A study of traveling wave solutions and modulation instability in the (3+1)-dimensional Sakovich equation employing advanced analytical techniques.

In this paper, we investigate the newly formulated (3+1)-dimensional Sakovich equation, highlighting its utility in describing the dynamics of nonlinear waves. This novel equation effectively incorporates increased dispersion and nonlinear effects, thereby enhancing its applicability across various physical scenarios. This model especially useful when modeling nonlinear phenomena in materials that simpler linear models would not accurately describe. Also serve as a founding model for numerical simulations in computational fluid dynamics and solid mechanics. We deploy both the Sardar Sub-Equation Method (SSEM) and the Simple Equation Method (SEM) to derive a broad spectrum of unique traveling wave solutions. These solutions have been thoroughly verified with Mathematica and include a wide variety of mathematical functions such as trigonometric hyperbolic and exponential forms. To provide a comprehensive visual representation of these solutions, we generate 3D, contour, density, and 2D graphs by meticulously setting the relevant parameters in Wolfram Mathematica. The solutions obtained illustrate various phenomena, such as dark, bright, kink, singular, periodic, periodic singular, and compacton solitons. The innovation of this work is in the systematic investigation and description of several types of soliton solution over a wide variety of nonlinear equations. Not only does this thorough study advance theoretical insight but also increase practical applications in areas like optical fiber communication and engineering. Additionally, we investigate the modulation instability (MI) of the proposed model, further elucidating its significance in the context of nonlinear wave propagation.