Water Waves

, Volume 1, Issue 1, pp 3–40 | Cite as

Numerical Study of the Second-Order Correct Hamiltonian Model for Unidirectional Water Waves

  • Jerry L. BonaEmail author
  • Hongqiu Chen
  • Youngjoon Hong
  • Ohannes Karakashian
Original Article


Second-order correct versions of the usual KdV–BBM models for unidirectional propagation of long-crested, surface water waves are considered here. The class of models studied here has a Hamiltonian structure and, in certain circumstances, is globally well posed. A fully discrete, numerical algorithm based on the Fourier-spectral method is developed and its convergence tested. We then use this algorithm to generate solitary-wave solutions to the model. While such waves are known to exist, exact formulas for them are not available. The heart of the paper is a sequence of numerical experiments aimed at understanding the stability of individual solitary waves, their interaction, and whether or not the model exhibits resolution of general initial data into solitary waves. A comparison is made between the first-order correct KdV–BBM models and the associated second-order correct equations. A number of tentative conjectures pertaining to the models are put forward on the basis of these experiments.


Higher-order water wave models Hamiltonian models Solitary Waves Spectral methods KdV-BBM models 



JB, HC, and YH are all grateful for support and excellent working conditions from the Mathematics Department of the University of Tennessee at Knoxville. Part of this work was done, while JB and HC were visiting professors at Victoria University inWellington, New Zealand. YH thanks the Simons Foundation for travel support during the period of this collaboration. OK was partially supported by NSF grant DMS-1620288.


