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

Exact parametric solutions for the first Painlevé nonlinear ODE

“…The coefficients j    In the present work we construct the exact parametric solutions of the first PE nonlinear ODE (redouble case), based on some of the results included in [8] and [9]. Introducing a series of Theorems, Propositions and Lemmas, several of which are new, we su cceed in constructing the exact parametric solutions (without any restrictive assumption) for the first PE.…”