Lie symmetries with optimal subalgebra, quasi self-adjointness condition, conservation laws and power series solutions for the Mikhailov-Shabat model in quantum theory

In this paper, the exact analytical solutions and conservation laws of the Mikhailov-Shabat model have been investigated by utilizing the corresponding infinitesimal generators. Two fundamental methods, namely the Lie symmetry method and the geometric method, are taken into consideration for finding the infinitesimal generators of the governing model. In Lie group theory, the invariance criteria of the Lie group of transformations are employed, while the extended Estabrook and Harrison's differential forms are considered in the geometric approach. To categorize the class of invariant solutions, a one-dimensional optimal subalgebra has been constructed. The conservation laws for the Mikhailov-Shabat model are generated using the 'new conservation theorem' introduced by Ibragimov. Additionally, the power series solutions corresponding to the Lie infinitesimal generators are established with their convergence analysis by adopting the implicit function theorem. Finally, the three-dimensional surfaces of the acquired solutions have been plotted with corresponding density and contour plots.