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

TL;DR

This paper introduces a comprehensive mathematical framework for classifying symmetric solutions of the generalized nonlinear Schr"odinger equation using angular pseudomomentum, linking symmetry, angular momentum, and pseudomomentum, with numerical and quantum considerations.

Contribution

It develops a novel theory based on angular pseudomomentum for classifying solutions of the nonlinear Schr"odinger equation with discrete rotational symmetry.

Findings

Solutions can be classified by irreducible representations of discrete groups.

The theory extends to non-stationary solutions and relates angular momentum to pseudomomentum.

Numerical examples support the theoretical framework.

Abstract

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schr\"odinger equation based on the new concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schr\"odinger 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. Finally, we explore the possibilities of the generalization of the previous framework to the quantum limit.