Real-World Applications of Analytic Travelling Wave Solutions of a (3+1)-Dimensional Hirota–Satsuma–Ito-Like System Via Polynomial Complete Discriminant System and Elementary Integral Technique

Abstract Models reveal the dynamic character of wave motion, delineated in shallow waters alongside fluid dynamics; an example is the Hirota–Satsuma–Ito model equation. Therefore, this paper showcases the detailed analytical investigations of a (3+1)-dimensional Hirota–Satsuma–Ito-like system. In order to reduce the equation to a nonlinear ordinary differential system of equations, a traveling plane wave transformation is engaged. Thereafter, the direct integration technique is adopted to solve the model, thus culminating in obtaining Jacobi elliptic integral function solutions. Moreover, to attain more various solitonic solutions of diverse structures, a standard approach called the polynomial complete discriminant system and elementary integral technique is engaged. This provides exact traveling wave solutions of diverse known functions in the form of periodic, trigonometric, dark, mixed bright, and topological kink, as well as singular soliton solutions. These are found to appear in the form of Jacobi elliptic, trigonometric, as well as hyperbolic functions. Furthermore, some of these solutions are further examined by investigating their wave nature via numerical simulations.