Some Remarks on the Riccati Equation Expansion Method for Variable Separation of Nonlinear Models

Abstract

Abstract Based on the Riccati equation expansion method, 11 kinds of variable separation solutions with different forms of (2+1)-dimensional modified Korteweg–de Vries equation are obtained. The following two remarks on the Riccati equation expansion method for variable separation are made: (i) a remark on the equivalence of different solutions constructed by the Riccati equation expansion method. From analysis, we find that these seemly independent solutions with different forms actually depend on each other, and they can transform from one to another via some relations. We should avoid arbitrarily asserting so-called “new” solutions; (ii) a remark on the construction of localised excitation based on variable separation solutions. For two or multi-component systems, we must be careful with excitation structures constructed by all components for the same model lest the appearance of some un-physical structures. We hope that these results are helpful to deeply study exact solutions of nonlinear models in physical, engineering and biophysical contexts.