Rational solutions in Grammian form for the (3+1)-dimensional generalized shallow water wave equation

In this paper, the (3+1)-dimensional generalized shallow water wave equation is investigated using the Hirota bilinear method and Kadomtsev-Petviashvili hierarchy reduction. The explicit rational solutions for such an equation have been presented in the Grammian form. Based on the Grammian form solution for the equation, the one-rational, two-rational and three-order rational solutions are obtained. When complex parameters pi with nonzero real and imaginary parts are chosen, the lump soliton solutions which are localized in all directions for the (3+1)-dimensional generalized shallow water wave equation can be derived from the corresponding rational solutions. As the figures illustrate, the one-lump soliton solution with one peak and one trough propagates stably on the (x,y) plane. The two-lump solitons with different velocities interact with each other and separate with their original shapes and propagation directions. Different from the case of two-lump solitons, the propagation directions of the third-order lump solitons change after the interaction.