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Symmetry
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Mathematical Physics: Topics and Advances
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Mathematical Physics: Topics and Advances

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

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 10583

Special Issue Editor

Dr. Anton A. Kutsenko

E-Mail Website
Guest Editor
Department of Mathematics & Logistics, Jacobs University, Campus Ring 1, 28759 Bremen, Germany
Interests: operator theory; spectral theory; wave propagation; phononic crystals; schrodinger operators

Special Issue Information

Dear Colleagues,

The concept of symmetry permeates all areas of science, and in fact, serves as the basis of any direction in scientific activity. Even in seemingly completely disordered and chaotic systems, the discovery of some regularities and symmetries is the greatest achievement. Most laws of nature are based on the observation of some sort of symmetry. Thus, symmetry is an absolutely interdisciplinary concept. However, the most expressive language for describing symmetry is mathematics. For example, algebras of operators acting on periodic lattices, or on fractal structures, are a powerful tool for revealing the symmetry properties of natural phenomena and physical processes. The present Special Issue is devoted to the application of various algebraic structures to problems of mechanics and physics. We are inviting authors working in such areas as:

- Spectral theory;

- Operator theory;

- Linear waves in heterogeneous structures;

- Non-linear waves, ocean waves;

- Homogenization theory;

- Representation theory.

Dr. Anton A. Kutsenko
Guest Editor

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

  • operators
  • algebras
  • phononic crystals
  • waves
  • fractals
  • Schrödinger equation
  • spectrum
  • perturbation theory

Published Papers (6 papers)

