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Instability of a class of dispersive solitary waves*

  • P. E. Souganidis (a1) and W. A. Strauss (a1)
Abstract
Synopsis

This paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.

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1J. Albert . Dispersion of low-energy waves for the generalized Benjamin–Bona–Mahony equation. J. Differential Equations 63 (1986), 117134.

3.T. Benjamin , J. Bona and J. Mahony . Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London Ser. A 272 (1972), 47.

4J. Bona , P. Souganidis and W. Strauss . Stability and instability of solitary waves of Korteweg-de Vries type. Proc. Roy. Soc. London Ser. A 411 (1987), 395412.

5M. Grillakis , J. Shatah and W. Strauss . Stability theory of solitary waves in the presence symmetry, I. J. Fund. Anal. 74 (1987), 160197.

7J. Shatah and W. Strauss . Instability of nonlinear bound states. Comm.Math. Phys. 100 (1985), 173190.

8M. Weinstein . Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation. Comm. Partial Differential Equations 12 (1987), 11331173.

9K. Yosida . Functional Analysis (Berlin: Springer, 1965)

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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