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

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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.

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References

  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.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. C. Antonopoulos. The Boussinesq system of equations: Theory and numerical analysis. Ph.D. thesis. University of Athens, 2000 (in Greek).

  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.

  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).

  5. T. B. Benjamin. The stability of solitary waves. Proc. R. Soc. Lond. Ser. A, 328:153–183, 1972.

    MathSciNet  Google Scholar 

  6. J. L. Bona. On the stability theory of solitary waves. Proc. R. Soc. Lond. Ser. A, 344:363–374, 1975.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. L. Bona and M. Chen. A Boussinesq system for two-way propagation of nonlinear dispersive waves. Physica D, 116:191–224, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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.

    MATH  MathSciNet  Google Scholar 

  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. 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. M. Chen. Exact traveling-wave solutions to bi-directional wave equations. Int. J. Theor. Phys., 37:1547–1567, 1998.

    Article  MATH  Google Scholar 

  18. M. Chen. Solitary-wave and multipulsed traveling-wave solutions of Boussinesq systems. Appl. Anal., 75:213–240, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  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. A. Durán and M. A. López-Marcos. Conservative numerical methods for solitary wave interactions. J. Phys. A: Math. Gen., 36:7761–7770, 2003.

    Article  MATH  Google Scholar 

  21. K. El Dika. Asymptotic stability of solitary waves for the Benjamin–Bona–Mahony equation. Discrete. Contin. Dyn. Syst., 13:583–622, 2005.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  23. T. Kato. Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin, 1980.

    MATH  Google Scholar 

  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.

    Article  Google Scholar 

  25. Y. Martel and F. Merle. Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Rat. Mech. Anal., 157:219–254, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  28. R. L. Pego and M. I. Weinstein. Asymptotic stability of solitary waves. Commun. Math. Phys., 164:305–349, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. L. Pego and M. I. Weinstein. Convective linear stability of solitary waves for Boussinesq equations. Stud. Appl. Math., 99:311–375, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  30. B. Pelloni. Spectral methods for the numerical solution of nonlinear dispersive wave equations. Ph.D. thesis, Yale University, 1996.

  31. B. Pelloni and V. A. Dougalis. Numerical modelling of two-way propagation of nonlinear dispersive waves. Math. Comput. Simul., 55:595–606, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  32. B. Pelloni and V. A. Dougalis. Numerical solutions of some nonlocal, nonlinear, dispersive wave equations. J. Nonlin. Sci., 10:1–22, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  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. M. Reed and B. Simon. Analysis of Operators IV. Academic Press, New York, 1978.

    MATH  Google Scholar 

  35. P. C. Schuur. Asymptotic Analysis of Soliton Problems: An Inverse Scattering Approach, volume 1232 of Lecture Notes in Mathematics. Springer, Berlin, 1986.

    MATH  Google Scholar 

  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. 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.

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. M. Weinstein. Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math., 39:51–68, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  42. M. I. Weinstein. Asymptotic stability of nonlinear bound states in conservative dispersive systems. Contemp. Math., 200:223–235, 1996.

    MathSciNet  Google Scholar 

  43. G. B. Whitham. Linear and Non-linear Waves. Wiley, New York, 1974.

    Google Scholar 

  44. R. Winther. A finite element method for a version of the Boussinesq equations. SIAM J. Numer. Anal., 19:561–570, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

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Correspondence to V. A. Dougalis.

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Communicated by J. Bona

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Dougalis, V.A., Durán, A., López-Marcos, M.A. et al. A Numerical Study of the Stability of Solitary Waves of the Bona–Smith Family of Boussinesq Systems. J Nonlinear Sci 17, 569–607 (2007). https://doi.org/10.1007/s00332-007-9004-8

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