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.

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.

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.

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

Weakly non-linear waves in rotating fluids

1970 ◽  
Vol 42 (4) ◽  
pp. 803-822 ◽  
Author(s):  
S. Leibovich
Keyword(s):  
Differential Equation ◽  
Singular Perturbation ◽  
Vortex Breakdown ◽  
Vries Equation ◽  
Tube Wall ◽  
Rotating Fluids ◽  
Integro Differential Equation ◽  
Stationary Flows ◽  
Linear Waves ◽  
De Vries

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to infinity, and the failure is corrected by singular perturbation procedures. The Korteweg–de Vries equation must then give way to an integro-differential equation. Also, critical stationary flows in tubes are considered with regard to Benjamin's vortex breakdown theories.

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.

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