New types of the traveling wave and numerical solutions for the modified (3+1)-dimensional Sakovich equation

In our current paper, we will construct new types of traveling wave solutions to the modified ([Formula: see text])-dimensional Sakovich Equation, which is one of the integrable nonlinear equations. The suggested model will contribute to modeling certain crucial phenomena across various scientific disciplines, such as atmospheric sciences, oceanography and related fields. Specifically, it is used to characterize the motion of nonlinear waves. We utilize two distinct analytical techniques to construct the traveling wave solutions and one numerical technique to perceive its corresponding numerical solutions. The considered semi-analytical methods are the Paul–Painleve approach method and the extended direct algebraic method which are employed and used for the first time to construct new visions of the soliton solutions to this model. Furthermore, the Haar Wavelet Method, whose initial conditions have been derived from the achieved soliton solutions, has been utilized to extract the numerical solutions of the achieved soliton solutions. The consistency between the achieved soliton solutions and each other’s as well as with the corresponding numerical solutions has been shown. The novelty of the explored solutions is clear when it’s compared with that explored before by other authors [Ma et al., Qual. Theory Dyn. Syst. 21, 158 (2022); Ali et al., J. Math. 2023, 4864334 (2023); Cortez et al., Results Phys. 55, 107131 (2023)] who solved this model by other techniques.