Although the modified tanh-function method was proposed to construct variable separation solutions many years ago, two important issues have not been emphasized: (1) the equivalence of variable separation solutions with different expressions by means of the modified tanh-function method; and (2) the universality of lack of physical meanings for some localized structures constructed by variable separation solutions. In this paper, we construct eleven types of variable separation solutions for the variable-coefficient breaking soliton system by means of the modified tanh-function method with three different ansätz, that is, positive-power ansatz, radical sign combined ansatz, and positive and negative power-symmetric ansatz. However, all other solutions with different forms can be re-derived from one of these solutions by means of some re-definitions of p and q. Therefore, solutions with different forms obtained by different ansätz of the modified tanh-function method are essentially equivalent. From this perspective, different ansätz are not really effective to construct so-called new solutions. Moreover, we study some localized coherent structures such as dromion, peakon and compacton for all components of the same model. We find if there is no divergent phenomenon for the other component, these localized coherent structures such as dromion, peakon and compacton are physical, otherwise, these localized coherent structures lose their values in the real application. We hope that these results have potential values to deeply investigate exact solutions and the related localized structures of nonlinear models in physics, engineering and biophysics.
Localized structures Modified tanh-function method Different ansätz Variable-coefficient breaking soliton model
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This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F050011), the National Natural Science Foundation of China (Grant No. 11375007) and the Zhejiang Province Undergraduate Scientific and Technological Innovation Project (Grant No. ). Dr. Chao-Qing Dai is also sponsored by the Foundation of New Century “151 Talent Engineering” of Zhejiang Province of China and Youth Top-notch Talent Development and Training Program of Zhejiang A&F University.
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