Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T17:22:01.248Z Has data issue: false hasContentIssue false
  • English
  • Français

Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

Published online by Cambridge University Press:  03 June 2015

Xiang Li ,
Weiguo Zhang  and
Yan Zhao
Xiang Li*
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Weiguo Zhang*
Affiliation:
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Yan Zhao*
Affiliation:
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China
*
Email: [email protected]
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

In this paper, we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems. Two critical values which can characterize the scale of dissipation effect are obtained. If dissipation effect is not less than a certain critical value, the traveling wave solutions appear as kink profile; while if it is less than this critical value, they appear as damped oscillatory. All expressions of bounded traveling wave solutions are presented, including exact expressions of bell and kink profile solitary wave solutions, as well as approximate expressions of damped oscillatory solutions. For approximate damped oscillatory solution, using homogenization principle, we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions. It can be seen that the error is an infinitesimal decreasing in the exponential form.

Keywords

35Q5137C29Nonlinear wave equationplanar dynamical systemexact solutionsapproximate damped oscillatory solutionserror estimate
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 559 - 574
Copyright
Copyright © Global-Science Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Wazwaz, A. M., Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations, Chaos. Soliton. Fract., 13 (2008), pp. 1005–1013.Google Scholar
[2]Jang, B., New exact travelling wave solutions of nonlinear Klein-Gordon equations, Chaos. Soliton. Fract., 41 (2009), pp. 646–654.Google Scholar
[3]Sirendaoreji, , Exact travelling wave solutions for four forms of nonlinear Klein-Gordon equations, Phys. Lett. A., 363 (2007), pp. 440–447.Google Scholar
[4]Ye, C. and Zhang, W. G., New explicit solutions for the Klein-Gordon equation with cubic nonlinearity, Appl. Math. Comput., 217 (2010), pp. 716–724.Google Scholar
[5]Farlow, S. J., Partial Differential Equations for Scientists and Engineers, Wiley Interscience, New York, 1982.Google Scholar
[6]Fan, E. G. and Zhang, H. Q., The solitary wave solution for a class of nonlinear wave equations, Acta. Phys. Sinica., 46 (1997), pp. 1254–1258.Google Scholar
[7]Shang, Y. D., Explicit and exact solutions for a class of nonlinear wave equations, Acta. Math. Appl. Sinica., 23(1) (2000), pp. 21–23.Google Scholar
[8]Zhang, W. G., Chang, Q. S. and Fan, E. G., Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order, J. Math. Appl., 12 (1992), pp. 325–331.Google Scholar
[9]Ma, W. X. and Fuchssteiner, B., Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation, Int. J. Nonlinear. Mech., 31(3) (1996), pp. 329–338.CrossRefGoogle Scholar
[10]Cohen, H., Nolinear Diffusion Problems in Studies in Applied Mathematics, The Mathematical Association of America, 1971.Google Scholar
[11]Zhang, Z. F., Ding, T. R., Huang, W. Z. and Dong, Z. X., Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, Volume 101, American Mathematical Society, Providence, 1992.Google Scholar
[12]Nemytskii, V. V. and Stepanov, V. V., Qualitative Theory of Differential Equations, Dover publications, New York, 1989.Google Scholar
[13]Ye, Q. X. and Li, Z. Y., Introduction of Reaction Diffusion Equations, Science Press, Beijing, 1990.Google Scholar
[14]Zhang, W. G., Chang, Q. S. and Jiang, B. G., Explicit exact solitary-wave solutions for compound KdV-type and compound KdV Burgers-type equations with nonlinear term of any order, Chaos. Soliton. Fract., 13 (2002), pp. 311–319.Google Scholar