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

2021;
: pp. 368–378
https://doi.org/10.23939/mmc2021.03.368
Received: March 05, 2021
Accepted: June 19, 2021
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

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 asymptotic 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.

  1. Korteweg D. J., de Vries G.  On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 39 (240), 422–443 (1895).
  2. Russel J. S.  Report on waves.  Reports Fourteenth Meeting of the British Association. 311 (1844).
  3. Novikov S., Manakov S. V., Pitaevskii L. P., Zakharov V. E.  Theory of Solitons. The Inverse Scattering Method.  Springer US, New York (1984).
  4. Zabusky N. J., Kruskal M. D.  Interaction of "solitons" in a collisionless plasma and the recurrence of initial states.  Phys. Rev. Lett. 15 (6), 240–243 (1965).
  5. Dodd R. K., Morris H. C., Eilbeck J. C., Gibbon J.D.  Solitons and nonlinear wave equations.  Academic Press, New York (1982).
  6. Blackmore D., Prykarpatsky A., Samoylenko V.  Nonlinear dynamical systems of mathematical physics. Spectral and symplectic integrability analysis.  World Scientific (2011).
  7. Bona J. L., Smith R.  The initial value problem for the Korteweg–de Vries equation.  Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 278, 555–601 (1975).
  8. Kato T.  On the Korteweg-de Vries equation.  Manuscripta Math. 28, 89–99 (1979).
  9. Faminskii A. V.  Cauchy problem for the Korteweg-de Vries equation and its generalizations.  Journal of Soviet Mathematics. 50, 1381–1420 (1990).
  10. Egorova I., Grunert K., Teschl G.  On Cauchy problem for the Korteweg-de Vries equation with step-like finite gap initial data. I. Schwartz-type perturbations.  Nonlinearity. 22 (6), 1431–1457 (2009).
  11. Faddeev L. D., Takhtadjan L. A.  Hamiltonian methods in the theory of solitons.  Springer, New York, Berlin (1986).
  12. Mitropol'skij A. Yu., Bogolyubov N. N. (jr.), Prikarpatskij A. K., Samojlenko V. G.  Integrable dynamical systems: spectral and differential-geometric aspects.  Naukova Dumka, Kiev (1986).
  13. Prykarpatsky A. K., Mykytyuk I. V.  Algebraic integrability of nonlinear dynamical systems on manifolds: classical and quantum aspects.  Kluwer Academic Publishers, the Netherlands (1998).
  14. Mokhov O. I.  Symplectic and Poisson geometry on loop spaces of smooth manifolds and integrable equations (Second Edition: Reviews in Mathematics and Mathematical Physics, Vol. 13, Part 2).  Cambridge Scientific Publishers Ltd, Cambridge (2009).
  15. Samoilenko V. G., Samoilenko Yu. I.  To the method of the boundary layer and the Hugoniot-type condition for the Korteweg-de Vries equation.  Vesnik of Brest University. Series 4. Physics. Mathematics.  2, 111–129 (2010), (in Russian).
  16. Olver P. J., Sokolov V. V.  Integrable evolution equations on associative algebras.  Communications in Mathematical Physics. 193, 245–268 (1998).
  17. Flaschka H., Forest M. G., McLaughlin D. W.  Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation.  Comm. Pure Appl. Math. 33 (6), 739–784 (1980).
  18. Prikarpatskij A. K., Samojlenko V. G.  Averaging method and equations of ergodic deformations for nonlinear evolution equations.  Preprint IM AN USSR 81.44, 31 p. Institute of Mathematics of AS of USSR, Kyiv (1981).
  19. 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).
  20. Maslov V. P., Omel'yanov G. A.  Asymptotic soliton-form solutions of equations with small dispersion.  Russian Mathematical Surveys. 36 (3), 73–149 (1981).
  21. Maslov V. P., Omel'yanov G. O.  Geometric asymptotics for PDE.  American Math. Society, Providence (2001).
  22. Lax P. D., Levermore C. D.  The small dispersion limit of the Korteweg–de Vries equation. I. Comm. Pure Appl. Math. 36 (3), 253–290 (1983).
  23. Lax P. D., Levermore C. D.  The small dispersion limit of the Korteweg–de Vries equation. ii. Comm. Pure Appl. Math. 36 (4), 571–593 (1983).
  24. Lax P. D., Levermore C. D.  The small dispersion limit of the Korteweg–de Vries equation. III. Comm. Pure Appl. Math. 36 (6), 809–829 (1983).
  25. Samoilenko V. H., Samoilenko Yu. I.  Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg-de Vries equation with variable coefficients. I. Ukrainian Mathematical Journal. 64 (7), 1109–1127 (2012).
  26. Samoilenko V. H., Samoilenko Yu. I. Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg-de Vries equation with variable coefficients. II. Ukrainian Mathematical Journal. 64 (8), 1241–1259 (2013).
  27. Samoilenko V., Samoilenko Yu.  Asymptotic soliton-like solutions to the singularly perturbed Benjamin–Bona–Mahony equation with variable coefficients.  Journal of Mathematical Physics. 60 (1), 011501-1–011501-13 (2019).
  28. Ablowitz M. J.  Nonlinear dispersive waves. Asymptotic analysis and solitons.  Cambridge University Press, Cambridge (2011).
  29. 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).
  30. 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).
  31. Kapustian O. A., Nakonechnyi O. G.  Approximate minimax estimation of functionals of solutions to the wave equation under nonlinear observations.  Cybernetices and Systems Analysis. 56 (5), 793–-801 (2020).
  32. Verovkina G., Gapyak I., Samoilenko V., Telyatnik T.  Asymptotic analysis of solutions to equations with regular perturbation.  Bulletin of Taras Shevchenko National University of Kyiv.  Mathematics. Mechanics. 40 (1), 11–15 (2019).
  33. 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).
Mathematical Modeling and Computing, Vol. 8, No. 3, pp. 368–378 (2021)