Abstract
The Lie group method analyses this study's two-mode Cahn-Allen (TMCAE) equation. The wave propagation form of TMCAE is defined by the mobility of two symmetric waves at the same instant. The Lie scheme is successfully employed on the considered model to find the infinitesimal generator using the invariance condition. Then, we convert the assumed nonlinear TMCAE into an ordinary differential equation utilizing obtained generators and similarity reduction concepts. Using the two analytical methods, the tanh method and the new auxiliary method, to find some new wave solutions. To show the importance of obtained results, we draw the graphical structure of some results by selecting some appropriate values of involving parameters. We find the multipliers depending on independent variables, dependent variables, and derivatives of dependent variables up to fixed order. The conserved quantities fluxes are found corresponding to each conservation law. The local conserved quantities are constructed for all understudying systems employing the multiplier approach.