Instability of solitary waves for generalized Boussinesq equations



An equation of Boussinesq-type of the formu tt -u xx +(f(u)+uxx)xx=0 is considered. It is shown that a traveling wave may be stable or unstable, depending on the range of the wave's speed of propagation and on the nonlinearity. Sharp conditions to that effect are given.

Key words

Boussinesq equation solitary wave stability theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, J. (1986). Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation.J. Diff. Eq. 63, 117–134.Google Scholar
  2. 2.
    Berestycki, H., and Lions, P. (1983). Nonlinear scalar field equation I.Arch. Rat. Mech. Anal. 82, 313–346.Google Scholar
  3. 3.
    Bona, J., and Sachs, R. (1988). Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation.Comm. Math. Phys. 118, 15–29.Google Scholar
  4. 4.
    Bona, J., Souganidis, P., and Strauss, W. (1987). Stability and instability of solitary waves of Korteweg-de Vries type.Proc. Roy. Soc. London Ser. A 411, 395–412.Google Scholar
  5. 5.
    Bona, J., and Smith, R. (1976). A model for the two-way propagation of water waves in a channel.Math. Proc. Camb. Phil. Soc. 79, 167–182.Google Scholar
  6. 6.
    Boussinesq, J. (1872). Theorie des ondes et de remous qui se propagent...J. Math. Pure Appl. Sect. 2 17, 55–108.Google Scholar
  7. 7.
    Deift, P., Tomei, C., and Trubowitz, E. (1982). Inverse scattering and the Boussinesqequation.Comm. Pure Appl. Math. 35, 567–628.Google Scholar
  8. 8.
    Grillakis, M., Shatah, J., and Strauss, W. (1987, 1990). Stability theory of solitary waves in the presence of symmetry I and II.J. Fund. Anal. 74, 160–197; 94, 308–348.Google Scholar
  9. 9.
    Kalantarov, V., and Ladyzhenskaya, O. (1978). The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types.J. Sov. Math. 10, 53–70.Google Scholar
  10. 10.
    Kato, T. (1974). Quasilinear equations of evolution, with applications to partial differential equations,Lecture Notes in Mathematics, Vol. 448, Springer, Berlin, Heidelberg, New York, pp. 25–70.Google Scholar
  11. 11.
    Pazy, A. (1983).Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.Google Scholar
  12. 12.
    Pego, R., and Weinstein, M. A class of eigenvalue problems with application to instability of solitary waves, Preprint.Google Scholar
  13. 13.
    Reed, M., and Simon, B. (1978).Methods of Modern Mathematical Physics, Vols. I, II, III, IV, Academic Press, New York.Google Scholar
  14. 14.
    Sachs, R., oral communication.Google Scholar
  15. 15.
    Shatah, J., and Strauss, W. (1985). Instability of nonlinear bound states.Comm. Math. Phys. 100, 173–190.Google Scholar
  16. 16.
    Souganidis, P., and Strauss, W. (1990). Instability of a class of dispersive solitary waves.Proc. Roy. Soc. Edinburgh 114A, 195–212.Google Scholar
  17. 17.
    Strauss, W. (1989).Nonlinear Wave Equations, A.M.S.Google Scholar
  18. 18.
    Weinstein, M. (1987). Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation.Comm. Part. Diff. Eq. 12, 1133–1177.Google Scholar
  19. 19.
    Weinstein, M. (1986). Lyapunov stability of ground states of nonlinear dispersive evolution equations.Comm. Pure Appl. Math. 39, 51–68.Google Scholar
  20. 20.
    Falk, F., Laedke, E. W., and Spatschek, K. H. (1987). Stability of solitary wave pulses in shape-memory alloys.Phys. Rev. B36, 3031–3041.Google Scholar

Copyright information

© Plenum Publishing Corporation 1993

Authors and Affiliations

  • Yue Liu
    • 1
  1. 1.Departement of MathematicsBrown UniversityProvidence

Personalised recommendations