Linear superposition formula of solutions for the extended (3+1)-dimensional shallow water wave equation

Active researches on the water waves have been done, and water waves are essentially complex waves controlled by gravity field and surface tension. Using the Hirota bilinear method, two bilinear auto-Bäcklund transformations of the extended (3+1)-dimensional shallow water wave equation are derived explicitly. The hyperbolic cosine-function solution and cosine-function solution are obtained by means of bilinear auto-Bäcklund transformations. Five linear superposition formulas of this equation are given and proved. All the results depend on the coefficients of the equation and the linear superposition relationship. Thereafter, we perform a numerical simulation to trace and study the dynamical behaviors of the linear superposition solutions via their three-dimensional profiles using symbolic calculation system Mathematica codes.