Kadomtsev–Petviashvili hierarchy reduction, soliton and semi-rational solutions for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation in a fluid. Issue 3 (4th March 2022) - Zhao, Xin
Abstract : In this paper, we study a (3+1)-dimensional generalized shallow water wave equation with variable coefficients, which describes the flow below a pressure surface in a fluid. We give the Kadomtsev–Petviashvili hierarchy reduction and construct the multi-soliton solutions and semi-rational solutions in terms of the Gramian. For the multi-soliton solutions, we conclude that: (1) v 1 ( t ) affects the directions for the two solitons to move; (2) there is the periodic interaction of the two solitons when v 2 ( t ) is a periodic function and (3) the magnitudes of the velocities for the two solitons increase as the amplitude of the periodic function v 2 ( t ) increases, where v 1 ( t ) represents the perturbed effect, v 2 ( t ) indicates the dispersion effect and t is an independent variable. For the first-order semi-rational solutions, we see that: (1) the fission with v 1 ( t ) < 0 and fusion with v 1 ( t ) > 0 appear; (2) there is the periodic interaction when v 2 ( t ) is a periodic function; (3) the magnitude of the velocity for the soliton increases as the amplitude of the periodic function v 2 ( t ) increases and (4) the lump becomes narrower as the amplitude of the periodic function v 4 ( t ) decreases, where v 4 ( t ) indicates the perturbed effect.