Skip to main content
SpringerLink
Log in

Collision of Solitons for a Non-homogenous Version of the KdV Equation: Asymptotics and Numerical Simulation

  • Conference paper
  • First Online:
Numerical Analysis and Its Applications (NAA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10187))

Included in the following conference series:

  • 1742 Accesses

Abstract

We consider a generalized KdV equation with a small dispersion and \(C^1\)-nonlinearity \(g'(u)\). We present sufficient conditions for \(g'(u)\) under which a soliton type solution exists and, moreover, pairs of solitary waves collide preserving in an asymptotic sense the KdV type scenario of interaction. Furthermore, we create a finite difference scheme to simulate the solution of the Cauchy problem and present some numerical results for the interaction problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Schamel, H.: A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons. J. Plasma Phys. 9, 377–387 (1973)

    Article  Google Scholar 

  2. Faminskii, A.V.: On an initial boundary-value problem in a bounded domain for the generalized Korteweg-de Vries equation. Funct. Differ. Equ. 8(1–2), 183–194 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Bona, J.L., Souganidis, P.E., Strauss, W.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. R. Soc. London Ser. A 411(1841), 395–412 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Danilov, V.G., Omel’yanov, G.A.: Weak asymptotics method and the interaction of infinitely narrow delta-solitons. Nonlinear Anal.: Theory Methods Appl. 54, 773–799 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Danilov, V.G., Omel’yanov, G.A., Shelkovich, V.M.: Weak asymptotics method and interaction of nonlinear waves. In: Karasev, M.V. (ed.) Asymptotic Methods for Wave and Quantum Problems. AMS Translations: Series 2, vol. 208, pp. 33–164. AMS, Providence (2003)

    Google Scholar 

  6. Garcia, M., Omel’yanov, G.: Interaction of solitary waves for the generalized KdV equation. Commun. Nonlinear Sci. Numer. Simul. 17(8), 3204–3218 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garcia, M., Omel’yanov, G.: Interaction of solitons and the effect of radiation for the generalized KdV equation. Commun. Nonlinear Sci. Numer. Simul. 19(8), 2724–2733 (2014)

    Article  MathSciNet  Google Scholar 

  8. Omel’yanov, G., Valdez-Grijalva, M.: Asymptotics for a \(C^1\)-version of the KdV equation. Nonlinear Phenom. Complex Syst. 17(2), 106–115 (2014)

    MATH  Google Scholar 

  9. Omel’yanov, G.: Soliton-type asymptotics for non-integrable equations: a survey. Math. Methods Appl. Sci. 38(10), 2062–2071 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Omel’yanov, G.: Collision of solitons for a non-homogenous version of the KdV equation, pp. 1–18 (2015). http://arxiv.org/abs/1511.09063

Download references

Acknowledgments

The research was supported by SEP-CONACYT under grant 178690 (Mexico).

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Sonora, Rosales y Blvd. Encinas s/n, 83000, Hermosillo, Mexico

    G. Omel’yanov

Authors
  1. G. Omel’yanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Omel’yanov .

Editor information

Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. Department of Applied Mathematics and Statistics, University of Rousse, Ruse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Omel’yanov, G. (2017). Collision of Solitons for a Non-homogenous Version of the KdV Equation: Asymptotics and Numerical Simulation. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science(), vol 10187. Springer, Cham. https://doi.org/10.1007/978-3-319-57099-0_58

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-57099-0_58

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57098-3

  • Online ISBN: 978-3-319-57099-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation