Discussions on localized structures based on equivalent solution with different forms of breaking soliton model

Abstract

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 ansatz, 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 ansatz of the modified tanh-function method are essentially equivalent. From this perspective, different ansatz 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.