Re-study on localized structures based on variable separation solutions from the modified tanh-function method

• Published in: Nonlinear dynamics Vol. 83; no. 3; pp. 1331 - 1339
• Main Authors: Wang, Yue-Yue, Zhang, Yu-Peng, Dai, Chao-Qing
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.02.2016; Springer Nature B.V
• Subjects: Automotive Engineering; Classical Mechanics; Coherence; Construction; Control; Dynamical Systems; Engineering; Equivalence; Exact solutions; Mathematical analysis; Mathematical models; Mathematical programming; Mechanical Engineering; Original Paper; Queuing theory; Radicals; Separation; Vibration; Different ansätz; Divergent structure; Modified tanh-function method; Korteweg–de Vries-type model
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-015-2406-5

We derive variable separation solutions of the ( 1 + 1 )-dimensional KdV-type model by means of the modified tan h -function method with three different ansätz. Superficially speaking, the positive and negative power-symmetric ansatz seems to be better than the positive-power ansatz and radical sign combined ansatz because the positive and negative power-symmetric ansatz can construct the most forms of variable separation solutions. Actually, we find that various “different” solutions obtained by the modified tanh-function method are not independent, and many of the so-called new solutions are equivalent to one another. Moreover, in the two- or multi-component system, if we construct localized coherent structures for a special component based on variable separation solutions, we must note the structure constructed by the other component for the same equation in order to avoid the appearance of some divergent and un-physical structures. We hope that these results above contribute to the analysis on exact solutions and the construction of localized structures for other nonlinear models.