Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation

In this work, the exact solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation are reported by adopting the He’s variational direct technique (HVDT). The analytic findings of solutions were obtained by semi-inverse scheme, and six form of supposed studies reveal that the solutions belong to soliton groups. The modulation instability is considered. The tan(Π(ξ))\tan \left(\Pi \left(\xi )) scheme on the suggested model is employed to study new rational solutions. The investigated properties of solutions were determined by graphic studies, which shows significantly values of the parameters and susceptibility of abundant solutions. The obtained results in this work are expected to open new perspectives for the traveling wave theory. For the aforementioned wave solutions, we graphically describe their dynamical properties. It is worth mentioning that our results not only enable us to understand the dynamic properties of such equations more intuitively but also provide some ideas for researchers to facilitate more in depth exploration. It is important to mention that our proposed method is highly effective, consistent, and impacting and can be utilized to solve different physical models.