A group-theory method to find stationary states in nonlinear discrete symmetry systems

In the field of nonlinear optics, the self-consistency method has been applied to searching optical solitons in different media. In this paper, we generalize this method to other systems, adapting it to discrete symmetry systems by using group theory arguments. The result is a new technique that incorporates symmetry concepts into the iterative procedure of the self-consistency method, that helps the search of symmetric stationary solutions. An efficient implementation of this technique is also presented, which restricts the computational work to a reduced section of the entire domain and is able to find different types of solutions by specifying their symmetry properties. As a practical application, we develop an efficient algorithm for solving the nonlinear Schrödinger equation with a discrete symmetry potential.