Exploration of wave dynamics and soliton behavior in a nonlinear 3D convection-diffusion wave equation

In this work, a two-dimensional nonlinear wave convection-diffusion equation is studied using the symmetry-based techniques, specifically Lie-symmetry approach. Through characteristic-based similarity transformations, the nonlinear wave convection-diffusion equation is simplified to ordinary differential equations, enabling the derivation of exact solutions. Moreover, traveling wave and periodic solutions are also determined using effective approaches such as tanh method, which describes localized wave structures, and the Jacobi elliptic function method, for capturing periodic waveforms. These solutions are further illustrated through 2D, 3D and contour graphical representations to highlight their physical interpretations. In addition, conservation laws are derived using the multiplier method. The flux fields corresponding to each conservation law are plotted for better understanding of their physical meaning. The results offer valuable insight into the underlying structure and behavior of solutions to nonlinear wave convection-diffusion equations and similar complex systems.