Generalized Bilinear Differential Operators Application in a (3+1)-Dimensional Generalized Shallow Water Equation

Abstract

The relations between Dp-operators and multidimensional binary Bell polynomials are explored and applied to construct the bilinear forms with Dp-operators of nonlinear equations directly and quickly. Exact periodic wave solution of a (3+1)-dimensional generalized shallow water equation is obtained with the help of the Dp-operators and a general Riemann theta function in terms of the Hirota method, which illustrate that bilinear Dp-operators can provide a method for seeking exact periodic solutions of nonlinear integrable equations. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.