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

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

MMC.
2021;
Volume 8, Number 3
: pp. 410–421
https://doi.org/10.23939/mmc2021.03.410
Received: March 05, 2021
Accepted: June 19, 2021

Mathematical Modeling and Computing, Vol. 8, No. 3, pp. 410–421 (2021)

Authors:
  1. S. I. Lyashko
  2. V. H. Samoilenko
  3. Yu. I. Samoilenko
  4. I. V. Gapyak
  5. M. S. Orlova
1
Taras Shevchenko National University of Kyiv
2
Taras Shevchenko National University of Kyiv
3
Taras Shevchenko National University of Kyiv
4
Taras Shevchenko National University of Kyiv
5
Borys Grinchenko Kyiv University

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.

Korteweg-de Vries equation
nonlinear WKB technique
asymptotic step-like solution
soliton
singular perturbation
asymptotic analysis

  1. Lyashko S. I., Samoilenko V. H., Samoilenko Yu. I., Lyashko N. I.  Asymptotic analysis of the Korteweg-de Vries equation by the nonlinear WKB method.  Mathematical Modelling and Computing. 8 (3), 410–420 (2021).

  2. Lyashko S. I., Samoilenko V. H., Samoilenko Yu. I., Lyashko N. I.  Stepwise asymptotic solutions to the Korteweg–de Vries equation with variable coefficients and a small parameter at the higher-order derivative.  Cybernetics and Systems Analysis.  56 (6), 934–942 (2020).

  3. Lyashko S. I., Samoilenko V. H., Samoilenko Yu. I., Gapyak I. V., Lyashko N. I., Orlova M. S.  Global asymptotic step-type solutions to singularly perturbed Korteweg-de Vries equation with variable coefficients.  Journal of Automation and Information Sciences. 52 (9), 27–38 (2020).

  4. Ablowitz M. J.  Nonlinear dispersive waves. Asymptotic analysis and solitons.  Cambridge University Press, Cambridge (2011).

  5. Miura R. M., Kruskal M. D.  Application of nonlinear WKB-method to the Korteweg–de Vries equation.  SIAM Appl. Math. 26 (2), 376–395 (1974).

  6. Maslov V. P., Omel'yanov G. O.  Geometric asymptotics for PDE.  American Math. Society, Providence (2001).

  7. Samoilenko V. H., Samoilenko Yu. I.  Asymptotic two-phase soliton-like solutions of the singularly perturbed Korteweg–de Vries equation with variable coefficients.  Ukrainian Mathematical Journal. 60 (3), 449–461 (2008).

  8. Samoilenko V. H., Samoilenko Yu. I., Limarchenko V. O., Vovk V. S., Zaitseva K. S.  Asymptotic solutions of soliton-type of the Korteweg–de Vries equation with variable coefficients and singular perturbation.  Mathematical Modeling and Computing. 6 (2), 374–385 (2019).

  9. Samoilenko V. H., Samoilenko Yu. I.  Asymptotic expansions for one-phase soliton-type solutions of the Korteweg-de Vries equation with variable coefficients.  Ukranian Mathematical Journal. 57 (1), 132–148 (2005).

  10. Krantz S. G., Parks H. R.  The implicit function theorem: history, theory and appications.  Birkhausers, Boston (2002).

  11. Samoilenko V. H., Kaplun Yu. I.  Existence and extendability of solutions of the equation $g(t,x)=0$.  Ukrainian Mathematical Journal. 53 (3), 427–437 (2001).

  12. Samoilenko V. H., Samoilenko Yu. I.  Existence of a solution to the inhomogeneous equation with the one-dimensional Schrodinger operator in the space of quickly decreasing functions.  Journal of Mathematical Sciences. 187 (1), 70–76 (2012).