A similarity reduction of the Grad-Shafranov equation

A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad–Shafranov equation. As an illustration of the method, the frequently used Solov’ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the Grad–Shafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions

Semi-analytical description of the modulator section of the coherent electron cooling

In the coherent electron cooling, the modern hadron beam cooling technique, each hadron receives an individual kick from the electric field of the amplified electron density perturbation created in the modulator by this hadron in a co-propagating electron beam. We developed a method for computing the dynamics of these density perturbations in an infinite electron plasma with any equilibrium velocity distribution -- a possible model for the modulator. We derived analytical expressions for the dynamics of the density perturbations in the Fourier-Laplace domain for a variety of 1D, 2D, and 3D equilibrium distributions of the electron beam. To obtain the space-time dynamics, we employed the fast Fourier transform (FFT) algorithm. We also found an analytical solution in the space-time domain for the 1D Cauchy equilibrium distribution, which serves as a benchmark for our general approach based on numerical evaluation of the integral transforms and as a fast alternative to the numerical computations. We tested the method for various distributions and initial conditions.Comment: 11 pages, 5 figure