Analysis of gas bubbling dynamics via lie symmetry approach of nonlinear wave equation

Bubbles formed by the introduction of gas into a liquid are a common phenomenon, known as gas bubbling in liquids. This process is widely utilized in various industries for aeration, mixing, and purification. An optimal system of Lie symmetry analysis is employed to investigate the generalized (3 + 1)-dimensional nonlinear wave equation (NLWE). Single, double, triple, and quadruple linear combinations are constructed to derive novel solutions that represent different dynamic and turbulent behaviors of the bubbles. This equation models a wide range of nonlinear phenomena occurring in liquids containing gas bubbles. The proposed methodology is used to obtain a diverse set of accurate soliton solutions to the equation. Furthermore, the resulting solutions are analyzed in terms of their physical interpretations.