Structural Stability for a Class of Nonlinear Wave Equations

Yıl 2009, Cilt: 22 Sayı: 2, 83 - 87, 22.03.2010

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Öz

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

 

u_t-u_txx+auu_x=bu_xu_xx+uu_xxx                                                                    (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

 

 

Anahtar Kelimeler

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Kaynakça

• 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).