Bilinear Auto-Bäcklund Transformation, Shock Waves, Breathers and X-Type Solitons for a (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation in a Fluid

There are abundant nonlinear phenomena in a fluid, such as the nonlinear waves and their interactions. In this paper, a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation in a fluid, which is used to describe the long waves and has the application in water percolation, is investigated. Via the Hirota method, we obtain a bilinear auto-B äcklund transformation as well as shock-wave, breather and X-type soliton solutions. We graphically show the shock waves and breathers, observe that the amplitudes and shapes of shock waves and breathers keep unchanged during the propagation, and show the X-type soliton on a periodic background. We also analysis the influence of the coefficients in the equation on the above waves.