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Under investigation, this paper constructs an improved and comprehensive class of analytical wave solutions to the (2+1)-dimensional Sakovich equation, a nonlinear evolution equation that plays a remarkable role in condensed physics, fiber optics, and fluid dynamics. By applying relatively, two renewed techniques named Lie symmetry analysis and extended Jacobian elliptic function expansion method, some standard class of new and wide-spectrum closed-form solutions are established in terms of trigonometric, hyperbolic, and Jacobi elliptic functions. These ascertained solutions contain several ascendant parameters that play a crucial role in describing the inner mechanism of a given physical model. Therefore, the obtained wave solutions are demonstrated graphically by three-and two-dimensional graphics using Mathematica. In addition, a couple of varieties of solutions, including multi soliton, periodic soliton, Bell-shaped, and parabolic profile, are depicted for the suitable values of the included parameters. Finally, it must be noted that the variation in obtained wave profiles by the change in subsidiary parameters reveals the parameter effects on the wave. This study ensures that the forgoing techniques are practical and may be used to seek the solitary solitons to a diversity of nonlinear evolution equations (NLEEs).

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko-Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closed-form invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, 3D, 2D combined line graph and their contour graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.