On the construction of the exact analytic or parametric closed-form solutions of standing waves concerning the cubic nonlinear Schr?dinger equation

We propose a new approach for the construction of the closed-form solutions of standing waves of the cubic nonlinear Schr?dinger equation (NLS). Through appropriate functional transformations, we reduce the radially symmetric NLS into an Emden–Fowler equation whose solution results to the derivation of the closed forms of the standing waves. We also derive the necessary restrictions under which the derived solutions are admissible.