数学物理学报(英文版) ›› 2014, Vol. 34 ›› Issue (3): 941-959.doi: 10.1016/S0252-9602(14)60061-8
张卫国*, 刘强, 李正明, 李想
ZHANG Wei-Guo*, LIU Qiang, LI Zheng-Ming, LI Xiang
摘要:
This article studies bounded traveling wave solutions of variant Boussinesq equa-tion with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and gen-eral shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r* which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r| ≥ r*; while they appear as damped oscillatory waves if |r| < r*. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory so-lutions according to the orbits evolution relation corresponding to the component u(ξ) in the global phase portraits, and then obtain the approximate solutions (u(ξ), H(ξ)). Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.
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