Analysis of a Mixed Dispersion Nonlinear Hydrodynamic Model Exhibiting Single and Periodic Solitary Wave Modes with Its Invariance Under Infinitesimal Transformation

Here, we consider a nonlinear hydrodynamic model with mixed dispersion–temporal evolution as the scalar version of the generalized shallow-water wave equation, which specifically provides a comprehensive and versatile framework for studying energy propagation in nonlinear fluids of constrained depth. This equation is acknowledged as an integrable model in the analysis of tidal wave dynamics and in simulations of weather variations, tsunami prediction, and irrigation flows. We also investigate a few of its singular and periodic solitary wave solutions by employing various Riccati-based ansatzes. These results highlight the necessity of studying various nonlinear wave phenomena, which may have potential applications in various domains of physics and applied mathematics. These results extend the variety of its solutions and also enrich the existing knowledge about its solutions with various profiles. To improve visual clarity and to facilitate structural understanding, the solution profiles are represented graphically using Maple software (version 2023.2) in 3D, 2D, and contour plots.We also discuss its invariance under infinitesimal transformations, which yields a one-dimensional Hamilton–Jacobi-like equation.