# American Institute of Mathematical Sciences

March  2013, 12(2): 1075-1090. doi: 10.3934/cpaa.2013.12.1075

## Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution

 1 School of Science, University of Shanghai for Science and Technology, Shanghai 200093, China, China, China

Received  August 2011 Revised  November 2012 Published  September 2012

In this paper, we apply the theory of planar dynamical systems to make a qualitative analysis to the traveling wave solutions of nonlinear Kakutani-Kawahara equation $u_t+uu_x+bu_{x x x}-a(u_t+uu_x)_x=0$ ($b>0, a\ge0$) and obtain the existent conditions of the bounded traveling wave solutions. In dispersion-dominant case, we find that the unique bounded traveling wave solution of this equation has not only oscillatory property but also damped property. Furthermore, according to the evolution of orbits in the global phase portraits, we present an approximate damped oscillatory solution for this equation by the undetermined coefficients method. Finally, by the idea of homogenization principles, we obtain an integral equation which reflects the relation between this approximate damped oscillatory solution and its exact solution, thereby having the error estimate. The error is an infinitesimal decreasing in exponential form. From the results in this paper, it can be seen that the damped oscillatory solution of Kakutani-Kawahara equation in dispersion-dominant case still remains some properties of solitary wave solution for KdV equation.
Citation: Weiguo Zhang, Yan Zhao, Xiang Li. Qualitative analysis to the traveling wave solutions of Kakutani-Kawahara equation and its approximate damped oscillatory solution. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1075-1090. doi: 10.3934/cpaa.2013.12.1075
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