  1. 1.
    Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J.: Comparisons between the BBM equation and a Boussinesq system. Adv. Differ. Equ. 11(2), 121–166 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Albert, J.P., Bona, J.L., Restrepo, J.M.: Solitary-wave solutions of the Benjamin equation. SIAM J. Appl. Math. 59, 2139–2161 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrose, D.M., Bona, J.L., Nichols, D.P.: On ill-posedness of truncated series models for water waves. Proc. R. Soc. Lond. Ser. A 470, 1–16 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benjamin, T.B.: Stability of solitary waves. Proc. R. Soc. Lond. Ser. A 328, 153–183 (1972)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bona, J.L.: On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A 349, 363–374 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A 272(1220), 47–78 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bona, J.L.: Convergence of periodic wave trains in the limit of large wavelength. Appl. Sci. Res. 37(1–2), 21–30 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bona, J.L., Carvajal, X., Panthee, M., Scialom, M.: Higher-order Hamiltonian model for unidirectional water waves. J. Nonlinear Sci. 28(2), 543–577 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bona, J.L., Chen, H.: Periodic traveling-wave solutions of nonlinear dispersive evolution equations. Discrete Cont. Dyn. Syst. 33, 4841–4873 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bona, J.L., Chen, H., Karakashian, O.A.: Stable solitary-wave solutions of systems of coupled dispersive equations. Appl. Math. Optim. 75, 27–53 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bona, J.L., Chen, H., Karakashian, O.A.: Unstable solitary-wave solutions of systems of coupled KdV-equations. In: preparationGoogle Scholar
  12. 12.
    Bona, J.L., Chen, H., Karakashian, O.A., Xing, Y.: Conservative, discontinuous Galerkin-methods for the generalized Korteweg-de Vries equation. Math. Comput. 82(283), 1401–1432 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bona, J.L., Chen, H., Sun, S., Zhang, B.-Y.: Approximating initial-value problems with two-point boundary-value problems: BBM equation. Bull. Iran. Math. Soc. 36, 1401–1432 (2010)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Bona, J.L., Chen, M.: A Boussinesq system for two-way propagation of nonlinear dispersive waves. Phys. D 116(1–2), 191–224 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12(4), 283–318 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory. Nonlinearity 17(3), 925–952 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178(3), 373–410 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A 351, 107–164 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bona, J.L., Pritchard, W.G., Scott, L.R.: Solitary–wave interaction. Phys. Fluids 23, 438–441 (1980)CrossRefzbMATHGoogle Scholar
  20. 20.
    Bona, J.L., Pritchard, W.G., Scott, L.R.: A comparison of solutions of two model equations for long waves. Lect. Appl. Math. 20, 235–267 (1981)MathSciNetGoogle Scholar
  21. 21.
    Boussinesq, J.: Essai sur la théorie des eaux courantes. Memoires présentes par divers savants. l’Acad. des Sci. Inst. Nat. Fr. XXIII, 1–680 (1877)Google Scholar
  22. 22.
    Craig, W.: An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. PDE 10, 787–1003 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Craig, W., Guyenne, P., Hammack, J., Henderson, D., Sulem, C.: Solitary water wave interactions. Phys. Fluids 18(5), 57–106 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  25. 25.
    Chen, H.: Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete Contin. Dyn. Syst. A 38(1), 397–429 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 36, 165–186 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Craik, A.D.D.: The origins of water wave theory. Ann. Rev. Fluid Mech. 36, 1–28 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Craik, A.D.D.: George Gabriel Stokes on water wave theory. Ann. Rev. Fluid Mech. 37, 23–42 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Dias, F., Milewski, P.: On the fully-nonlinear shallow-water generalized Serre equations. Phys. Lett. A 374(8), 1049–1053 (2010)CrossRefzbMATHGoogle Scholar
  30. 30.
    Fenton, J.: A ninth-order solution for the solitary wave. J. Fluid Mech. 53(2), 257–271 (1972)CrossRefzbMATHGoogle Scholar
  31. 31.
    Fermi, E., Pasta, J., Ulam, S., Tsingou, M.: Studies of nonlinear problems. Los Alamos Natl. Lab. Rep. LA–1940, 1–20 (1955)Google Scholar
  32. 32.
    Fokas, A.S., Liu, Q.M.: Asymptotic integrability of water waves. Phys. Rev. Lett. 77(12), 2347–2351 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Gardner, C.S., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-deVries equation. Phys. Rev. Lett. 19, 1095–1097 (1967)CrossRefzbMATHGoogle Scholar
  34. 34.
    Hammack, J.L.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60(4), 769–799 (1973)CrossRefzbMATHGoogle Scholar
  35. 35.
    Hammack, J.L., Segur, H.: The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments. J. Fluid Mech. 65(2), 289–314 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Holger, R., Dullin, G., Gottwald, G.A., Holm, D.D.: Camassa–Holm, Korteweg-de Vries-5 and other asymptotically equivalent equations for shallow water waves. Fluid Dyn. Res. 33(1–2), 73–95 (2003)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kraenkel, R.A., Manna, M.A., Merle, V., Montero, J.C., Pereira, J.G.: Multiple-time higher-order perturbation analysis of the regularized long-wavelength equation. Phys. Rev. E 54, 2976–2981 (1996)CrossRefGoogle Scholar
  38. 38.
    Kodama, Y.: On integrable systems with higher order corrections. Phys. Lett. A 107(6), 245–249 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Kodama, Y.: On solitary-wave interaction. Phys. Lett. A 123(6), 276–282 (1987)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Korteweg, D.J., deVries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. (5th series) 39, 422443 (1895)MathSciNetGoogle Scholar
  41. 41.
    Lannes, D.: The water waves problem. Mathematical analysis and asymptotics. Volume 188 of Mathematical Surveys and Monographs. American Math. Soc., Providence, RI (2013)Google Scholar
  42. 42.
    Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Lin, Z., Zeng, C.: Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs. arXiv:1703.04016 (submitted)
  44. 44.
    Liu, P.L.-F., Cheng, Y.: A numerical study of the evolution of a solitary wave over a shelf. Phys. Fluids 13(6), 1660–1667 (2001)CrossRefzbMATHGoogle Scholar
  45. 45.
    Marchant, T.R.: Solitary wave interaction for the extended BBM equation. Proc. R. Soc. Lond. Ser. A 456(1994), 433–453 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Madsen, O.S., Mei, C.C.: The transformation of a solitary wave over an uneven bottom. J. Fluid Mech. 39(4), 781–791 (1969)CrossRefGoogle Scholar
  47. 47.
    Olver, P.J.: Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Philos. Soc. 85, 143–160 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Olver, P.J.: Hamiltonian and non-Hamiltonian models for water waves. In: Trends and applications of pure mathematics to mechanics (Palaiseau, 1983). Volume 195 of Lecture Notes Phys., pp. 273–290. Springer, Berlin (1984)Google Scholar
  49. 49.
    Peregrine, D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25(2), 321–330 (1966)CrossRefMathSciNetGoogle Scholar
  50. 50.
    Pöschel, J., Trubowitz, E.: Inverse spectral theory. Pure and Applied Mathematics Series, vol. 130. Academic Press, Elsevier, Amsterdam (1987)zbMATHGoogle Scholar
  51. 51.
    Russell, J.S.: “Report on Waves”. Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844, London: John Murray, pp 311–390, Plates XLVII–LVII (1845)Google Scholar
  52. 52.
    Strusinska-Correia, A., Oumeraci, H.: Nonlinear behaviour of tsunami-like solitary wave over submerged impermeable structures of finite width. Proc. Coast Eng. 1(33), 6 (2012)CrossRefGoogle Scholar
  53. 53.
    Seabra-Santos, F.J., Renouard, D.P., Temperville, A.M.: Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle. J. Fluid Mech. 176, 117–134 (1987)CrossRefGoogle Scholar
  54. 54.
    Shen, J.: A new dual-Petrov–Galerkin method for third and higher odd-order differential equations: application to the KdV equation. SIAM J. Numer. Anal. 41(5), 1595–1619 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Shen, J., Tang, T., Wang, L.L.: Spectral methods. Algorithms, analysis and applications. Volume 41 of Springer Series in Computational Mathematics. Springer, Heidelberg (2011)Google Scholar
  56. 56.
    Trefethen, L.N.: Spectral methods in MATLAB. Volume 10 of Software, Environments, and Tools. SIAM, Philadelphia, PA (2000)Google Scholar
  57. 57.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley-Interscience (Wiley), New York (1974)zbMATHGoogle Scholar
  58. 58.
    Yuan, J.-M., Chen, H., Sun, S.-M.: Existence and orbital stability of solitary-wave solutions for higher-order BBM equations. J. Math. Study 49(3), 293–318 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Zabusky, N.J., Galvin, C.J.: Shallow-water waves, the Korteweg-deVries equation and solitons. J. Fluid Mech. 47(4), 811–824 (1971)CrossRefGoogle Scholar
  60. 60.
    Zabusky, N.J., Kruskal, M.D.: Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240–243 (1965)CrossRefzbMATHGoogle Scholar
  61. 61.
    Zou, Q., Su, C.-H.: Overtaking collision between two solitary waves. Phys. Fluids 29(7), 2113–2123 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jerry L. Bona
    • 1
    Email author
  • Hongqiu Chen
    • 2
  • Youngjoon Hong
    • 3
  • Ohannes Karakashian
    • 4
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  3. 3.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA
  4. 4.Department of MathematicsUniversity of TennesseesKnoxvilleUSA

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