Instability of solitary waves for generalized Boussinesq equations. (English) Zbl 0784.34048

A nonlinear equation of Boussinesq type is considered. It is investigated the stability of the solitary wave solutions proving that a traveling wave may be stable or unstable, depending on the range of the wave’s speed of propagation and on the nonlinearity.


34G20 Nonlinear differential equations in abstract spaces
35Q35 PDEs in connection with fluid mechanics
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35L75 Higher-order nonlinear hyperbolic equations
Full Text: DOI


[1] Albert, J. (1986). Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation.J. Diff. Eq. 63, 117-134. · Zbl 0596.35109 · doi:10.1016/0022-0396(86)90057-4
[2] Berestycki, H., and Lions, P. (1983). Nonlinear scalar field equation I.Arch. Rat. Mech. Anal. 82, 313-346. · Zbl 0533.35029
[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. · Zbl 0654.35018 · doi:10.1007/BF01218475
[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. · Zbl 0648.76005 · doi:10.1098/rspa.1987.0073
[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. · Zbl 0332.76007 · doi:10.1017/S030500410005218X
[6] Boussinesq, J. (1872). Theorie des ondes et de remous qui se propagent...J. Math. Pure Appl. Sect. 2 17, 55-108.
[7] Deift, P., Tomei, C., and Trubowitz, E. (1982). Inverse scattering and the Boussinesqequation.Comm. Pure Appl. Math. 35, 567-628. · doi:10.1002/cpa.3160350502
[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. · Zbl 0656.35122 · doi:10.1016/0022-1236(87)90044-9
[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. · Zbl 0388.35039 · doi:10.1007/BF01109723
[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.
[11] Pazy, A. (1983).Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York. · Zbl 0516.47023
[12] Pego, R., and Weinstein, M. A class of eigenvalue problems with application to instability of solitary waves, Preprint. · Zbl 0872.76042
[13] Reed, M., and Simon, B. (1978).Methods of Modern Mathematical Physics, Vols. I, II, III, IV, Academic Press, New York. · Zbl 0401.47001
[14] Sachs, R., oral communication.
[15] Shatah, J., and Strauss, W. (1985). Instability of nonlinear bound states.Comm. Math. Phys. 100, 173-190. · Zbl 0603.35007 · doi:10.1007/BF01212446
[16] Souganidis, P., and Strauss, W. (1990). Instability of a class of dispersive solitary waves.Proc. Roy. Soc. Edinburgh 114A, 195-212. · Zbl 0713.35108
[17] Strauss, W. (1989).Nonlinear Wave Equations, A.M.S.
[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. · Zbl 0657.73040 · doi:10.1080/03605308708820522
[19] Weinstein, M. (1986). Lyapunov stability of ground states of nonlinear dispersive evolution equations.Comm. Pure Appl. Math. 39, 51-68. · Zbl 0594.35005 · doi:10.1002/cpa.3160390103
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.