Angular pseudomomentum theory for the generalized nonlinear Schrِdinger equation in discrete rotational symmetry media

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schrِdinger equation based on the concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schrِdinger equation with a nonlinearity depending on the modulus of the field. We provide a rigorous proof of a set of mathematical results justifying that these solitons can be classified according to the irreducible representations of a discrete group. Then we extend this theory to non-stationary solutions and study the relationship between angular momentum and pseudomomentum. We illustrate these theoretical results with numerical examples.