From Grad-Shafranov equations set to a pseudo-general form of the non-linear Schroedinger equation

TL;DR

This paper explores the connection between Grad-Shafranov equations and a generalized nonlinear Schrödinger equation, demonstrating conditions under which both previous transformation approaches are valid in plasma physics.

Contribution

It introduces a critical condition for the poloidal flux function that unifies different field transformation approaches to derive a pseudo-general NLSE from GSE.

Findings

Identifies a critical condition for the flux function psi.

Establishes a class of pseudo-general NLSEs.

Validates previous transformation approaches.

Abstract

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.