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Asymptotic Σ-Solutions of a Singularly Perturbed Benjamin–Bona–Mahony Equation with Variable Coefficients

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Ukrainian Mathematical Journal Aims and scope

We study the problem of construction of asymptotic Σ-solutions of a singularly perturbed Benjamin–Bona–Mahony equation with variable coefficients and propose an algorithm for the construction of these solutions. We determine the main and first terms of the asymptotic solution. The theorems on the accuracy with which the indicated asymptotic solution satisfies the analyzed equation are also proved.

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References

  1. D. J. Korteweg and G. de Vries, “On the change in form of long waves advancing in a rectangular canal and a new type of long stationary waves,” Phil. Mag., No. 39, 422–433 (1895).

  2. N. J. Zabusky and M. D. Kruskal, “Interaction of solitons in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett., 15, 240–243 (1965).

    Article  Google Scholar 

  3. C. S. Gardner, J. M. Green, M. D. Kruskal, and R. M. Miura, “Method for solving the Korteweg–de Vries equation,” Phys. Rev. Lett., 19, 1095–1097 (1967).

    Article  Google Scholar 

  4. S. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method, Springer, Berlin (1984).

    MATH  Google Scholar 

  5. F. Calogero and A. Degasperis, Spectral Transform and Solitons, North-Holland, Amsterdam (1982).

    MATH  Google Scholar 

  6. M. Wadati, “The modified Korteweg–de Vries equation,” J. Phys. Soc. Japan, 34, No. 6, 1289–1296 (1973).

    Article  MathSciNet  Google Scholar 

  7. M. Toda, Nonlinear Waves and Solitons, Kluwer Acad. Publ., Tokyo (1989).

    MATH  Google Scholar 

  8. D. J. Kaup and A. C. Newell, “An exact solution for a derivative nonlinear Schrödinger equation,” J. Math. Phys., 19, No. 4, 798–801 (1978).

    Article  Google Scholar 

  9. B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakly dispersive media,” Sov. Phys. Dokl., 15, 539–541 (1970).

    MATH  Google Scholar 

  10. D. J. Kaup, “A higher-order water-wave equation and the method for solving it,” Prog. Theor. Phys., 54, 396–408 (1975).

    Article  MathSciNet  Google Scholar 

  11. A. C. Newell, Nonlinear Wave Motion, American Mathematical Society, Providence, RI (1974).

    MATH  Google Scholar 

  12. G. R. Lamb, Jr., Elements of Soliton Theory, Wiley, New York (1980).

    MATH  Google Scholar 

  13. D. H. Peregrin, “Calculations of the development of an undular bore,” J. Fluid Mech., 25, No. 2, 321–330 (1966).

    Article  Google Scholar 

  14. T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Phil. Trans. Roy. Soc. London, Ser. A, 272, 47–78 (1972).

    Article  MathSciNet  Google Scholar 

  15. Kh. O. Abdulloyev, I. L. Bogolubsky, and V. G. Makhankov, “One more example of inelastic soliton interaction,” Phys. Lett. A, 56, 427–428 (1976).

    Article  MathSciNet  Google Scholar 

  16. A. R. Santarelli, “Numerical analysis of the regularized long-wave equation: inelastic collision of solitary waves,” Nuova Cim. B, 46, 179–188 (1978).

    Article  Google Scholar 

  17. J. L. Bona, W. G. Pritchard, and L. R. Scott, “Solitary wave interaction,” Phys. Fluids, 23, No. 3, 438–441 (1980).

    Article  Google Scholar 

  18. J. L. Bona, W. G. Pritchard, and L. R. Scott, “An evolution of a model equation for water waves,” Phyl. Trans. Roy. Soc. A, 302, 457–510 (1981).

    Article  Google Scholar 

  19. A. R. Seadway and A. Sayed, “Travelling wave solutions of the Benjamin–Bona–Mahony water wave equations,” Abstr. Appl. Anal., 2014 (2014), Article ID 926838.

  20. J. C. Eilback and G. R. McGruire, “Numerical studies of the regularized long wave equation. I: Numerical methods,” J. Comput. Phys., 19, 43–57 (1975).

    Article  Google Scholar 

  21. J. C. Eilback and G. R. McGruire, “Numerical studies of the regularized long wave equation. II: Interaction of solitary waves,” J. Comput. Phys., 23, 63–73 (1977).

    Article  Google Scholar 

  22. R. K. Dodd, J. C. Eilback, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London (1982).

    Google Scholar 

  23. N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1963).

    MATH  Google Scholar 

  24. N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).

  25. R. M. Miura and M. D. Kruskal, “Application of nonlinear WKB-method to the Korteweg–de Vries equation,” SIAM Appl. Math., 26, No. 2, 376–395 (1974).

