SOLITON SOLUTIONS
FOR THE MANHATTAN LATTICE
Levon Beklaryan1, Armen Beklaryan2,
Andranik Akopov3
1 Central Economics and Mathematics Institute RAS
47, Nakhimovsky Pr.
Moscow – 117418, RUSSIA
2 HSE University
26-28, Shabolovka Str.
Moscow – 119049, RUSSIA
3 Central Economics and Mathematics Institute RAS
Nakhimovsky pr., 47
Moscow – 117418, RUSSIA

Abstract

The article is devoted to the study of the aggregated model of motion in the Manhattan lattice. For such a model, the existence and uniqueness theorem of a soliton solution is established, the ranges of characteristics for which the statements of the theorem are valid are indicated, and the asymptotics of the possible growth of such solutions are obtained. A complete family of bounded soliton solutions is constructed.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 4
Year: 2023

DOI: 10.12732/ijam.v36i4.10

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, Hindawi Publishing Corporation, New York (2007).
  2. [2] L.A. Beklaryan, A boundary value problem for a differential equation with deviating argument, Dokl. Akad. Nauk SSSR, 291, No 1 (1986), 19-22.
  3. [3] L.A. Beklaryan, A differential equation with deviating argument as an infinite-dimensional dynamical system, In: Reports in Applied Mathematics, Vychisl. Tsentr, Moscow (1989), 18.
  4. [4] L.A. Beklaryan, A method for the regularization of boundary value problems for differential equations with deviating argument, Soviet Math. Dokl., 43, No 2 (1991), 567-571.
  5. [5] L.A. Beklaryan, An optimal control problem for systems with deviating argument and its connection with the finitely generated group of homeomorphisms R generated by deviation functions, Soviet Math. Dokl., 43, No 2 (1991), 600-605.
  6. [6] L.A. Beklaryan, Group singularities of differential equations with deviating arguments and metric invariants related to them, Journal of Mathematical Sciences, 105, No 1 (2001), 1799-1811.
  7. [7] L.A. Beklaryan, Equations of advanced-retarded type and solutions of traveling-wave type for infinite-dimensional dynamic systems, Journal of Mathematical Sciences, 124, No 4 (2004), 5098-5109.
  8. [8] L.A. Beklaryan, N.K. Khachatryan, Traveling wave type solutions in dynamic transport models, Functional Differential Equations, 13, No 2 (2006), 125-155.
  9. [9] L.A. Beklaryan, Introduction to the Theory of Functional Differential Equations. Group Approach, Factorial Press, Moscow (2007).
  10. [10] L.A. Beklaryan, Quasitravelling waves, Sbornik: Mathematics, 201, No 12 (2010), 1731-1775.
  11. [11] L.A. Beklaryan, N.K. Khachatryan, On one class of dynamic transportation models, Comput. Math. Math. Phys., 53, No 10 (2013), 1466-1482.
  12. [12] L.A. Beklaryan, Quasi-travelling waves as natural extension of class of traveling waves, Tambov University Reports. Series Natural and Technical Sciences, 19, No 2 (2014), 331-340.
  13. [13] L.A. Beklaryan, A.L. Beklaryan, A.Yu. Gornov, Solutions of traveling wave type for Korteweg-de Vries-type system with polynomial potential, In: Optimization and Applications 9th International Conference, OPTIMA 2018, Springer, Switzerland (2019), 291-305.
  14. [14] L.A. Beklaryan, A.L. Beklaryan, Existence of bounded soliton solutions in the problem of longitudinal vibrations of an infinite elastic rod in a field with a strongly nonlinear potential, Comput. Math. Math. Phys., 61, No 12 (2021), 1980-1994.
  15. [15] L.A. Beklaryan, A.L. Beklaryan, Existence of bounded soliton solutions in the problem of longitudinal oscillations of an elastic infinite rod in a field with a nonlinear potential of general form, Comput. Math. Math. Phys., 62, No 6 (2022), 904-919.
  16. [16] L.E. El’sgol’ts, S.B. Norkin, Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York (1973).
  17. [17] Ya.I. Frenkel, T.A. Contorova, On the theory of plastic deformation and twinning, Journal of Experimental and Theoretical Physics, 8, No 1 (1938), 89-95.
  18. [18] A.Yu. Gornov, T.S. Zarodnyuk, T.I. Madzhara, A.V. Daneeva, I.A. Veyalko, A collection of test multiextremal optimal control problems, In: Optimization, Simulation, and Control, Springer, New York (2013), 257274.
  19. [19] J.K. Hale, Theory of Functional Differential Equations, Springer, New York (1977).
  20. [20] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, New York (1993).
  21. [21] J.P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM Journal on Applied Mathematics, 47, No 3 (1987), 556-572.
  22. [22] J.W. Cahn, J. Mallet-Paret, E.S. van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM Journal on Applied Mathematics, 59, No 2 (1998), 455-493.
  23. [23] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, Journal of Dynamics and Differential Equations, 11, No 1 (1999), 49-127.
  24. [24] J. Mallet-Paret, The Fredholm alternative for functional-differentional equations mixed type, Journal of Dynamics and Differential Equations, 11, No 1 (1999), 1-47.
  25. [25] T. Miwa, M. Jimbo, E. Date, Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambridge University Press, UK (2000).
  26. [26] L.D. Pustyl’nikov, Infinite-dimensional non-linear ordinary differential equations and the KAM theory, Russian Mathematical Surveys, 52, No 3 (1997), 551-604.
  27. [27] M. Toda, Theory of Nonlinear Lattices, Springer, Berlin, Heidelberg (1989).
  28. [28] B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, Journal of Differential Equations, 96, No 1 (1992), 1-27.