Journal of Nonlinear Science

, Volume 17, Issue 6, pp 569–607

A Numerical Study of the Stability of Solitary Waves of the Bona–Smith Family of Boussinesq Systems

  • V. A. Dougalis
  • A. Durán
  • M. A. López-Marcos
  • D. E. Mitsotakis
Article

Abstract

In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the behavior of solitary waves of these systems under small and large perturbations with the aim of illuminating their long-time asymptotic stability properties and, in the case of large perturbations, examining, among other, phenomena of possible blow-up of the perturbed solutions in finite time.

Keywords

Boussinesq systems Stability of solitary waves 

Mathematics Subject Classification (2000)

35Q53 65M60 76B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. P. Albert, J. L. Bona, and D. Henry. Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Physica D, 24:343–366, 1987. MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. C. Antonopoulos. The Boussinesq system of equations: Theory and numerical analysis. Ph.D. thesis. University of Athens, 2000 (in Greek). Google Scholar
  3. 3.
    D. C. Antonopoulos and V. A. Dougalis. Numerical approximation of Boussinesq systems. In A. Bermudez et al., editors, Proceedings of the 5th International Conference on Mathematical and Numerical Aspects of Wave Propagation, pages 265–269. SIAM, Philadelphia, 2000. Google Scholar
  4. 4.
    D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis. Theory and numerical analysis of the Bona–Smith type systems of Boussinesq equations (to appear). Google Scholar
  5. 5.
    T. B. Benjamin. The stability of solitary waves. Proc. R. Soc. Lond. Ser. A, 328:153–183, 1972. MathSciNetGoogle Scholar
  6. 6.
    J. L. Bona. On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A, 344:363–374, 1975. MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    J. L. Bona and M. Chen. A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D, 116:191–224, 1998. MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. L. Bona, M. Chen, and J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and linear theory. J. Nonlinear Sci., 12:283–318, 2002. MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. L. Bona, M. Chen, and J.-C. Saut. Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: II. The nonlinear theory. Nonlinearity, 17:925–952, 2004. MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. L. Bona, V. A. Dougalis, O. A. Karakashian, and W. R. McKinney. Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. R. Soc. Lond. Ser. A, 351:107–164, 1995. MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. L. Bona, W. R. McKinney, and J. M. Restrepo. Unstable solitary-wave solutions of the generalized regularized long-wave equation. J. Nonlinear Sci., 10:603–638, 2000. MATHMathSciNetGoogle Scholar
  12. 12.
    J. L. Bona and R. L. Sachs. The existence of internal solitary waves in a two-fluid system near the KdV limit. Geophys. Astrophys. Fluid Dyn., 48:25–51, 2000. CrossRefMathSciNetGoogle Scholar
  13. 13.
    J. L. Bona and R. Smith. A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc., 79:167–182, 1976. MATHMathSciNetGoogle Scholar
  14. 14.
    J. L. Bona, P. E. Souganidis, and W. A. Strauss. Stability and instability of solitary waves of KdV type. Proc. R. Soc. Lond. Ser. A, 411:395–412, 1987. MATHMathSciNetGoogle Scholar
  15. 15.
    J. V. Boussinesq. Théorie des ondes et des remous qui se propagent le long d’ un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pure Appl., 17:55–108, 1872. Google Scholar
  16. 16.
    J. V. Boussinesq. Essai sur la théorie des eaux courants. Mém. prés. div. sav. Acad. des Sci. Inst. Fr. (sér. 2), 23:1–680, 1877. Google Scholar
  17. 17.
    M. Chen. Exact traveling-wave solutions to bi-directional wave equations. Int. J. Theor. Phys., 37:1547–1567, 1998. MATHCrossRefGoogle Scholar
  18. 18.
    M. Chen. Solitary-wave and multipulsed traveling-wave solutions of Boussinesq systems. Appl. Anal., 75:213–240, 2000. MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    V. A. Dougalis and D. E. Mitsotakis. Solitary waves of the Bona–Smith system. In D. Fotiadis and C. Massalas, editors, Advances in Scattering Theory and Biomedical Engineering, pages 286–294. World Scientific, River Edge, 2004. Google Scholar
  20. 20.
    A. Durán and M. A. López-Marcos. Conservative numerical methods for solitary wave interactions. J. Phys. A: Math. Gen., 36:7761–7770, 2003. MATHCrossRefGoogle Scholar
  21. 21.
    K. El Dika. Asymptotic stability of solitary waves for the Benjamin–Bona–Mahony equation. Discrete. Contin. Dyn. Syst., 13:583–622, 2005. MATHMathSciNetGoogle Scholar
  22. 22.
