Symbolically Computing the Shallow Water via a (2+1)-Dimensional Generalized Modified Dispersive Water-Wave System: Similarity Reductions, Scaling and Hetero-Bäcklund Transformations

For the water waves, people consider some dispersive systems. Describing the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, we now symbolically compute a (2+1)-dimensional generalized modified dispersive water-wave system. With respect to the height of the water surface and horizontal velocity of the water wave, with symbolic computation, we work out (1) a set of the scaling transformations, (2) a set of the hetero-Bäcklund transformations, from that system to a known linear partial differential equation, and (3) four sets of the similarity reductions, each of which is from that system to a known ordinary differential equation. We pay attention that our hetero-Bäcklund transformations and similarity reductions rely on the coefficients in that system.