The computation of Lie point symmetry generators, modulational instability, classification of conserved quantities, and explicit power series solutions of the coupled system

The well-known Chaffee-Infante reaction hierarchy is examined in this article along with its reaction-diffusion coupling. It has numerous variety of applications in modern sciences, such as electromagnetic wave fields, fluid dynamics, high-energy physics, ion-acoustic waves in plasma physics, coastal engineering, and optical fibres. The physical processes of mass transfer and particle diffusion might be expressed in this way. The Lie invariance criteria is taken into consideration while we determine the symmetry generators. The suggested approach produces the six dimensional Lie algebra, where translation symmetries in space and time are associated to mass conservation and conservation of energy respectively, the other symmetries are scaling or dilation. Additionally, similarity reductions are performed, and the optimal system of the sub-algebra should be quantified. There are an enormous number of exact solutions can construct for the traveling waves when the governing system is transformed into ordinary differential equations using the similarity transformation technique. The power series approach is also utilized for ordinary differential equations to obtain closed -form analytical solutions for the proposed diffusive coupled system. The stability of the model under the limitations is ensured by the modulation instability analysis. The reaction diffusion hierarchy's conserved vectors are calculated using multiplier methods using Lie Backlund symmetries. The acquired results are presented graphically in 2-D and 3-D to demonstrate the wave propagation behavior.