    M. Grillakis, J. Shatah, and W. A. Strauss. Stability of solitary waves in the presence of symmetry: I. J. Funct. Anal., 74:170–197, 1987. CrossRefMathSciNetGoogle Scholar
  23. 23.
    T. Kato. Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin, 1980. MATHGoogle Scholar
  24. 24.
    Yi A. Li. Hamiltonian structure and linear stability of solitary waves of the Green–Naghdi equation. J. Nonlinear Math. Phys., 9:99–105, 2002. Suppl. I. CrossRefGoogle Scholar
  25. 25.
    Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rat. Mech. Anal., 157:219–254, 2001. MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    J. R. Miller and M. I. Weinstein. Asymptotic stability of solitary waves for the Regularized Long-Wave equation. Commun. Pure Appl. Math., 49:399–441, 1996. MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    R. L. Pego, P. Smereka, and M. I. Weinstein. Oscillatory instability of solitary waves in a continuum model of lattice vibrations. Nonlinearity, 8:921–941, 1995. MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    R. L. Pego and M. I. Weinstein. Asymptotic stability of solitary waves. Commun. Math. Phys., 164:305–349, 1994. MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    R. L. Pego and M. I. Weinstein. Convective linear stability of solitary waves for Boussinesq equations. Stud. Appl. Math., 99:311–375, 1997. MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    B. Pelloni. Spectral methods for the numerical solution of nonlinear dispersive wave equations. Ph.D. thesis, Yale University, 1996. Google Scholar
  31. 31.
    B. Pelloni and V. A. Dougalis. Numerical modelling of two-way propagation of nonlinear dispersive waves. Math. Comput. Simul., 55:595–606, 2001. MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    B. Pelloni and V. A. Dougalis. Numerical solutions of some nonlocal, nonlinear, dispersive wave equations. J. Nonlin. Sci., 10:1–22, 2000. MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    D. H. Peregrine. Equations for water waves and the approximations behind them. In R. E. Meyer, editor, Waves on Beaches and Resulting Sediment Transport, pages 95–121. Academic Press, New York, 1972. Google Scholar
  34. 34.
    M. Reed and B. Simon. Analysis of Operators IV. Academic Press, New York, 1978. MATHGoogle Scholar
  35. 35.
    P. C. Schuur. Asymptotic Analysis of Soliton Problems: An Inverse Scattering Approach, volume 1232 of Lecture Notes in Mathematics. Springer, Berlin, 1986. MATHGoogle Scholar
  36. 36.
    P. Smereka. A remark on the solitary wave stability for a Boussinesq equation. In L. Debnath, editor, Nonlinear Dispersive Wave Systems, pages 255–263. World Scientific, Singapore, 1992. Google Scholar
  37. 37.
    J. F. Toland. Solitary wave solutions for a model of the two-way propagation of water waves in a channel. Math. Proc. Camb. Philos. Soc., 90:343–360, 1981. MATHMathSciNetGoogle Scholar
  38. 38.
    J. F. Toland. Uniqueness and a priori bounds for certain homoclinic orbits of a Boussinesq system modelling solitary water waves. Commun. Math. Phys., 94:239–254, 1984. MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    J. F. Toland: Existence of symmetric homoclinic orbits for systems of Euler–Lagrange equations. In Proceedings of Symposia in Pure Mathematics, volume 45, Part 2, pages 447–459. Am. Math. Soc., Providence, 1986. Google Scholar
  40. 40.
    M. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math., 39:51–68, 1986. MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    M. Weinstein. Existence and dynamic stability of solitary-wave solutions of equations arising in long wave propagation. Commun. Partial Differ. Eq., 12:1133–1173, 1987. MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    M. I. Weinstein. Asymptotic stability of nonlinear bound states in conservative dispersive systems. Contemp. Math., 200:223–235, 1996. MathSciNetGoogle Scholar
  43. 43.
    G. B. Whitham. Linear and Non-linear Waves. Wiley, New York, 1974. Google Scholar
  44. 44.
    R. Winther. A finite element method for a version of the Boussinesq equations. SIAM J. Numer. Anal., 19:561–570, 1982. MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    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. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • V. A. Dougalis
    • 1
    • 2
  • A. Durán
    • 3
  • M. A. López-Marcos
    • 3
  • D. E. Mitsotakis
    • 1
    • 2
  1. 1.Mathematics DepartmentUniversity of AthensZographouGreece
  2. 2.Institute of Applied and Computational MathematicsF.O.R.T.H.HeraklionGreece
  3. 3.Applied Mathematics Department, Faculty of SciencesUniversity of ValladolidValladolidSpain

Personalised recommendations