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Nonlinear Stability of a One-Dimensional Boussinesq Equation

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Abstract

We study nonlinear orbital stability and instability of the set of ground state solitary wave solutions of a one-dimensional Boussinesq equation or one-dimensional Benney–Luke equation. It is shown that a solitary wave (traveling wave with finite energy) may be orbitally stable or unstable depending on the range of the wave's speed of propagation.

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References

  • Alexander, J., and Sachs, R. Linear stability of solitary waves for Boussinesq-type equation. A Computer Assisted Computation, preprint.

  • Benney, D. J., and Luke, J. C. (1964). Interactions of permanent waves of finite amplitude. J. Math. Phys. 43, 309–313.

    Google Scholar 

  • Bona, J., and Sachs, R. (1988). Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm. Math. Phys. 188, 15–29.

    Google Scholar 

  • Liu, Y. (1993). Instability of solitary waves for generalized Boussinesq equations. J. Dynam. Differential Equations 5, 537–558.

    Google Scholar 

  • Liu, Y. (1995). Instability and blow-up of solutions to a generalized Boussinesq equation. SIAM J. Math. Anal. 26, 1527–1546.

    Google Scholar 

  • Grillakis, M., Shatah, J., and Strauss, W. (1987). Stability theory of solitary waves in presence of symmetry, I. Funct. Anal. 74, 160–197.

    Google Scholar 

  • Milewski, P. A., and Keller, J. B. (1996). Three dimensional water waves. Stud. Appl. Math. 37, 149–166.

    Google Scholar 

  • Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.

    Google Scholar 

  • Pego, R. L., and Quintero, J. L. (1999). Two-dimensional solitary Waves for a Benney–Luke equation. Physica D 132, 476–496.

    Google Scholar 

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

    Google Scholar 

  • Pego, R. L., Smereka, P., and Weinstein, M. (1995). Oscillatory instability of solitary waves in a continuum model of lattice vibrations. Nonlinearity 8, 921–941.

    Google Scholar 

  • Smereka, P. (1992). A remark on the solitary waves stability for a Boussinesq equation. In Debnath, L. (ed), In Nonlinear Dispersive Wave Systems, World Scientific, Singapore, pp. 255–263.

    Google Scholar 

  • Struwe, W. (1990). Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, New York.

    Google Scholar 

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  1. Departamento Matemáticas, Universidad del Valle, A.A, 25360, Cali-Colombia, South America

    José Rául Quintero

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  1. José Rául Quintero
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Quintero, J.R. Nonlinear Stability of a One-Dimensional Boussinesq Equation. Journal of Dynamics and Differential Equations 15, 125–142 (2003). https://doi.org/10.1023/A:1026109529292

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  • DOI: https://doi.org/10.1023/A:1026109529292

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