The purpose of this study is to highlight the shallow water wave patterns along the ocean shore or in lakes with the higher-order Boussinesq–Burgers system possessing a fractional derivative operator. A generic fractional transformation is used, which turns the proposed model into an nonlinear ordinary differential equations (NLODEs) system. For the construction of new solitons of the mentioned coupled system, the auxiliary equation technique is employed. This approach produced numerous soliton solutions such as bright, singular and w-shaped solitons of the aforesaid model successfully. These results are expressed graphically to exemplify their physical appearance with the help of soft computations in Mathematica. All the solutions yielded by this method are novel and have not been derived yet.

The current research is about the optical solitons of the Kundu–Mukherjee–Naskar (KMN) equation that are obtained by implementing the two proficient approaches named: the extended Jacobi’s elliptic expansion function method and the [Formula: see text] function method. The aforesaid methods are used for the first time in the KMN equation to obtain novel soliton solutions in terms of Jacobi’s elliptic function solutions, which turn into dark, bright, and periodic solutions later. Also, the rational function solutions of the above-mentioned equation are obtained. The obtained solutions are also graphed and verified with the use of symbolic soft computations. The obtained results may be applied to illustrate the substantial concept of the studious structures as well as other related nonlinear physical structures.