Sawtooth-like oscillations and steady states caused by the m/n=2/1 double tearing mode

The sawtooth-like oscillations resulting from the m/n=2/1 double tearing mode (DTM) are numerically investigated through the three-dimensional, toroidal, nonlinear resistive-MHD code (CLT). We find that the nonlinear evolution of the m/n=2/1 DTM can lead to sawtooth-like oscillations, which are similar to those driven by the kink mode. The perpendicular thermal conductivity and the external heating rate can significantly alter the behaviors of the DTM driven sawtooth-like oscillations. With a high perpendicular thermal conductivity, the system quickly evolves into a steady state with m/n=2/1 magnetic islands and helical flow. However, with a low perpendicular thermal conductivity, the system tends to exhibit sawtooth-like oscillations. With a sufficiently high or low heating rate, the system exhibits sawtooth-like oscillations, while with an intermediate heating rate, the system quickly evolves into a steady state. At the steady state, there exist the non-axisymmetric magnetic field and strong radial flow, and both are with helicity of m/n=2/1. Like the steady state with m/n=1/1 radial flow, which is beneficial for preventing the Helium ash accumulation in the core, the steady state with m/n=2/1 radial flow might also be a good candidate for the advanced steady-state operations in future fusion reactors. We also find that the behaviors of the sawtooth-like oscillations are almost independent of Tokamak geometry, which implies that the steady state with saturated m/n=2/1 islands might exist in different Tokamaks.

Bäcklund Transformation and Exact Solutions to a Generalized (3 + 1)-Dimensional Nonlinear Evolution Equation

In this article, a generalized (3 + 1)-dimensional nonlinear evolution equation (NLEE), which can be obtained by a multivariate polynomial, is investigated. Based on the Hirota bilinear method, the N-soliton solution and bilinear Bäcklund transformation (BBT) with explicit formulas are successfully constructed. By using BBT, two traveling wave solutions and a mixed solution of the generalized (3 + 1)-dimensional NLEE are obtained. Furthermore, the lump and the interaction solutions for the equation are constructed. Finally, the dynamic properties of the lump and the interaction solutions are described graphically.

Analytical solutions of the fifth-order time fractional nonlinear evolution equations by the unified method

This key purpose of this study is to investigate soliton solution of the fifth-order Sawada–Kotera and Caudrey–Dodd–Gibbon equations in the sense of time fractional local [Formula: see text]-derivatives. This important goal is achieved by employing the unified method. As a result, a number of dark and rational soliton solutions to the nonlinear model are retrieved. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behavior. The results demonstrate that the presented approach is more effective in solving issues in mathematical physics and other fields.

Influence of aspect ratio, plasma viscosity, and radial position of the resonant surfaces on the plasmoid formation in the low resistivity plasma in Tokamak

In the present paper, we systematically investigate the nonlinear evolution of the resistive kink mode in the low resistivity plasma in Tokamak geometry. We find that the aspect ratio of the initial equilibrium can significantly influence the critical resistivity for plasmoid formation. With the aspect ratio of 3/1, the critical resistivity can be one magnitude larger than that in cylindrical geometry due to the strong mode-mode coupling. We also find that the critical resistivity for plasmoid formation decreases with increasing plasma viscosity in the moderately low resistivity regime. Due to the geometry of Tokamaks, the critical resistivity for plasmoid formation increases with the increasing radial location of the resonant surface.

Collisions of Local and Spiritual in State and Public Activities of Metropolitan Ilarion (Ivan Ohienko)

The article clarifies the views of one of the brightest and most significant figures of the Ukrainian church — Metropolitan Ilarion (Ivan) Ohienko on the spiritual and secular service to Ukraine and his practical activities, which naturally effectively combined these two aspects. This article notes that an important element that united the two ministries and substantiated them was the deep level of their interpenetration, where Orthodoxy acquired a national character based on traditions. The article concludes that during this ministry his views on the church did not undergo nonlinear evolution, but only acquired depth and system. Even when Ivan Ohienko was in public office or abroad, he attached great importance to moral, ethical and ecclesiastical issues. Despite the ideological closeness with the views of another prominent Ukrainian church figure Andrei Sheptytsky on church-state relations, education and revival of the Ukrainian nation, language and culture as factors of Ukrainian identity, Ivan Ohienko was still skeptical of the Ukrainian Greek Catholic Church, seeing in it is an instrument of Catholicization of the Ukrainian people. Ohienko believed that in reality only an autocephalous church could be Ukrainian, which relied exclusively on the traditions and needs of the people. This was the criterion of the truth of Orthodoxy for him.

Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form 0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.

Degeneration of Solitons for a the (3+1)-dimensional Generalized Nonlinear Evolution Equation for the Shallow Water Waves

A the (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves is investigated with different methods. Based on symbolic computation and Hirota bilinear form, Nsoliton solutions are constructed. In the process of degeneration of N-soliton solutions, T-breathers are derived by taking complexication method. Then rogue waves will emerge during the degeneration of breathers by taking the parameter limit method. Through full degeneration of N-soliton, M-lump solutions are derived based on long wave limit approach. In addition, we also find out that the partial degeneration of N-soliton process can generate the hybrid solutions composed of soliton, breather and lump.