Some physical structures for the (2+1) -dimensional Boussinesq water equation with positive and negative exponents

In this paper, a mathematical method is constructed to study two variants of the two-dimensional Boussinesq water equation with positive and negative exponents. In terms of travelling wave solutions, the partial differential equations are transformed to nonlinear ordinary differential equations. Exact solutions are then derived for various cases to describe the different physical structures such as compactons, solitons, solitary patterns and periodic solutions. The exponent of the wave function u and the ratio of the two coefficients a and b in the Boussinesq equation are shown to qualitatively determine the physical structures of the solutions.