Be careful with variable separation solutions via the extended tanh-function method and periodic wave structures
Articles
Chaoqing Dai
Zhejiang A & F University, China
Qin Liu
Zhejiang Lishui University, China
Published 2015-10-27
https://doi.org/10.15388/NA.2015.4.2
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Keywords

extended tanh-function method
uniform variable separation solutions
general (2 1)-dimensional Korteweg–de Vries system
periodic localized coherent structures

How to Cite

Dai C. and Liu Q. (2015) “Be careful with variable separation solutions via the extended tanh-function method and periodic wave structures”, Nonlinear Analysis: Modelling and Control, 20(4), pp. 487-500. doi: 10.15388/NA.2015.4.2.

Abstract

We analyze the extended tanh-function method to realize variable separation, however, we find that various "different" solutions obtained by this method are seriously equivalent to the general solution derived by the multilinear variable separation approach. In order to illustrate this point, we take a general (2 + 1)-dimensional Korteweg–de Vries system in water for example. Eight kind of variable separation solutions for a general (2 + 1)-dimensional Korteweg–de Vries system are derived by means of the extended tanh-function method and the improved tanh-function method. By detailed investigation, we find that these seemly independent variable separation solutions actually depend on each other. It is verified that many of so-called "new" solutions are equivalent to one another. Based on the uniform variable separation solution, abundant localized coherent structures can be constructed. However, we must pay our attention to the solution expression of all components to avoid the appearance of some un-physical related and divergent structures: seemly abundant structures for a special component are obtained while the divergence of the corresponding other component for the same equation appears.

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