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Structural Stability for a Class of Nonlinear Wave Equations

Year 2009, Volume: 22 Issue: 2, 83 - 87, 22.03.2010

Abstract

In this paper we discuss the structural stability of an initial value problem defined for the equation

 

ut-utxx+auux=buxuxx+uuxxx                                                                    (i.1)

 where  a, b are constants, x Ğ„ ℝ , t Ğ„ ℝ.  For the choices of a and b , (i.1) describe the nonlinear shallow water waves. Upper and lower bounds are derived for energy decay rate in every finite interval [0,T] which reveals that only the lower bound of the energy decays exponentially.

 

Key Words: Degasperis-Procesi equation, Camassa-Holm equation, traveling wave

 

 

References

  • Degasperis, A., Procesi, integrability ”, Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, 23-37 (1999).
  • Camassa, R., Holm, D., “An integrable shallow water equation with peaked solution”, Phys. Rev. Lett., 71: 1661-1664 (1993).
  • Lenells, J. S, “Traveling wave solutions of the Degasperis-Procesi equation”, J. Math. Anal. Appl., 306: 72-82 (2005).
  • Lenell, J.S, “Traveling wave solutions of the equation”, Camassa-Holm Equations, 217: 393-430 (2005). J. Differential
  • Degasperis, A., Holm, D., Hone, A.N.W., “A new integrable equation with peakon solutions”, Theoretical and Mathematical Physics, 133 (2): 1474 (2002).
Year 2009, Volume: 22 Issue: 2, 83 - 87, 22.03.2010

Abstract

References

  • Degasperis, A., Procesi, integrability ”, Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, 23-37 (1999).
  • Camassa, R., Holm, D., “An integrable shallow water equation with peaked solution”, Phys. Rev. Lett., 71: 1661-1664 (1993).
  • Lenells, J. S, “Traveling wave solutions of the Degasperis-Procesi equation”, J. Math. Anal. Appl., 306: 72-82 (2005).
  • Lenell, J.S, “Traveling wave solutions of the equation”, Camassa-Holm Equations, 217: 393-430 (2005). J. Differential
  • Degasperis, A., Holm, D., Hone, A.N.W., “A new integrable equation with peakon solutions”, Theoretical and Mathematical Physics, 133 (2): 1474 (2002).
There are 5 citations in total.

Details

Primary Language English
Journal Section Mathematics
Authors

Ülkü Dinlemez

Publication Date March 22, 2010
Published in Issue Year 2009 Volume: 22 Issue: 2

Cite

APA Dinlemez, Ü. (2010). Structural Stability for a Class of Nonlinear Wave Equations. Gazi University Journal of Science, 22(2), 83-87.
AMA Dinlemez Ü. Structural Stability for a Class of Nonlinear Wave Equations. Gazi University Journal of Science. March 2010;22(2):83-87.
Chicago Dinlemez, Ülkü. “Structural Stability for a Class of Nonlinear Wave Equations”. Gazi University Journal of Science 22, no. 2 (March 2010): 83-87.
EndNote Dinlemez Ü (March 1, 2010) Structural Stability for a Class of Nonlinear Wave Equations. Gazi University Journal of Science 22 2 83–87.
IEEE Ü. Dinlemez, “Structural Stability for a Class of Nonlinear Wave Equations”, Gazi University Journal of Science, vol. 22, no. 2, pp. 83–87, 2010.
ISNAD Dinlemez, Ülkü. “Structural Stability for a Class of Nonlinear Wave Equations”. Gazi University Journal of Science 22/2 (March 2010), 83-87.
JAMA Dinlemez Ü. Structural Stability for a Class of Nonlinear Wave Equations. Gazi University Journal of Science. 2010;22:83–87.
MLA Dinlemez, Ülkü. “Structural Stability for a Class of Nonlinear Wave Equations”. Gazi University Journal of Science, vol. 22, no. 2, 2010, pp. 83-87.
Vancouver Dinlemez Ü. Structural Stability for a Class of Nonlinear Wave Equations. Gazi University Journal of Science. 2010;22(2):83-7.