Notes on the equivalence of different variable separation approaches for nonlinear evolution equations

• Note on the equivalence of extended projective Ricatti equation method and multilinear variable separation approach.
• Note on the equivalence of improved tanh-function method and multilinear variable separation approach.
• Be careful with localized structures from variable separation solutions.

The equivalence of multilinear variable separation approach, the extended projective Ricatti equation method and the improved tanh-function method is firstly reported when these three popular methods are used to realize variable separation for nonlinear evolution equations. We take the (2 + 1)-dimensional modified Broer–Kaup system for an example to illustrate this point. All solutions obtained by the extended projective Ricatti equation method and the improved tanh-function method coincide with the one obtained by the multilinear variable separation approach. Moreover, based on one of variable separation solutions, we also find that although abundant localized coherent structures can be constructed for a special component, we must pay our attention to the solution expression of the corresponding other component for the same equation lest many un-physical related structures might be obtained.