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Applied Mathematics and Mechanics (English Edition) ›› 2014, Vol. 35 ›› Issue (1): 117-132.doi: https://doi.org/10.1007/s10483-014-1777-7

• 论文 • 上一篇    

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

李想1，张卫国2，李正明1   

  1. 1. Business School, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China;
    2. College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China
  • 收稿日期:2013-05-20 修回日期:2013-09-11 出版日期:2014-01-20 发布日期:2013-12-27

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term

 LI Xiang1, ZHANG Wei-Guo2, LI Zheng-Meng1   

  1. 1. Business School, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China;
    2. College of Science, University of Shanghai for Science and Technology, Shanghai 200093, P. R. China
  • Received:2013-05-20 Revised:2013-09-11 Online:2014-01-20 Published:2013-12-27

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被引次数

2

摘要： This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing
the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

关键词: nonlinear wave equation, bounded traveling wave solution, shape analysis, approximate damped oscillatory solution, error estimate, null

Abstract: This paper aims at analyzing the shapes of the bounded traveling wave solutions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and conditions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approximate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing
the integral equation, which reflects the relation between the exact and approximate solutions. It can be seen that the error is infinitesimal decreasing in the exponential form.

Key words: nonlinear wave equation, bounded traveling wave solution, shape analysis, approximate damped oscillatory solution, error estimate, null

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