Caution with respect to “new” variable separation solutions and their corresponding localized structures

Abstract

Caution is advised regarding so-called “new” variable separation solutions obtained by the generalizing Riccati equation mapping method and the improved projective Riccati equation method because many seemingly independent variable separation solutions actually depend on each other. To illustrate this point, we employ the (2+1)-dimensional Bogoyavlenskii–Schiff model as an example and we derive 10 different variable separation solutions. Based on a detailed investigation, we show that many of the so-called “new” solutions are equivalent to each other. Thus, when we discuss localized structures based on variable separation solution, we should consider all of the field components to avoid the appearance of false unphysical related structures, where seemingly abundant structures are obtained for a special component, while unphysical (non-localized) or even divergent structures may appear for other corresponding components of the same equation.