Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one Tau-function

In the framework of Bell-polynomial manipulations, under investigation hereby are three single-field bilinearizable equations: the (1+1)-dimensional shallow water wave model, Boiti–Leon–Manna–Pempinelli model, and (2+1)-dimensional Sawada–Kotera model. Based on the concept of scale invariance, a direct and unifying Bell-polynomial scheme is employed to achieve the Bäcklund transformations and Lax pairs associated with those three soliton equations. Note that the Bell-polynomial expressions and Bell-polynomial-typed Bäcklund transformations for those three soliton equations can be, respectively, cast into the bilinear equations and bilinear Bäcklund transformations with symbolic computation. Consequently, it is also shown that the Bell-polynomial-typed Bäcklund transformations can be linearized into the corresponding Lax pairs.