Skip to main content
SpringerLink
Log in

Model Reduction of a Higher-Order KdV Equation for Shallow Water Waves

  • Conference paper
  • First Online:
Book cover Coping with Complexity: Model Reduction and Data Analysis

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 75))

Abstract

We present novel results on a non-integrable generalized KdV equation proposed by Fokas [A.S. Fokas, Physica D87, 145 (1995)], aiming to describe unidirectional solitary water waves with greater accuracy than the standard KdV equation. The profile of the solitary wave solutions is determined via a reduction of the partial differential equation (PDE) to a set of ordinary differential equations (ODEs). Subsequently, we study the stability of the wave using this profile as initial condition for the PDE. In the case of the standard KdV equation it is well-known that the solitary wave solutions are always stable, irrespective of their height. However, in the case of our higher-order KdV equation we find that the stability of the solutions breaks down beyond a certain critical height, just like solitary waves in real water experiments.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Korteweg, D.J., 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. Phil. Mag. 39 (1895) 422–443

    Google Scholar 

  2. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  3. Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)

    MATH  Google Scholar 

  4. Novikov, S., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons: The inverse Scattering Method. Consultants Bureau, New York (1984)

    Google Scholar 

  5. Drazin, P.J., Johnson, R.S.: Solitons: an Introduction. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  6. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  7. Remoissenet, M.: Waves Called Solitons: Concepts and Experiments, 3rd Ed. Springer, Berlin (1999)

    MATH  Google Scholar 

  8. Billingham, J., King, A.C.: Wave Motion. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  9. Russell, J.S.: Report on Waves. In: Rep. 14th Meet. British Assoc. Adv. Sci. York. John Murray, London (1845) 11–390

    Google Scholar 

  10. Hammack, J.L., Segur, H.: The Korteweg-de Vries equation and water waves, Part 2: Comparison with Experiments. J. Fluid Mech. 65 (1974) 289–314

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  12. Amick, C.J., Toland, J.F.: On solitary water-waves of finite amplitude. Arch. Rat. Mech. Anal. 76 (1981) 9–95

    Article  MATH  MathSciNet  Google Scholar 

  13. Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  14. Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44 (2007) 423-431

    Article  MATH  MathSciNet  Google Scholar 

  15. Nelson, R.C.: Depth limited design wave heights in very flat regions. Coastal Engng. 23 (1994) 43–59

    Article  Google Scholar 

  16. Alvarez-Samaniego, B., Lannes, D.: Large time existence for 3D water-waves and asymptotics, Invent. Math. 171 (2008) 485-541

    Article  MATH  MathSciNet  Google Scholar 

  17. Constantin, A., Johnson, R.S.: On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves. J. Nonl. Math. Phys. 15 (2008) 58–73

    Article  MathSciNet  Google Scholar 

  18. Kodama, Y.: On integrable systems with higher order corrections. Phys. Lett. A 107 (1985) 245–249

    Article  MATH  MathSciNet  Google Scholar 

  19. Marchant, T.R., Smyth, N.F.: The extended Korteweg-de Vries equation and the resonant flow over topography. J. Fluid Mech. 221 (1990) 263–288

    Article  MATH  MathSciNet  Google Scholar 

  20. Kichenassamy, S., Olver, P.J.: Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J. Math. Anal. 23 (1992) 1141–1161

    Article  MATH  MathSciNet  Google Scholar 

  21. Marchant, T.R., Smyth, N.F.: Soliton solutions for the extended Korteweg-de Vries equation. IMA J. Appl. Math. 56 (1996) 157–176

    Article  MATH  MathSciNet  Google Scholar 

  22. Fokas, A.S., Liu, Q.M.: Asymptotic integrability of water waves. Phys. Rev. Lett. 77 (1996) 2347–2351

    Article  MATH  MathSciNet  Google Scholar 

  23. Kraenkel, R.A.: First-order perturbed Korteweg-de Vries solitons. Phys. Rev. E 57 (1998) 4775–4777

    Article  MathSciNet  Google Scholar 

  24. Marchant, T.R.: Asymptotic solitons of the extended Korteweg-de Vries equation. Phys. Rev. E 59 (1999) 3745–3748

    Article  MathSciNet  Google Scholar 

  25. Andriopoulos, K., Bountis, T., van der Weele, K., Tsigaridi, L.: The shape of soliton-like solutions of a higher-order KdV equation describing water waves. J. Nonlin. Math. Phys. 16, s-1 (2009) 1–12

    Google Scholar 

  26. Marchant, T.R.: High-order interaction of solitary waves on shallow water. Studies in Applied Math. 109 (2002) 1–17

    Article  MATH  MathSciNet  Google Scholar 

  27. Marchant, T.R.: Asymptotic solitons for a third-order Korteweg-de Vries equation. Chaos, Solitons and Fractals 22 (2004) 261–270

    Google Scholar 

  28. Fokas, A.S.: On a class of physically important integrable equations. Physica D 87 (1995) 145–150

    Article  MATH  MathSciNet  Google Scholar 

  29. Tzirtzilakis, E., Xenos, M., Marinakis, V., Bountis, T.C.: Interactions and stability of solitary waves in shallow water. Chaos, Solitons and Fractals 14 (2002) 87–95

    Google Scholar 

  30. Tzirtzilakis, E., Marinakis, V., Apokis, C., Bountis, T.C.: Soliton-like solutions of higher order wave equations of the Korteweg-de Vries type. J. Math. Phys. 43 (2002) 6151–6165

    Article  MATH  MathSciNet  Google Scholar 

  31. McCowan, J.: On the highest wave of permanent type, Phil. Mag. 38 (1894) 351–358

    Google Scholar 

  32. Massel, S.R.: On the largest wave height in water of constant depth. Ocean Engng. 23 (1996) 553–573

    Article  Google Scholar 

  33. Abohadima, S., Isobe, M.: Limiting criteria of permanent progressive waves. Coastal Engng. 44 (2002) 231–237

    Article  Google Scholar 

  34. Toland, J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7 (1996) 1–48

    MATH  MathSciNet  Google Scholar 

  35. Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166 (2006) 523–535

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

TB, KvdW and GK acknowledge support from the Carathéodory Programme of the University of Patras, grant number C167. KA thanks the Hellenic Scholarships Foundation (IKY) for partial financial support of this work.

Author information

Authors and Affiliations

  1. Department of Mathematics and Centre for Research and Applications of Nonlinear Systems, University of Patras, 26500, Patras, Greece

    Tassos Bountis, Ko van der Weele, Giorgos Kanellopoulos & Kostis Andriopoulos

  2. Centre for Education in Science, 26504, Platani – Rio, Greece

    Tassos Bountis & Kostis Andriopoulos

Authors
  1. Tassos Bountis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ko van der Weele
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giorgos Kanellopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kostis Andriopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tassos Bountis .

Editor information

Editors and Affiliations

  1. Dept. Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  2. Dept. Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200a, Leuven, 3001, Belgium

    Dirk Roose

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bountis, T., van der Weele, K., Kanellopoulos, G., Andriopoulos, K. (2011). Model Reduction of a Higher-Order KdV Equation for Shallow Water Waves. In: Gorban, A., Roose, D. (eds) Coping with Complexity: Model Reduction and Data Analysis. Lecture Notes in Computational Science and Engineering, vol 75. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14941-2_15

Download citation

Publish with us

Policies and ethics

Search

Navigation