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Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients

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Abstract

We propose an algorithm for the construction of asymptotic two-phase solitonlike solutions of the Korteweg-de Vries equation with a small parameter at the higher derivative.

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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).

    Google Scholar 

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

    Article  Google Scholar 

  3. P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math., 21, No. 15, 467–490 (1968).

    Article  MATH  MathSciNet  Google Scholar 

  4. 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  MATH  Google Scholar 

  5. V. P. Maslov and G. A. Omel’yanov, “Asymptotic solitonlike solutions of equations with small dispersion,” Usp. Mat. Nauk, Issue 36 (219), No. 2, 63–124 (1981).

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

    MATH  Google Scholar 

  7. Yu. I. Samoilenko, “Asymptotical expansions for one-phase soliton-type solution to perturbed Korteweg-de Vries equation,” in: Proc. of the Fifth Internat. Conf. “Symmetry in Nonlinear Mathematical Physics,” Vol. 3, Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (2004), pp. 1435–1441.

    Google Scholar 

  8. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotic expansions for one-phase solitonlike solutions of the Korteweg-de Vries equation with variable coefficients,” Ukr. Mat. Zh., 58, No. 1, 111–124 (2005).

    Google Scholar 

  9. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotical expansion of solution to Cauchy problem for Korteweg-de Vries equation with varying coefficients and a small parameter,” in: Comm. CERMCS Int. Conf. Young Sci., Moldova State University, Chisinau (2006), pp. 186–192.

    Google Scholar 

  10. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotic expansions of the Cauchy problem for the singularly perturbed Korteweg-de Vries equation with variable coefficients,” Ukr. Mat. Zh., 59, No. 1, 122–132 (2007).

    Article  MATH  Google Scholar 

  11. V. H. Samoilenko and Yu. I. Samoilenko, “Asymptotic solution to Cauchy problem for Korteweg-de Vries equation with varying coefficients and a small dispersion,” in: Proc. of the Fourth Internat. Workshop “Computer Algebra Systems in Teaching and Research” (CASTR 2007) (Siedlce, Poland, January 31–February 3, 2007) (2007), pp. 272–280.

  12. Yu. I. Samoilenko, “Asymptotic expansions for one-phase solitonlike solutions of the Cauchy problem for the Korteweg-de Vries equation with variable coefficients,” Nauk. Visn. Cherniv. Univ., Ser. Mat., Issues 336, 337, 170–177 (2007).

  13. A. V. Faminskii, “Cauchy problem for the Korteweg-de Vries equation and its generalizations,” Tr. Seminara im. Petrovskogo, Issue 13, 56–105 (1998).

    Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 388–397, March, 2008.

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Samoilenko, V.H., Samoilenko, Y.I. Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients. Ukr Math J 60, 449–461 (2008). https://doi.org/10.1007/s11253-008-0067-y

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  • DOI: https://doi.org/10.1007/s11253-008-0067-y

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