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24 pages, 7113 KB

Open AccessArticle

Non-Axisymmetric Tokamak Plasma Equilibrium by 3-D Multi-Layers Method

by Jingting Wang and Hiroaki Tsutsui

Appl. Sci. 2025, 15(18), 10037; https://doi.org/10.3390/app151810037 - 14 Sep 2025

Viewed by 519

A three-dimensional (3-D) Multi-Layers Method (MLM) of an extension of the axisymmetric version has been developed to compute non-axisymmetric tokamak plasma equilibria with a separatrix. Conventional axisymmetric tokamak control codes cannot simulate non-axisymmetric effects, while stellarator equilibrium solvers such as VMEC do not include the effects of conducting structures. Moreover, VMEC cannot obtain equilibria with separatrices since it uses magnetic coordinates. The 3-D MLM removes these limitations by using a deformable circuit model of a magnetic confinement system. Plasma is modeled by multiple current layers coinciding with magnetic surfaces, and equilibria are obtained as solutions of a variational problem of a free energy functional with current sources. Validations of equilibrium solutions against a stellarator vacuum field and a VMEC solution for a small non-axisymmetric tokamak show good agreement in magnetic configurations, pressure profile, and plasma current. By incorporating conducting structures and extension to dynamic simulations, the 3-D MLM establishes a method for simulating tokamak plasma control under non-axisymmetric magnetic fields. Full article

(This article belongs to the Special Issue Plasma Physics: Theory, Methods and Applications)

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12 pages, 611 KB

Open AccessEditor’s ChoiceArticle

Non-Associative Structures and Their Applications in Differential Equations

by Yakov Krasnov

Mathematics 2023, 11(8), 1790; https://doi.org/10.3390/math11081790 - 9 Apr 2023

Cited by 4 | Viewed by 2658

Abstract

This article establishes a connection between nonlinear DEs and linear PDEs on the one hand, and non-associative algebra structures on the other. Such a connection simplifies the formulation of many results of DEs and the methods of their solution. The main link between these theories is the nonlinear spectral theory developed for algebra and homogeneous differential equations. A nonlinear spectral method is used to prove the existence of an algebraic first integral, interpretations of various phase zones, and the separatrices construction for ODEs. In algebra, the same methods exploit subalgebra construction and explain fusion rules. In conclusion, perturbation methods may also be interpreted for near-Jordan algebra construction. Full article

(This article belongs to the Special Issue Non-associative Structures and Their Applications in Physics and Geometry)

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11 pages, 2544 KB

Open AccessArticle

Stochastic Bifurcations and Excitement in the ZS-Model of a Thermochemical Reaction

by Lev Ryashko and Irina Bashkirtseva

Mathematics 2022, 10(6), 960; https://doi.org/10.3390/math10060960 - 17 Mar 2022

Cited by 2 | Viewed by 1382

Abstract

The Zeldovich–Semenov model of the continuous stirred tank reactor with parametric random disturbances in temperature is considered. We study a phenomenon of noise-induced transformation of the equilibrium mode into the mixed-mode oscillatory stochastic regime with alternations between small and large amplitudes. In the parametric analysis of the stochastic excitement, we use the analytical method of confidence domains based on the stochastic sensitivity technique. Analyzing a mutual arrangement of the confidence ellipses and separatrices, we estimate the critical intensity of the noise that causes the excitation. The phenomena of stochastic P-bifurcations and coherence resonances are discovered and studied by probability density functions and the statistics of interspike intervals. Full article

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25 pages, 891 KB

Open AccessArticle

Asymptotic Solutions of a Generalized Starobinski Model: Kinetic Dominance, Slow Roll and Separatrices

by Elena Medina and Luis Martínez Alonso

Universe 2021, 7(12), 500; https://doi.org/10.3390/universe7120500 - 15 Dec 2021

Cited by 1 | Viewed by 2681

Abstract

We consider a generalized Starobinski inflationary model. We present a method for computing solutions as generalized asymptotic expansions, both in the kinetic dominance stage (psi series solutions) and in the slow roll stage (asymptotic expansions of the separatrix solutions). These asymptotic expansions are derived in the framework of the Hamilton-Jacobi formalism where the Hubble parameter is written as a function of the inflaton field. They are applied to determine the values of the inflaton field when the inflation period starts and ends as well as to estimate the corresponding amount of inflation. As a consequence, they can be used to select the appropriate initial conditions for determining a solution with a previously fixed amount of inflation. Full article

(This article belongs to the Special Issue Cosmological Models, Quantum Theories and Astrophysical Observations)

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14 pages, 323 KB

Open AccessEditor’s ChoiceArticle

Two-Parametric, Mathematically Undisclosed Solitary Electron Holes and Their Evolution Equation

by Hans Schamel

Plasma 2020, 3(4), 166-179; https://doi.org/10.3390/plasma3040012 - 30 Sep 2020

Cited by 6 | Viewed by 3251

Abstract

The examination of the mutual influence of the two main trapping scenarios, which are characterized by B and D and which in isolation yield the known

{sech}^{4}

(

D = 0

) and Gaussian (

B = 0

) electron holes, show [...] Read more.

The examination of the mutual influence of the two main trapping scenarios, which are characterized by B and D and which in isolation yield the known

{sech}^{4}

(

D = 0

) and Gaussian (

B = 0

) electron holes, show generalized, two-parametric solitary wave solutions. This increases the variety of hole solutions considerably beyond the two cases previously discussed, but at the expense of their mathematical disclosure, since

ϕ (x)

, the electrical wave potential, can no longer be expressed analytically by known functions. Therefore, they belong to a variety with a partially hidden mathematical background, a hitherto unexplored world of structure formation, the origin of which is the chaotic individual particle dynamics at resonance in the coherent wave particle interaction. A third trapping scenario

Γ

, being independent of (B, D) and representing the perturbative trapping scenarios in lowest order, provides a broad, continuous band of associated phase velocities

v_{0}

. For structures propagating near

C_{S E A} = 1.307

, the

s l o w

e l e c t r o n

a c o u s t i c

s p e e d

, a Generalized Schamel equation is derived:

φ_{τ} + [A - B \frac{15}{8} \sqrt{φ} + D ln φ] φ_{x} - φ_{x x x} = 0

, which governs their evolution.

A

is associated with the phase speed and

τ : = C_{S E A} t

and

φ : = ϕ / ψ \geq 0

are the renormalized time and electric potential, respectively, where

ψ

is the amplitude of the structure. Full article

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