  In the year 2003 a paper by G. Lapenta demonstrated that there is a new class
of soliton-like solutions for the Grad-Shafranov Equations (GSE).The author
determined an appropriate pair of transformations of the free fields p (fluid
field of hydrodynamical pressure) and Bz (z- component of magnetic induction
field) that leads from the Helmholtz Equation to the Non-Linear Schroedinger
Equation (NLSE) with cubic non- linearity. In the following year (2004), the
work of Lapenta was opposed by G.N. Throumoulopoulos et al.,who criticized his
idea of the field transformations as a mathematically incoherent choice;
contextually the authors suggested a new point of view for this one. In his
response, in the same year, G. Lapenta carried out numerical simulations that
showed the existence of solitonic structures in a Magnetohydrodynamical (MHD)
plasma-context. In the present work I want to demonstrate a critical condition
for a complex poloidal flux function psi in a plane framework (x;y) that leads
to a class of pseudo-general NLSEs which establishes the validity of both
G.Lapenta and G.N. Throumoulopoulos et al. choices for the field transformation
functionals.
