Asian Journal of Mathematics and Computer Research

In this paper, we investigate the newly formulated (2+1)-dimensional Sakovich equation, highlighting its utility in describing the dynamics of nonlinear waves. This new equation effectively incorporates increased dispersion and nonlinear effects, thereby enhancing its applicability across various physical scenarios. This model is especially useful when modeling nonlinear phenomena in materials that simpler linear models would not accurately describe. It also serves as a founding model for numerical simulations in computational fluid dynamics and solid mechanics. We employ the Clarkson-Kruskal (CK) direct method to investigate exact solutions of the extended (2+1)-dimensional Sakovich equation. A review of the relevant literature indicates that the CK direct method has not yet been applied to solve the extended (2+1)-dimensional Sakovich equation. In this work, we successfully perform the complex and tedious computations required by the CK direct method. The results are classified into two distinct cases. In the first case, the obtained solutions include rational functions, a Weierstrass elliptic function, and new similarity reductions leading to Painlev´e I and II equations. The second case yields new solutions involving trigonometric functions and hyperbolic function solutions. To the best of our knowledge, some of the solutions obtained have not been previously reported. All of these solutions manifest diverse wave phenomena, such as the soliton, bright, and dark solitons. Some of them reveal new wave phenomena governed by the extended (2+1)-dimensional Sakovich equation.