    Article  MathSciNet  Google Scholar 

  26. P. D. Lax and C. D. Levermore, “The small dispersion limit of the Korteweg–de Vries equation. I,” Comm. Pure Appl. Math., 36, No. 3, 253–290 (1983).

    Article  MathSciNet  Google Scholar 

  27. P. D. Lax and C. D. Levermore, “The small dispersion limit of the Korteweg–de Vries equation. II,” Comm. Pure Appl. Math., 36, No. 5, 571–593 (1983).

    Article  MathSciNet  Google Scholar 

  28. P. D. Lax and C. D. Levermore, “The small dispersion limit of the Korteweg–de Vries equation. III,” Comm. Pure Appl. Math., 36, No. 6, 809–829 (1983).

    Article  MathSciNet  Google Scholar 

  29. H. Flaschka, M. G. Forest, and D.W. McLaughlin, “Multiphase averaging and the inverse spectral solution of the Korteweg–de Vries equation,” Comm. Pure Appl. Math., 33, No. 6, 739–784 (1980).

    Article  MathSciNet  Google Scholar 

  30. A. M. Il’in and L. A. Kalyakin, “Perturbation of finite-soliton solutions of the Korteweg–de Vries equation,” Dokl. Ros. Akad. Nauk, 336, No. 5, 595–598 (1994).

    MathSciNet  MATH  Google Scholar 

  31. A. M. Samoilenko and Ya. A. Prykarpats’kyi, Algebraic-Analytic Aspects of the Theory of Completely Integrable Dynamical Systems and Their Perturbations [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2002).

  32. F. de Kerf, Asymptotic Analysis of a Class of Perturbed Korteweg–de Vries Initial Value Problems, Centrum voor Wiskunde en Informatica, Amsterdam (1988).

    MATH  Google Scholar 

  33. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotic expansions for one-phase soliton-type solutions of the Korteweg–de-Vries equation with variable coefficients,” Ukr. Mat. Zh., 57, No. 1, 111–124 (2005); English translation : Ukr. Math. J., 57, No. 1, 132–148 (2005).

  34. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg–de-Vries equation with variable coefficients,” Ukr. Mat. Zh., 64, No. 7, 970–987 (2012); English translation : Ukr. Math. J., 64, No. 7, 1109–1127 (2012).

  35. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotic m-phase soliton-type solutions of a singularly perturbed Korteweg–de-Vries equation with variable coefficients. II,” Ukr. Mat. Zh., 64, No. 8, 1089–1105 (2012); English translation : Ukr. Math. J., 64, No. 8, 1241–1259 (2013).

  36. V. Samoilenko and Yu. Samoilenko, Asymptotic Soliton-Like Solutions to the Singularly Perturbed Benjamin–Bona–Mahony Equation with Variable Coefficients, Preprint arXiv:1703.01265 (2017).

  37. V. Hr. Samoylenko and Yu. I. Samoylenko, “Asymptotic multiphase Σ-solutions to the singularly perturbed Korteweg–de-Vries equation with variable coefficients,” J. Math. Sci., 200, No. 3, 358–373 (2014).

    Article  MathSciNet  Google Scholar 

  38. V. P. Maslov and G. A. Omel’yanov, Geometric Asymptotics for PDE. I, American Mathematical Society, Providence, RI (2001).

  39. Yu. I. Samoilenko, “Existence of solution of the Cauchy problem for the Hopf equation with variable coefficients in the space of rapidly decreasing functions,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], Kyiv, 9, No. 1 (2012), pp. 293–300.

  40. Yu. I. Samoilenko, “Existence of solution of the Cauchy problem for a linear partial differential equation of the first order with variable coefficients in the space of rapidly decreasing functions,” Bukov. Mat. Zh., 1, No. 1-2, 120–124 (2013).

    MATH  Google Scholar 

  41. V. Hr. Samoylenko and Yu. I Samoylenko, “Existence of a solution to the inhomogeneous equation with the one-dimensional Schrödinger operator in the space of quickly decreasing functions,” J. Math. Sci., 187, No. 1, 70–76 (2012).

    Article  MathSciNet  Google Scholar 

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  1. T. Shevchenko Kyiv National University, Kyiv, Ukraine

    V. H. Samoilenko & Yu. I. Samoilenko

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  1. V. H. Samoilenko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 2, pp. 236–254, February, 2018.

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Samoilenko, V.H., Samoilenko, Y.I. Asymptotic Σ-Solutions of a Singularly Perturbed Benjamin–Bona–Mahony Equation with Variable Coefficients. Ukr Math J 70, 266–287 (2018). https://doi.org/10.1007/s11253-018-1500-5

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  • DOI: https://doi.org/10.1007/s11253-018-1500-5

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