Soliton-Like Solutions of the Grad-Shafranov Equation

A new class of soliton-like solutions is derived for the Grad-Shafranov (GS) equations. A mathematical analogy between the GS equation for MHD equilibria and the cubic Schr\"odinger (CS) equation for non-linear wave propagation forms the basis to derive the new class of solutions. The soliton-like solutions are considered for their possible relevance to astrophysics and solar physics problems. We discuss how a soliton-like solution can be generated by a repetitive process of magnetic arcade stretching and plasmoid formation induced by the differential rotation of the solar photosphere or of an accretion disk.