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We develop a complete mathematical theory for the symmetrical solutions ofthe generalized nonlinear Schr\"odinger equation based on the new concept ofangular pseudomomentum. We consider the symmetric solitons of a generalizednonlinear Schr\"odinger equation with a nonlinearity depending on the modulusof the field. We provide a rigorous proof of a set of mathematical resultsjustifying that these solitons can be classified according to the irreduciblerepresentations of a discrete group. Then we extend this theory tonon-stationary solutions and study the relationship between angular momentumand pseudomomentum. We illustrate these theoretical results with numericalexamples. Finally, we explore the possibilities of the generalization of theprevious framework to the quantum limit.

Angular Pseudomomentum Theory for the Generalized Nonlinear Schr\"{o}dinger Equation in Discrete Rotational Symmetry Media