Qualitative analysis to the traveling wave solutions ofKakutani-Kawahara equation and its approximate damped oscillatorysolution
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.
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