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

Abstract 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 p i 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.

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