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The Stability of Solitary Waves of Depression

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Analysis and Mathematical Physics

Part of the book series: Trends in Mathematics ((TM))

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Abstract

We provide a sufficient condition for the orbital stability of negative solitary-wave solutions of the regularized long-wave equation. In particular, it is found that solitary waves with speed \( c < - \tfrac{1} {6} \) are orbitally stable.

This work was supported in part by the Research Council of Norway and the ESF Research Networking Programme HCAA.

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References

  1. J.P. Albert, Concentration compactness and the stability of solitary-wave solutions to nonlocal equations. Applied analysis (Baton Rouge, LA, 1996), 1–29, Contemp. Math. 221, Amer. Math. Soc, Providence, RI, 1999.

    Google Scholar 

  2. J.P. Albert and J.L. Bona Total positivity and the stability of internal waves in stratified fluids of finite depth. The Brooke Benjamin special issue (University Park, PA, 1989). IMA J. Appl. Math. 46 (1991), 1–19.

    Google Scholar 

  3. J.P. Albert and J.L. Bona Comparisons between model equations for long waves. J. Nonlinear Sci. 1 (1991), 345–374.

    Article  MATH  MathSciNet  Google Scholar 

  4. J.P. Albert and J.L. Bona and D.B. Henry, Sufficient conditions for stability of solitary-wave solutions of model equations for long waves. Phys. D 24 (1987), 343–366.

    Article  MATH  MathSciNet  Google Scholar 

  5. T.B. Benjamin, The stability of solitary waves. Proc. Roy. Soc. London A 328 (1972), 153–183.

    Article  MathSciNet  Google Scholar 

  6. T.B. Benjamin, J.B. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. Roy. Soc. London A 272 (1972), 47–78.

    Article  MATH  MathSciNet  Google Scholar 

  7. J.L. Bona On the stability theory of solitary waves. Proc. Roy. Soc. London A 344 (1975), 363–374.

    Article  MATH  MathSciNet  Google Scholar 

  8. J.L. Bona P.E. Souganidis and W.A. Strauss, Stability and instability of solitary waves of Korteweg-de Vries type. Proc. Roy. Soc. London A 411 (1987), 395–412.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulalre horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17 (1872), 55–108.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. I.D. Iliev, E.Kh. Khristov and K.P. Kirchev, Spectral methods in soliton equations. Pitman Monographs and Surveys in Pure and Applied Mathematics 73. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994.

    Google Scholar 

  12. H. Kalisch, Solitary Waves of Depression. J. Computational Analysis and Application 8 (2006), 5–24.

    MATH  MathSciNet  Google Scholar 

  13. N. Nguyen and H. Kalisch, Orbital Stability of Negative Solitary Waves, to appear in Math. Comput. Simulation.

    Google Scholar 

  14. P.G. Peregrine, Calculations of the development of an undular bore. J. Fluid Mech. 25 (1966), 321–330.

    Article  Google Scholar 

  15. J. Shatah and W. Strauss, Instability of nonlinear bound states. Commun. Math. Phys. 100 (1985), 173–190.

    Article  MATH  MathSciNet  Google Scholar 

  16. P.E. Souganidis and W.A. Strauss, Instability of a class of dispersive solitary waves. Proc. Roy. Soc. Edinburgh 114A (1990), 195–212.

    MathSciNet  Google Scholar 

  17. G.B. Whitham, Linear and Nonlinear Waves. Wiley, New York, 1974.

    MATH  Google Scholar 

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Nguyen, N.T., Kalisch, H. (2009). The Stability of Solitary Waves of Depression. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_22

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