The superposition solution of positive quadratic function and arbitrary positive function of a (3+1)-dimensional soliton equation under vector representation

The Hirota bilinear method was utilized to study a (3+1)-dimensional soliton equation, and we achieved success in obtaining a variety of solutions to the equation. Successfully yielding various solutions, such as lump solutions, rogue wave solutions, and interaction solutions. The first step in research is to transform the orginal equation into Hirota bilinear form. Through the symbolic calculation and the Cole-Hopf transformation, we obtain the solution of the original equation. Especially, we introduced vectors as tools to get the rational solutions of the equation. We make plots according to select different values of parameters, and the plots of various forms of solutions are dynamically analyzed to understand their physical significance. For the selection of trial functions, the first type is the positive quadratic functions, which can be used to obtain lump solutions and rogue wave solutions. The second type is the superposition of positive quadratic functions and positive arbitrary functions, resulting in an interaction solution consisting of both rational solution and arbitrary function solutions. We will provide examples to illustrate the interaction solutions formed by the superposition of positive quadratic and exponential functions, the superposition of positive quadratic, exponential and trigonometric functions, and the superposition of positive quadratic, exponential, trigonometric and hyperbolic functions. In short, we constructed different trial functions, so various new superposition solutions and wave motion were obtained.