Oceanic shallow water waves via two versions of the (2+1)-dimensional Akinyemi shallow water wave equations

New shallow-water-wave contributions have been documented in IJMPC. Recently, describing the shallow water waves in the oceans, there have appeared two versions of the (2[Formula: see text]1)-dimensional Akinyemi shallow water wave equations, which we study in this paper via a non-characteristic movable singular manifold. In oceanic shallow water dynamics, common special cases of both versions are listed out. For the first equation, symbolic computation helps us work out an auto-Bäcklund transformation under some coefficient constraints, along with five families of the solitonic solutions and three families of the analytic solutions. For the second equation, symbolic computation helps us work out an auto-Bäcklund transformation under some coefficient constraints, along with two families of the solitonic solutions and one family of the analytic solutions. Our results have some structures determined by the values of the coefficients in both equations. By selecting different parameter combinations of those coefficients, one might gain new insights into the oceanic dynamic characteristics described by the first and/or the second equations.