The generalized shallow water like equation is investigated in this research paper. Exact solutions of generalized shallow water like equation are extracted using modified auxiliary equation (MAE) method and extended (G′G2)-expansion method. Many novel soliton solutions are obtained using these methods. The retrieved solution of governing model include rational, trigonometric and hyperbolic functions. The 3D graphs, 2D contour graphs and line graphs of obtained solutions are plotted using symbolic software such as Maple. The aim of plotting graphs is to demonstrate the dynamical behavior of acquired solutions. Thus, this study investigate the exact soliton solutions of generalized shallow water like using proposed methods.

The KdV–mKdV equation is investigated in this study. This equation is a useful tool to model many nonlinear phenomena in the fields of fluid dynamics, quantum mechanics, and soliton wave theory. The exact soliton solutions of the KdV–mKdV equation are extracted using three distinct exact methods, namely, the generalized projective Riccati equation method, the modified auxiliary equation method, and the generalized unified method. Many novel soliton solutions, including kink, periodic, bright, dark, and singular dark–bright soliton solutions, are obtained. Rational functions, exponential functions, trigonometric functions, and hyperbolic functions are contained in the acquired nontrivial exact solutions. The graphical simulation of some obtained solutions is depicted using 3D plots, 2D contour plots, density plots, and 2D line plots. For the first time, the KdV–mKdV equation is investigated using the proposed three exact methods, and many novel solutions, such as dark, bright, and dark–bright singular soliton solutions, are determined, which have never been reported in the literature.

The Laksmanan–Porsezian–Daniel model is one of the useful models used in nonlinear optics. The extended G′G2-expansion method is used to discuss the dynamical behavior of the proposed model. Many novel solitary wave solutions are obtained using the considered model. To deal with the nonlinearity of this model, three laws of nonlinearity are used, namely, Kerr law, parabolic law, and anti-cubic law. Three-dimensional surface, two-dimensional contour, density, and two-dimensional-line plots of some retrieved solutions are drawn using Maple software. The graphical simulations show the shape and structure of the wave profile corresponding to the obtained results.