Abundant Solutions and Superposition Behavior Analysis of a (3+1)‐Dimensional Boiti–Leon–Manna–Pempinelli Equation

ABSTRACT By employing the Hirota bilinear (HB) method and starting from the solution with arbitrary functions, we independently analyze the relationship between abundant wave behavior and the superposition formulas of solutions for a (3+1)‐dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. Concurrently, we correlate findings with the behaviors of oceanic waves, yielding some valuable insights. By taking the arbitrary functions as exponential functions, polynomial functions, hyperbolic cosine functions, trigonometric functions, and mixed function, numerous solutions to the equation are obtained as follows: the rogue wave solution, the multikink wave solution, the periodic wave solution, and interaction solutions. Through asymptotic analysis, the superposition of exponential functions leads to the increase in the kink wave numbers. For the rogue wave as the rational function solution, reformulating parameter relations using vectors facilitates the precise determination of its emergence time. Additionally, the impacts of parameters and the order of the rational function on the rogue‐wave shape are independently analyzed. Finally, we examine amplitude variations in rogue and periodic waves under the gradual superposition of kink waves. All types of wave profiles are visually displayed in the form of graphics by selecting appropriate parameters.