Abstract
We study ground-state traveling wave solutions of a fourth-order wave equation. We find conditions on the speed of the waves which imply stability and instability of the solitary waves. The analysis depends on the variational characterization of the ground states rather than information about the linearized operator.
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REFERENCES
Berestycki, H., and Lions, P. L. (1983). Nonlinear scalar field equations, I. Existence of a ground state. Arch. Rat. Mech Anal. 82, 313–375.
Bona, J. L., Souganidis, P. E., and Strauss, W. A. (1987). Stability and instability of solitary waves of Korteweg-de Vries type. Proc. Roy. Soc. London A 411, 395–412.
Cazenave, T., and Lions, P. L. (1982). Orbital stability of standing waves for some nonlinear Schrodinger equations. Commun. Math. Phys. 85, 549–561.
Champneys, A. R. and Groves, M. D. (1996). A global investigation of solitary-wave solutions to a two-parameter model for water waves (preprint).
Evans, L. C. (1990). Weak Convergence Methods for Nonlinear Partial Differential Equations. CBMS Regional Conference Series, American Mathematics Society, pp. 34–37.
Grillakis, M., Shatah, J., and Strauss, W. (1987). Stability theory of solitary waves in the presence of symmetry, I. J. Funct. Anal. 74, 160–197.
Grillakis, M., Shatah, J., and Strauss, W. (1990). Stability theory of solitary waves in the presence of symmetry, II. J. Funct. Anal. 94, 308–348.
Levandosky, S., (1997). Stability of solitary-waves for fifth-order KdV equations (in preparation).
Lions, P. L. (1984). The concentration-compactness principle in the calculus of variations. The locally compact case. Part I and Part II. Ann. Inst. Henri Poincaré Sect. A (N.S.) 1, 109–145, 223–283.
Lions, P. L. (1985). The concentration-compactness principle in the calculus of variations. The limit case, Part I and Part II. Revista Mat. Iberoam, 1 (1, 2), 145–201, 45–121.
Liu, Yue (1993). Instability of solitary waves for generalized Boussinesq equations. J. Dynam. Diff. Eqs. 5, 537–558.
McKenna, P. J., and Walter, W. (1990). Traveling waves in a susp. bridge. SIAM J. Appl. Math. 50, 703–715.
McKenna, P. J. (1995). Oral communication.
Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.
Pego, R., and Weinstein, W. (1992). Eigenvalues and instability of solitary waves. Phil. Trans. Roy. Soc. Lond. A 340, 47–94.
Segal, I. (1963). Non-linear semi-groups. Ann. Math. 78, 339–364.
Shatah, J. (1983). Stable Standing waves of nonlinear Klein Gordon equations. Commun. Math. Phys. 91, 313–327.
Shatah, J. (1985). Unstable ground states of nonlinear Klein Gordon equations. Trans. AMS 290, 701–710.
Shatah, J., and Strauss, W. (1985). Instability of nonlinear bound states. Commun. Math. Phys. 100, 173–190.
Souganidis, P. E., and Strauss, W. A. (1990). Instability of a class of dispersive solitary waves. Proc. Roy. Soc. Edinburgh 114A, 195–212.
Strauss, W. (1977). Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162.
Weinstein, M. (1986). Lyapunov stability of ground states of nonlinear dispersive wave equations. CPAM 39, 51–68.
Weinstein, M. (1985). Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16, 472–491.
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Levandosky, S. Stability and Instability of Fourth-Order Solitary Waves. Journal of Dynamics and Differential Equations 10, 151–188 (1998). https://doi.org/10.1023/A:1022644629950
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DOI: https://doi.org/10.1023/A:1022644629950