Volume 4, Issue 5
Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term

Xiang Li, Weiguo Zhang & Yan Zhao

Adv. Appl. Math. Mech., 4 (2012), pp. 559-574.

Published online: 2012-04

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  • 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

Nonlinear wave equation planar dynamical system exact solutions approximate damped oscillatory solutions error estimate

  • AMS Subject Headings

35Q51 37C29

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COPYRIGHT: © Global Science Press

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@Article{AAMM-4-559, author = {Xiang Li, Weiguo Zhang and Yan Zhao}, title = {Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {5}, pages = {559--574}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m11181}, url = {http://global-sci.org/intro/article_detail/aamm/136.html} }
TY - JOUR T1 - Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term AU - Xiang Li, Weiguo Zhang & Yan Zhao JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 559 EP - 574 PY - 2012 DA - 2012/04 SN - 4 DO - http://dor.org/10.4208/aamm.10-m11181 UR - https://global-sci.org/intro/aamm/136.html KW - Nonlinear wave equation KW - planar dynamical system KW - exact solutions KW - approximate damped oscillatory solutions KW - error estimate AB -

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.

Xiang Li, Weiguo Zhang & Yan Zhao. (1970). Shape Analysis and Solution to a Class of Nonlinear Wave Equation with Cubic Term. Advances in Applied Mathematics and Mechanics. 4 (5). 559-574. doi:10.4208/aamm.10-m11181
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