×

Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. (English) Zbl 0654.35018

Summary: Certain generalizations of one of the classical Boussinesq-type equations, \[ u_{tt}=u_{xx}-(u\quad 2+u_{xx})_{xx},\quad (*) \] 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.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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) · Zbl 0634.35079
[2] Benjamin, T. B.: The stability of solitary waves. Proc. Roy. Soc. Lond.A328, 153-183 (1972)
[3] 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) · Zbl 0574.76028
[4] Berryman, J.: Stability of solitary waves in shallow water. Phys. Fluids19, 771-777 (1976) · Zbl 0351.76023
[5] Bona, J. L.: On the stability theory of solitary waves. Proc. Roy. Soc. Lond.A344, 363-374 (1975) · Zbl 0328.76016
[6] 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) · Zbl 0332.76007
[7] Bona, J. L., Souganidis, P., Strauss, W.: Stability and instability of solitary waves of KdV type. Proc. Roy. Soc. Lond.A411, 395-412 (1987) · Zbl 0648.76005
[8] Boussinesq, J.: Théorie des ondes et de remous qui se propagent....J. Math. Pures Appl., Sect. 2,17, 55-108 (1872)
[9] Boussinesq, J.: Essai sur la théorie des eaux courantes. Mem. prés. div. Sav. Acad. Sci. Inst. Fr.23, 1-680 (1877)
[10] Courant, R., Hilbert, D.: Methods of mathematical physics Vol. 1. New York: Interscience 1953 · Zbl 0051.28802
[11] Deift, P., Tomei, C., Trubowitz, E.: Inverse scattering and the Boussinesq equation. Commun. Pure Appl. Math.35, 567-628 (1982) · Zbl 0489.35073
[12] 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). · Zbl 0656.35122
[13] 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) · Zbl 0388.35039
[14] 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
[15] 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) · Zbl 0549.34001
[16] Lions, J.-L., Magenes, E.: Problèmes aux limites non homogènes et applications, Vol. I. Paris: Dunod 1968 · Zbl 0165.10801
[17] Weinstein, M.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun. Pure Appl. Math.39, 51-68 (1986) · Zbl 0594.35005
[18] Zakharov, V. E.: On the stochastization of one-dimensional chains of nonlinear oscillators. Sov. Phys. JETP38, 108-110 (1974)
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.