Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water

The Boussinesq equation can describe wave motions in media with damping mechanism, e.g., the propagation of long waves in shallow water and the oscillations of nonlinear elastic strings. To study the propagation of gravity waves on the surface of water, a second spatial variable (say, y) is weakly dependent, and an alternative form of generalized two-dimensional Boussinesq equation is investigated in this paper. Four families of lump solutions are derived by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee the analyticity and rational localization of the lumps, some conditions are posed on both the lump parameters and the coefficients of the generalized two-dimensional Boussinesq equation. Localized structures and energy distribution of the lumps are analyzed as well.