Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods: the Lie symmetry analysis, the generalized Riccati equation mapping approach, and the modified Kudryashov approach, we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation (NCDR) with power-law nonlinearity and density-dependent diffusion. Lie symmetry analysis is one of the powerful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables. By the Lie group technique, we obtain one-parameter invariant transformations, determining equations and corresponding vectors for the considered convection-diffusion-reaction equation. By treating the parameters of the governing equation as constants, the applied methods yield a variety of new closed-form solutions, including inverse function solutions, periodic solutions, exponential function solutions, dark solitons, singular solitons, combo bright-singular solitons, and the combine of bright-dark solitons and dark-bright solitons. Moreover, using the Bäcklund transformation of the generalized Riccati equation and modified Kudryashov method, we can construct multiple solitons and other solutions of the considered equation. The obtained new solutions of this work demonstrate that the used approaches are powerful and effective in dealing with nonlinear equations, and that these solutions are required to explain many biological and physical phenomena. Comparing our obtained solutions of this paper with the ones obtained in the literature, we see that our solutions are new and not reported elsewhere. These newly formed soliton solutions will be more beneficial in the various disciplines of ocean engineering, plasma physics, and nonlinear sciences.