We construct analytic solutions to the generalized Grad-Shafranov (GS) equation, adopting the general linearizing ansatz for the free-function terms it contains, by expanding the genaralized Solov'ev solution [Ch. Simintzis, G. N. Throumoulopoulos, G. Pantis and H. Tasso, Phys. Plasmas 8, 2641 (2001)]. On the basis of these solutions, we examine how the genaralized Solov'ev configuration is modified as the values of the free parameters associated with the additional pressure, poloidal-current and electric-field terms are changed. Thus, a variety of equilibria of tokamak, spherical tokamak and spheromak pertinence are constructed including D-shaped configurations with positive and negative triangularity and diverted configurations with either a couple of X-points or a single X-point.

The Burgers hierarchy consists of nonlinear evolutionary partial differential equations (PDEs) with progressively higher-order dispersive and nonlinear terms. Notable members of this hierarchy are the Burgers equation and the Sharma-Tasso-Olver equation, which are widely applied in fields such as plasma physics, fluid mechanics, optics, and biophysics to describe nonlinear waves in inhomogeneous media. Various soliton and multi-soliton solutions to these equations have been identified and the fission and fusion of solitons have been studied using analytical and numerical techniques. Recently, deep learning methods, particularly Physics Informed Neural Networks (PINNs), have emerged as a new approach for solving PDEs. These methods use deep neural networks to minimize PDE residuals while fitting relevant data. Although PINNs have been applied to equations like Burgers' and Korteweg-de Vries, higher-order members of the Burgers hierarchy remain unexplored in this context. In this study, we employ a PINN algorithm to approximate multi-soliton solutions of linear combinations of equations within the Burgers hierarchy. This semi-supervised approach encodes the PDE and relevant data, determining PDE parameters and resolving the linear combination to discover the PDE that describes the data. Additionally, we employ gradient-enhanced PINNs (gPINNs) and a conservation law, specific to the generic Burgers' hierarchy, to improve training accuracy. The results demonstrate the effectiveness of PINNs in describing multi-soliton solutions within the generic Burgers' hierarchy, their robustness to increased levels of data noise, and their limited yet measurable predictive capabilities. They also verify the potential for training refinement and accuracy improvement using enhanced approaches in certain cases, while enabling the discovery of the PDE model that describes the observed solitary structures.

We derive axisymmetric equilibrium equations in the context of the hybrid Vlasov model with kinetic ions and massless fluid electrons, assuming isothermal electrons and deformed Maxwellian distribution functions for the kinetic ions. The equilibrium system comprises a Grad–Shafranov partial differential equation and an integral equation. These equations can be utilized to calculate the equilibrium magnetic field and ion distribution function, respectively, for given particle density or given ion and electron toroidal current density profiles. The resulting solutions describe states characterized by toroidal plasma rotation and toroidal electric current density. Additionally, due to the presence of fluid electrons, these equilibria also exhibit a poloidal current density component. This is in contrast to the fully kinetic Vlasov model, where axisymmetric Jeans equilibria can only accommodate toroidal currents and flows, given the absence of a third integral of the microscopic motion.

We extend previous work [Y. E. Litvinenko, Phys. Plasmas 17, 074502 (2010)] on a direct method for finding similarity reductions of partial differential equations such as the Grad–Shafranov equation, to the case of the generalized Grad–Shafranov equation (GGSE) with arbitrary incompressible flow. Several families of analytic solutions are constructed, the generalized Solovév solution being a particular case, which contain both the classical and non-classical group-invariant solutions to the GGSE. Those solutions can describe a variety of equilibrium configurations pertinent to toroidal magnetically confined plasmas and planetary magnetospheres.

We extend previous work [Y. E. Litvinenko, Phys. Plasmas 17, 074502 (2010)] on a direct method for finding similarity reductions of partial differential equations such as the Grad-Shafranov equation (GSE), to the case of the generalized Grad-Shafranov equation (GGSE) with arbitrary incompressible flow. Several families of analytic solutions are constructed, the generalized Solovév solution being a particular case, which contain both the classical and non-classical group-invariant solutions to the GGSE. Those solutions can describe a variety of equilibrium configurations pertinent to toroidal magnetically confined plasmas and planetary magnetospheres. https://arxiv.org/abs/2401.09061