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Research

16 pages, 4004 KiB  
Article
New Localized and Periodic Solutions to a Korteweg–de Vries Equation with Power Law Nonlinearity: Applications to Some Plasma Models
by Samir A. El-Tantawy, Alvaro H. Salas and Wedad Albalawi
Symmetry 2022, 14(2), 197; https://doi.org/10.3390/sym14020197 - 20 Jan 2022
Cited by 15 | Viewed by 1976
Abstract
Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, [...] Read more.
Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on. Full article
(This article belongs to the Special Issue Mathematical Physics: Topics and Advances)
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17 pages, 708 KiB  
Article
Linear and Nonlinear Electrostatic Excitations and Their Stability in a Nonextensive Anisotropic Magnetoplasma
by Muhammad Khalid, Ata-ur-Rahman, Ali Althobaiti, Sayed K. Elagan, Sadah A. Alkhateeb, Ebtehal A. Elghmaz and Samir A. El-Tantawy
Symmetry 2021, 13(11), 2232; https://doi.org/10.3390/sym13112232 - 22 Nov 2021
Cited by 9 | Viewed by 1578
Abstract
In the present work, the propagation of (non)linear electrostatic waves is reported in a normal (electron–ion) magnetoplasma. The inertialess electrons follow a non-extensive q-distribution, while the positive inertial ions are assumed to be warm mobile. In the linear regime, the dispersion relation [...] Read more.
In the present work, the propagation of (non)linear electrostatic waves is reported in a normal (electron–ion) magnetoplasma. The inertialess electrons follow a non-extensive q-distribution, while the positive inertial ions are assumed to be warm mobile. In the linear regime, the dispersion relation for both the fast and slow modes is derived, whose properties are analyzed parametrically, focusing on the effect of nonextensive parameter, component of parallel anisotropic ion pressure, component of perpendicular anisotropic ion pressure, and magnetic field strength. The reductive perturbation technique is employed for reducing the fluid equation of the present plasma model to a Zakharov–Kuznetsov (ZK) equation. The parametric role of physical parameters on the characteristics of the symmetry planar structures such solitary waves is investigated. Furthermore, the stability of the pulse soliton solution of the ZK equation against the oblique perturbations is investigated. Furthermore, the dependence of the instability growth rate on the related physical parameters is examined. The present investigation could be useful in space and astrophysical plasma systems. Full article
(This article belongs to the Special Issue Mathematical Physics: Topics and Advances)
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17 pages, 924 KiB  
Article
Symmetry Solutions and Conservation Laws for the 3D Generalized Potential Yu-Toda-Sasa-Fukuyama Equation of Mathematical Physics
by Chaudry Masood Khalique, Karabo Plaatjie and Oageng Lawrence Diteho
Symmetry 2021, 13(11), 2058; https://doi.org/10.3390/sym13112058 - 1 Nov 2021
Cited by 3 | Viewed by 1515
Abstract
In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are [...] Read more.
In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass. Full article
(This article belongs to the Special Issue Mathematical Physics: Topics and Advances)
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17 pages, 5941 KiB  
Article
Electron-Acoustic (Un)Modulated Structures in a Plasma Having (r, q)-Distributed Electrons: Solitons, Super Rogue Waves, and Breathers
by Wedad Albalawi, Rabia Jahangir, Waqas Masood, Sadah A. Alkhateeb and Samir A. El-Tantawy
Symmetry 2021, 13(11), 2029; https://doi.org/10.3390/sym13112029 - 27 Oct 2021
Cited by 19 | Viewed by 1701
Abstract
The propagation of electron-acoustic waves (EAWs) in an unmagnetized plasma, comprising (r,q)-distributed hot electrons, cold inertial electrons, and stationary positive ions, is investigated. Both the unmodulated and modulated EAWs, such as solitary waves, rogue waves (RWs), and breathers [...] Read more.
The propagation of electron-acoustic waves (EAWs) in an unmagnetized plasma, comprising (r,q)-distributed hot electrons, cold inertial electrons, and stationary positive ions, is investigated. Both the unmodulated and modulated EAWs, such as solitary waves, rogue waves (RWs), and breathers are discussed. The Sagdeev potential approach is employed to determine the existence domain of electron acoustic solitary structures and study the perfectly symmetric planar nonlinear unmodulated structures. Moreover, the nonlinear Schrödinger equation (NLSE) is derived and its modulated solutions, including first order RWs (Peregrine soliton), higher-order RWs (super RWs), and breathers (Akhmediev breathers and Kuznetsov–Ma soliton) are presented. The effects of plasma parameters and, in particular, the effects of spectral indices r and q, of distribution functions on the characteristics of both unmodulated and modulated EAWs, are examined in detail. In a limited cases, the (r,q) distribution is compared with Maxwellian and kappa distributions. The present investigation may be beneficial to comprehend and predict the modulated and unmodulated electron acoustic structures in laboratory and space plasmas. Full article
(This article belongs to the Special Issue Mathematical Physics: Topics and Advances)
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13 pages, 310 KiB  
Article
Classification of Integrodifferential C∗-Algebras
by Anton A. Kutsenko
Symmetry 2021, 13(10), 1900; https://doi.org/10.3390/sym13101900 - 9 Oct 2021
Viewed by 1161
Abstract
The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli [...] Read more.
The infinite product of matrices with integer entries, known as a modified Glimm–Bratteli symbol n, is a new, sufficiently simple, and very powerful tool for the characterization of approximately finite-dimensional (AF) algebras. This symbol provides a convenient algebraic representation of the Bratteli diagram for AF algebras in the same way as was previously performed by J. Glimm for more simple uniformly hyperfinite (UHF) algebras. We apply this symbol to characterize integrodifferential algebras. The integrodifferential algebra FN,M is the C∗-algebra generated by the following operators acting on L2([0,1)N→CM): (1) operators of multiplication by bounded matrix-valued functions, (2) finite-difference operators, and (3) integral operators. Most of the operators and their approximations studying in physics belong to these algebras. We give a complete characterization of FN,M. In particular, we show that FN,M does not depend on M, but depends on N. At the same time, it is known that differential algebras HN,M, generated by the operators (1) and (2) only, do not depend on both dimensions N and M; they are all ∗-isomorphic to the universal UHF algebra. We explicitly compute the Glimm–Bratteli symbols (for HN,M, it was already computed earlier) which completely characterize the corresponding AF algebras. This symbol n is an infinite product of matrices with nonnegative integer entries. Roughly speaking, all the symmetries appearing in the approximation of complex infinite-dimensional integrodifferential and differential algebras by finite-dimensional ones are coded by a product of integer matrices. Full article
(This article belongs to the Special Issue Mathematical Physics: Topics and Advances)
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8 pages, 371 KiB  
Article
Minimizing Curvature in Euclidean and Lorentz Geometry
by Martin Tamm
Symmetry 2021, 13(8), 1433; https://doi.org/10.3390/sym13081433 - 5 Aug 2021
Cited by 1 | Viewed by 1354
Abstract
In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The [...] Read more.
In this paper, an interesting symmetry in Euclidean geometry, which is broken in Lorentz geometry, is studied. As it turns out, attempting to minimize the integral of the square of the scalar curvature leads to completely different results in these two cases. The main concern in this paper is about metrics in R3, which are close to being invariant under rotation. If we add a time-axis and let the metric start to rotate with time, it turns out that, in the case of (locally) Euclidean geometry, the (four-dimensional) scalar curvature will increase with the speed of rotation as expected. However, in the case of Lorentz geometry, the curvature will instead initially decrease. In other words, rotating metrics can, in this case, be said to be less curved than non-rotating ones. This phenomenon seems to be very general, but because of the enormous amount of computations required, it will only be proved for a class of metrics which are close to the flat one, and the main (symbolic) computations have been carried out on a computer. Although the results here are purely mathematical, there is also a connection to physics. In general, a deeper understanding of Lorentz geometry is of fundamental importance for many applied problems. Full article
(This article belongs to the Special Issue Mathematical Physics: Topics and Advances)
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