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Symmetry
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Application of Symmetry in Equations
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Application of Symmetry in Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 30 November 2024 | Viewed by 5204

Special Issue Editors

Dr. Jin Zhang

E-Mail Website
Guest Editor
School of Mathematics, Jilin University, Changchun 130012, China
Interests: ordinary differential equations; dynamic systems; integrability
Dr. Wenlei Li

E-Mail Website
Guest Editor
School of Mathematics, Jilin University, Changchun 130012, China
Interests: differential galois theory; integrability; dynamical system; singularly renormalization group theory
Prof. Dr. Fengde Chen

E-Mail Website
Guest Editor
College of Mathematics and Computer Science, Fuzhou University, Fuzhou 350116, China
Interests: mutation; control systems; population dynamics; control theory; MATLAB simulation; mathematical modelling; difference equations; math

Special Issue Information

Dear Colleagues,

Symmetry is an important property by which we identify the structure of objects. The symmetry properties of equations can be used to determine solutions with particular properties or to classify equations with the same intrinsic properties. The study of symmetry is important for both applied and pure mathematics, as well as physics, engineering and other mathematically based sciences.

Symmetry analysis based on Lie group theory is one of the most important methods for solving nonlinear problems. This method can be used to find the symmetries of almost any system of differential equations, and these symmetries can be used to reduce the complexity of physical problems governed by equations.

Therefore, this Special Issue titled “Application of Symmetry in Equations” aims to collect outstanding research results in the field of symmetry in differential equations and its related fields. We encourage revolutionary research results, including new approaches, techniques and perspectives. Please note that all submitted papers should be within the scope of the journal and there should be no conflicts of interest.

Dr. Jin Zhang
Dr. Wenlei Li
Prof. Dr. Fengde Chen
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • lie group
  • galois group
  • symmetry
  • generalized symmetry
  • rotational symmetry
  • first integral
  • integrability
  • hamiltonian system
  • dynamical system

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (5 papers)

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Research

14 pages, 273 KiB  
Article
Symmetries and Invariant Solutions of Higher-Order Evolution Systems
by Rita Tracinà
Symmetry 2024, 16(8), 1023; https://doi.org/10.3390/sym16081023 - 10 Aug 2024
Viewed by 735
Abstract
In this paper, we investigate evolution systems in two components, characterized by higher-order spatial derivatives and the presence of two arbitrary functions. Our study begins with an analysis of a fourth-order system. We perform a detailed group classification and identify specific forms of [...] Read more.
In this paper, we investigate evolution systems in two components, characterized by higher-order spatial derivatives and the presence of two arbitrary functions. Our study begins with an analysis of a fourth-order system. We perform a detailed group classification and identify specific forms of the constitutive functions that allow the system to exhibit additional symmetries in addition to spatial and temporal translations. We extend these results to nth-order systems. Moreover, we derive invariant solutions for these systems. Finally, for each order n, we are able to find non-negative solutions. Full article
(This article belongs to the Special Issue Application of Symmetry in Equations)
24 pages, 345 KiB  
Article
Gauge Symmetry of Magnetic and Electric Two-Potentials with Magnetic Monopoles
by Rodrigo R. Cuzinatto, Pedro J. Pompeia and Marc de Montigny
Symmetry 2024, 16(7), 914; https://doi.org/10.3390/sym16070914 - 17 Jul 2024
Viewed by 1063
Abstract
We generalize the U(1) gauge transformations of electrodynamics by means of an analytical extension of their parameter space and observe that this leads naturally to two gauge potentials, one electric, one magnetic, which permit the writing of local Lagrangians describing elementary particles with [...] Read more.
We generalize the U(1) gauge transformations of electrodynamics by means of an analytical extension of their parameter space and observe that this leads naturally to two gauge potentials, one electric, one magnetic, which permit the writing of local Lagrangians describing elementary particles with electric and magnetic charges. Gauge invariance requires a conformal transformation of the metric tensor. We apply this approach, which borrows from Utiyama’s methodology, to a model with a massless scalar field and a model with a massless spinor field. We observed that for spinor models non-symmetrized Lagrangians can enable the existence of magnetic monopoles, but this is not possible with symmetrized Lagrangian. Such restrictions do not occur for spinless fields, but the model does not allow spin-one fields interacting with monopoles. Full article
(This article belongs to the Special Issue Application of Symmetry in Equations)
15 pages, 2686 KiB  
Article
A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension
by Amal S. Alali, Muhammad Ahsan Binyamin and Maria Mehtab
Symmetry 2024, 16(7), 854; https://doi.org/10.3390/sym16070854 - 5 Jul 2024
Viewed by 1003
Abstract
A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph [...] Read more.
A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7. Full article
(This article belongs to the Special Issue Application of Symmetry in Equations)
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26 pages, 15223 KiB  
Article
Construction of Soliton Solutions of Time-Fractional Caudrey–Dodd–Gibbon–Sawada–Kotera Equation with Painlevé Analysis in Plasma Physics
by Khadija Shakeel, Alina Alb Lupas, Muhammad Abbas, Pshtiwan Othman Mohammed, Farah Aini Abdullah and Mohamed Abdelwahed
Symmetry 2024, 16(7), 824; https://doi.org/10.3390/sym16070824 - 1 Jul 2024
Viewed by 1272
Abstract
Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE [...] Read more.
Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. Full article
(This article belongs to the Special Issue Application of Symmetry in Equations)
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17 pages, 722 KiB  
Article
Dynamics of Symmetrical Discontinuous Hopfield Neural Networks with Poisson Stable Rates, Synaptic Connections and Unpredictable Inputs
by Marat Akhmet, Zakhira Nugayeva and Roza Seilova
Symmetry 2024, 16(6), 740; https://doi.org/10.3390/sym16060740 - 13 Jun 2024
Viewed by 671
Abstract
The purpose of this paper is to study the dynamics of Hopfield neural networks with impulsive effects, focusing on Poisson stable rates, synaptic connections, and unpredictable external inputs. Through the symmetry of impulsive and differential compartments of the model, we follow and extend [...] Read more.
The purpose of this paper is to study the dynamics of Hopfield neural networks with impulsive effects, focusing on Poisson stable rates, synaptic connections, and unpredictable external inputs. Through the symmetry of impulsive and differential compartments of the model, we follow and extend the principal dynamical ideas of the founder. Specifically, the research delves into the phenomena of unpredictability and Poisson stability, which have been examined in previous studies relating to models of continuous and discontinuous neural networks with constant components. We extend the analysis to discontinuous models characterized by variable impulsive actions and structural ingredients. The method of included intervals based on the B-topology is employed to investigate the networks. It is a novel approach that addresses the unique challenges posed by the sophisticated recurrence. Full article
(This article belongs to the Special Issue Application of Symmetry in Equations)
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