Shallow water in an open sea or a wide channel: Auto- and non-auto-Bäcklund transformations with solitons for a generalized (2+1)-dimensional dispersive long-wave system

•1. Oceanic water waves are actively studied. Taking into account the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth. 3. Investigating a generalized (2+1)-dimensional dispersive long-wave system. 4. With symbolic computation while with respect to the horizontal velocity and wave elevation above the undisturbed water surface. 5. working out two non-auto-Backlund transformations and two auto-Backlund transformations with solitons. Oceanic water waves are actively studied. Hereby, taking into account the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth, we investigate a generalized (2+1)-dimensional dispersive long-wave system. With symbolic computation while with respect to the horizontal velocity and wave elevation above the undisturbed water surface, we work out two non-auto-Bäcklund transformations and two auto-Bäcklund transformations with solitons. All of our results are dependent on the constant coefficients in the original system.