New exact solution forms and stability aspects to Drinfel’d–Sokolov–Wilson model by using extended Jacobi elliptic rational function approach

In this study, the travelling wave solutions of the general Drinfel’d–Sokolov–Wilson (DSW)-system, introduced as a model of water waves, are obtained and wave dynamics are investigated. Jacobi elliptic functions are valuable mathematical tools that can be used to various aspects of physics engineering. The method used is effective in generating periodic solutions. It has been observed that the periodic solutions obtained by using Jacobi elliptic function expansions containing different Jacobi elliptic functions may be different, and some new periodic solutions can be obtained. To examine the behaviour of the solutions obtained for appropriate values of the parameters, three-dimensional simulations were conducted using Maple \(^\textrm{TM}\) ${}^{\text{TM}}$ . Additionally, 2D simulations are presented to physicists for easy observation of wave motion. The usage of these simulations aids in better understanding and predicting the behaviour of waves, tidal forces and other phenomena, ultimately contributing to the development of safer and more efficient structures and systems. The stability property of the obtained solutions was tested to demonstrate their reliability. The novelty of this study is that the rich solution group obtained with Jacobi elliptic functions is different from the solutions in the literature and thus provides new physical interpretations. The findings are expected to contribute to the development of new analytical and numerical tools for solving other nonlinear systems in the future, containing a large number of travelling wave solutions crucial for explaining certain scientific phenomena in fluid media.