УДК 517.9

АСИМПТОТИЧЕСКИЕ РЕШЕНИЯ СТУПЕНЧАТОГО ТИПА
ДЛЯ УРАВНЕНИЯ КОРТЕВЕГА–де ФРИЗА С ПЕРЕМЕННЫМИ
КОЭФФИЦИЕНТАМИ И МАЛЫМ ПАРАМЕТРОМ ПРИ СТАРШЕЙ ПРОИЗВОДНОЙ

Ляшко Сергей Иванович,
чл.-кор. НАН Украины, доктор физ.-мат. наук, профессор, заведующий кафедрой Киевского национального университета имени Тараса Шевченко,  lyashko.serg@gmail.com

Самойленко Валерий Григорьевич,
доктор физ.-мат. наук, профессор, заведующий кафедрой Киевского национального университета имени Тараса Шевченко,  valsamyul@gmail.com

Самойленко Юлия Ивановна,
доктор физ.-мат. наук, старший научный сотрудник Киевского национального университета
имени Тараса Шевченко,  yusam@univ.kiev.ua

Ляшко Наталья Ивановна,
кандидат техн. наук, научный сотрудник Института кибернетики им. В.М. Глушкова НАН Украины, Киев,  dept165@insyg.kiev.ua; lyashko.natali@gmail.com

СПИСОК ЛИТЕРАТУРЫ

1. Lyashko I.I., Lyashko S.I., Semenov V.V. Control of pseudo-hyperbolic systems by concentrated impacts. Journal of Automation and Information Sciences. 2000. Vol. 32, N 12. P. 23–36. https://doi.org/10.1615/JAutomatInfScien.v32.i12.40.


2. Lyashko S.I., Nomirovskij D.A., Sergienko T.I. Trajectory-final controllability in hyperbolic and pseudo-hyperbolic systems with generalized actions. Cybernetics and System Analysis. 2001. Vol. 37, N 5. P. 756–763. https://doi.org/10.1023/a:1013871026026.


3. Whitham G.B. Linear and nonlinear waves. New Jersey: John Wiley & Sons, 1999. 636 p. https://doi.org/10.1002/9781118032954.


4. Newel A.C. Solitons in mathematics and physics. SIAM: Philadelphia, 1985. 260 p.


5. Selezov I.T. Diffraction of elastic waves by a sphere in the semi-bounded region. Cybernetics and System Analysis. 2019. Vol. 55, N 3. P. 393–399. https://doi.org/10.1007/s10559-019-00146-3.


6. Davydov A.S. Solitons in biology. In: Modern Problems in Condensed Matter Sciences. Ch. 1. Amsterdam: North-Holland Physics Publishing, 1986. Vol. 17. P. 1–51. https://doi.org/10.1016/ B978-0-444-87002-5.50007-2.


7. Haus H.A., Wong W.S. Solitons in optical communications. Reviews of Modern Physics. 1996. Vol. 68, N 2. P. 423–444. https://doi.org/10.1103/RevModPhys.68.423.


8. Zabusky N.J., Kruskal M.D. Interaction of solitons in a collisionless plasma and recurrence of initial states. Phys. Review Lett. 1965. Vol. 15. P. 240–243. http://dx.doi.org/10.1103/PhysRevLett.15.240.


9. Blacmore D., Prykarpatsky A.K., Samoylenko V.H. Nonlinear dynamical systems of mathematical physics. Spectral and integrability analysis. Singapore: World Scientific, 2011. 564 p.


10. Zadiraka V.K. Using reserves of computing optimization to solve complex problems. Cybernetics and Systems Analysis. 2019. Vol. 55. N 1. P. 40–54. https://doi.org/10.1007/s10559-019-00111-0.


11. Galba E.F., Deineka V.S., Sergienko I.V. Weighted pseudoinverses and weighted normal pseudosolutions with singular weights. Computational Mathematics and Mathematical Physics. 2009. Vol. 49, N 8. P. 1281–1297. https://doi.org/10.1134/S0965542509080016.


12. Pokhozhaev S.I. On the singular solutions of the Korteweg-de Vries equation. Mathematical Notes. 2010. Vol. 88, N 5. P. 741–747. https://doi.org/10.1134/S0001434610110131.


13. Novikov S.P., Manakov S.V., Pitaevskii L.P., Zakharov V.E. Theory of solitons. The inverse scattering method. Springer US, 1984. 288 p.


14. Nakonechnyi O.G., Kapustian O.A., Chikrii A.O. Approximate guaranteed mean square estimates of functionals on solutions of parabolic problems with fast oscillating coefficients under nonlinear observations. Cybernetics and Systems Analysis. 2019. Vol. 55, N 5. P. 785–795. https://doi.org/10.1007/s10559-019-00189-6.


15. Ji J., Zhang L., Wang L. et al. Variable coefficient KdV equation with time-dependent variable coefficient topographic forcing term and atmospheric blocking. Advance Difference Equation. 2019. Vol. 320. https://doi.org/10.1186/s13662-019-2045-0.


16. Maslov V.P., Omel’yanov G.A. Geometric asymptotics for PDE. I. Providence: American Mathematical Society, 2001. 243 p.


17. Prikazchikov V.G., Khimich A.N. Asymptotic estimates of the accuracy of eigenvalues of fourth order elliptic operator with mixed boundary conditions. Cybernetics and Systems Analysis. 2017. Vol. 53, N 3. P. 358–365. https://doi.org/10.1007/s10559-017-9935-5.


18. Samoilenko V.Hr., Samoilenko Yu.I. Asymptotic expansions for one-phase soliton-type solutions of the Korteweg–de Vries equation with variable coefficients. Ukrainian Mathematical Journal. 2005. Vol. 57, N 1. P. 132–148.https://doi.org/10.1007/s11253-005-0176-9.


19. Samoilenko V.Hr., Samoilenko Yu.I. Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg-de Vries equation with variable coefficients. Ukrainian Mathematical Journal. 2012. Vol. 64, N 7. P. 1109–1127. https://doi.org/10.1007/s11253-012-0702-5.


20. 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. 2019. Vol. 6, N 2. P. 374–384. https://doi.org/10.23939/mmc2019.02.374.


21. Samoilenko V.Hr., 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. 2012. Vol. 187, N 1. P. 70–76. https://doi.org/10.1007/s10958-012-1050-6.