Solutions of Traveling Wave Type for Korteweg-de Vries-Type System with Polynomial Potential

  • Levon A. Beklaryan
  • Armen L. BeklaryanEmail author
  • Alexander Yu. Gornov
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 974)


This paper deals with the implementation of numerical methods for searching for traveling waves for Korteweg-de Vries-type equations with time delay. Based upon the group approach, the existence of traveling wave solution and its boundedness are shown for some values of parameters. Meanwhile, solutions constructed with the help of the proposed constructive method essentially extend the class of systems, possessing solutions of this type, guaranteed by theory. The proposed method for finding solutions is based on solving a multiparameter extremal problem. Several numerical solutions are demonstrated.


Korteweg-de Vries equation Functional differential equations Traveling waves 



This work was partially supported by Russian Science Foundation, Project 17-71-10116. Also, the reported study was partially funded by RFBR according to the research project 16-01-00110 A.


  1. 1.
    Abell, K., Elmer, C., Humphries, A., Van Vleck, E.: Computation of mixed type functional differential boundary value problems. SIAM J. Appl. Dyn. Syst. 4(3), 755–781 (2005). Scholar
  2. 2.
    Baotong, C.: Functional differential equations mixed type in banach spaces. Rendiconti del Seminario Matematico della Università di Padova, 94, 47–54 (1995).
  3. 3.
    Beklaryan, L.: A method for the regularization of boundary value problems for differential equations with deviating argument. Soviet Math. Dokl. 43, 567–571 (1991)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Beklaryan, L.: Introduction to the Theory of Functional Differential Equations. Group Approach. Factorial Press, Moscow (2007)Google Scholar
  5. 5.
    Beklaryan, L.: Quasitravelling waves. Sbornik: Math. 201(12), 1731–1775 (2010). Scholar
  6. 6.
    Beklaryan, L.: Quasi-travelling waves as natural extension of class of traveling waves. Tambov Univ. Reports. Ser. Nat. Tech. Sci. 19(2), 331–340 (2014)Google Scholar
  7. 7.
    Beklaryan, L., Beklaryan, A.: Traveling waves and functional differential equations of pointwise type. what is common? In: Proceedings of the VIII International Conference on Optimization and Applications (OPTIMA-2017), Petrovac, Montenegro, October 2–7. (2017).
  8. 8.
    Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, USA (2003)CrossRefGoogle Scholar
  9. 9.
    Bellman, R., Cooke, K.: Differential-Difference Equations. Academic Press, New York (1963)zbMATHGoogle Scholar
  10. 10.
    El’sgol’ts, L., Norkin, S.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic Press, New York (1973)zbMATHGoogle Scholar
  11. 11.
    Ford, N., Lumb, P.: Mixed-type functional differential equations: a numerical approach. J. Comput. Appl. Math. 229(2), 471–479 (2009). Scholar
  12. 12.
    Frenkel, Y., Contorova, T.: On the theory of plastic deformation and twinning. J. Exp. Theor. Phys. 8(1), 89–95 (1938)Google Scholar
  13. 13.
    Gardner, C., Greene, J., Kruskal, M.: Method for solving the korteweg-de vries equation. Phys. Rev. Lett. 19(19), 1095–1097 (1967). Scholar
  14. 14.
    Gornov, A., Zarodnyuk, T., Madzhara, T., Daneeva, A., Veyalko, I.: A Collection of Test Multiextremal Optimal Control Problems, pp. 257–274. Springer, New York (2013). Scholar
  15. 15.
    Hale, J.: Theory of Functional Differential Equations. Springer, New York (1977)CrossRefGoogle Scholar
  16. 16.
    Hale, J., Lunel, S.V.: Introduction to Functional Differential Equations. Applied Mathematical Sciences, vol. 99. Springer, New York (1993). Scholar
  17. 17.
    Kortweg, D., 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. Philos. Mag. 5(39), 422–443 (1895)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, X., Wu, B.: A continuous method for nonlocal functional differential equations with delayed or advanced arguments. J. Math. Anal. Appl. 409(1), 485–493 (2014). Scholar
  19. 19.
    Lima, P., Teodoro, M., Ford, N., Lumb, P.: Analytical and numerical investigation of mixed-type functional differential equations. J. Comput. Appl. Math. 234(9), 2826–2837 (2010). Scholar
  20. 20.
    Lima, P., Teodoro, M., Ford, N., Lumb, P.: Finite element solution of a linear mixed-type functional differential equation. Numer. Algorithms 55(2–3), 301–320 (2010). Scholar
  21. 21.
    Maset, S.: Numerical solution of retarded functional differential equations as abstract cauchy problems. J. Comput. Appl. Math. 161(2), 259–282 (2003). Scholar
  22. 22.
    Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  23. 23.
    Myshkis, A.: Linear Differential Equations with Retarded Arguments. Nauka, Moscow (1972)Google Scholar
  24. 24.
    Sun, D., Chen, D., Zhao, M., Liu, W., Zheng, L.: Linear stability and nonlinear analyses of traffic waves for the general nonlinear car-following model with multi-time delays. Phys. A: Stat. Mech. Appl. 501, 293–307 (2018). Scholar
  25. 25.
    Toda, M.: Theory of Nonlinear Lattices, vol. 20. Springer, Berlin, Heidelberg (1989). Scholar
  26. 26.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, vol. 119. Springer, New York (1996). Scholar
  27. 27.
    Xu, L., Lo, S., Ge, H.: The korteweg-de vires equation for bidirectional pedestrian flow model. Procedia Eng. 52, 495–499 (2013). Scholar
  28. 28.
    Yu, L., Shi, Z., Li, T.: A new car-following model with two delays. Phys. Lett. A 378(4), 348–357 (2014). Scholar
  29. 29.
    Zarodnyuk, T., Anikin, A., Finkelshtein, E., Beklaryan, A., Belousov, F.: The technology for solving the boundary value problems for nonlinear systems of functional differential equations of pointwise type. Mod. Technol. Syst. Anal. Model. 49(1), 19–26 (2016)Google Scholar
  30. 30.
    Zarodnyuk, T., Gornov, A., Anikin, A., Finkelstein, E.: Computational technique for investigating boundary value problems for functional-differential equations of pointwise type. In: Proceedings of the VIII International Conference on Optimization and Applications (OPTIMA-2017), Petrovac, Montenegro, October 2–7. (2017).
  31. 31.
    Zhao, Z., Rong, E., Zhao, X.: Existence for korteweg-de vries-type equation with delay. Adv. Differ. Equ. 2012(1), 64 (2012). Scholar
  32. 32.
    Zhao, Z., Xu, Y.: Solitary waves for korteweg-de vries equation with small delay. J. Math. Anal. Appl. 368(1), 43–53 (2010). Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Central Economics and Mathematics Institute RASMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.Institute for System Dynamics and Control Theory of SB RASIrkutskRussia

Personalised recommendations