Abstract
Certain generalizations of one of the classical Boussinesq-type equations,
are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow-up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz.
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References
Albert, J., Bona, J. L., Henry, D.: Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Physica24D, 343–366 (1987)
Benjamin, T. B.: The stability of solitary waves. Proc. Roy. Soc. Lond.A328, 153–183 (1972)
Bennett, D. P., Bona, J. L., Brown, S. E., Stansfield, D. D., Stroughair. J. D.: The stability of internal solitary waves. Math. Proc. Camb. Phil. Soc.94, 351–379 (1983)
Berryman, J.: Stability of solitary waves in shallow water. Phys. Fluids19, 771–777 (1976)
Bona, J. L.: On the stability theory of solitary waves. Proc. Roy. Soc. Lond.A344, 363–374 (1975)
Bona, J. L., Smith, R.: A model for the two-way propagation of water waves in a channel. Math. Proc. Camb. Phil. Soc.79, 167–182 (1976)
Bona, J. L., Souganidis, P., Strauss, W.: Stability and instability of solitary waves of KdV type. Proc. Roy. Soc. Lond.A411, 395–412 (1987)
Boussinesq, J.: Théorie des ondes et de remous qui se propagent....J. Math. Pures Appl., Sect. 2,17, 55–108 (1872)
Boussinesq, J.: Essai sur la théorie des eaux courantes. Mem. prés. div. Sav. Acad. Sci. Inst. Fr.23, 1–680 (1877)
Courant, R., Hilbert, D.: Methods of mathematical physics Vol. 1. New York: Interscience 1953
Deift, P., Tomei, C., Trubowitz, E.: Inverse scattering and the Boussinesq equation. Commun. Pure Appl. Math.35, 567–628 (1982)
Grillakis, J., Shatah, J., Strauss, W. A.: Stability theory of solitary waves in the presence of symmetry I. J. Funct. Anal.74, 160–197 (1987).
Kalantarov, V.K., Ladyzhenskaya, O.A.: The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types. J. Sov. Math.10, 53–70 (1978)
Kato, T.: Quasilinear equations of evolution, with applications to partial differential equations. Lecture Notes in Mathematics Vol.448, pp. 25–70. Berlin, Heidelberg, New York: Springer 1974
Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de-Vries equation. Studies in Applied Mathematics. Ad. Math. Suppl. Stud.8, 93–128 (1983)
Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, Vol. I. Paris: Dunod 1968
Weinstein, M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math.39, 51–68 (1986)
Zakharov, V. E.: On the stochastization of one-dimensional chains of nonlinear oscillators. Sov. Phys. JETP38, 108–110 (1974)
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Communicated by C. H. Taubes
Work partially supported by the National Science Foundation
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Bona, J.L., Sachs, R.L. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun.Math. Phys. 118, 15–29 (1988). https://doi.org/10.1007/BF01218475
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DOI: https://doi.org/10.1007/BF01218475