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

Lyashko, S. I.; Samoilenko, Valeriy H.; Samoilenko, Yulia I.; Gapyak, I. V.; Lyashko, N. I.; Orlova, Mariya S.

doi:10.1615/jautomatinfscien.v52.i9.30

Cited by 4 publications

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“…It means that higher terms of the singular part of the asymptotic tend to zero as τ → +∞ and tend to non-zero as τ → −∞. Such asymptotic solutions are called the asymptotic step-like solutions [33].…”

Section: Introductionmentioning

confidence: 99%

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

Lyashko¹,

Самойленко²,

Samoilenko³

et al. 2021

Math. Model. Comput.

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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.

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Section: Introductionmentioning

confidence: 99%

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

Lyashko¹,

Самойленко²,

Samoilenko³

et al. 2021

Math. Model. Comput.

View full text Add to dashboard Cite

show abstract

“…In [1][2][3] we began to study new type of asymptotic solutions to the Korteweg-de Vries equation with variable coefficients and a small parameter of the following form…”

Section: Introductionmentioning

confidence: 99%

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

Lyashko¹,

Самойленко²,

Samoilenko³

et al. 2021

Math. Model. Comput.

View full text Add to dashboard Cite

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.

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

Самойленко

Samoilenko

2021

J Math Sci

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

Cited by 4 publications

References 0 publications

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

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

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

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

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