New optical phenomena can be obtained if light is propagated in periodic or discrete 
rotational symmetry media in the nonlinear limit. In this work, to study these 
phenomena arising from the combination of symmetry and nonlinearity, the 
mathematical tools provided by discrete group theory, solid state and nonlinear 
dynamics are used. The behavior of light is modeled by a partial differential equation of 
the Schrödinger type, with a linear and nonlinear periodicity or discrete rotational 
symmetry. For the systems showing discrete rotational symmetry the study is based on 
the key concept of angular pseudomomentum. For the systems showing discrete 
translational symmetry the analogy with solid state physics is deeply explored.

Additionally, numerical methods to solve that partial differential equation in two-
dimensions and in the nonlinear limit have been developed. These methods have been 
used to simulate the control of light by using discrete symmetry and other phenomena. 
New optical devices based on these effects related to discrete rotational symmetry or in 
solitonic periodic structures can be designed. The ultimate goal of this work is to 
simulate experimentally feasible optical devices based on these phenomena.