020 ◽  
Vol 476 (2240) ◽  
pp. 20200300
Author(s):  
Mark J. Ablowitz ◽  
Justin T. Cole ◽  
Igor Rumanov
Keyword(s):  
Small Parameter ◽  
Perturbation Analysis ◽  
Vries Equation ◽  
Phase Shifts ◽  
Higher Order ◽  
Numerical Calculations ◽  
Leading Order ◽  
Order Solution ◽  
Whitham Equations ◽  
De Vries

The semi-classical Korteweg–de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations.

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.

Gauge transformation and the higher order Korteweg–de Vries equation

1988 ◽  
Vol 29 (2) ◽  
pp. 308-314 ◽  
Author(s):  
Yu‐kun Zheng ◽  
W. L. Chan
Keyword(s):  
Gauge Transformation ◽  
Vries Equation ◽  
Higher Order ◽  
De Vries
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