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Symmetry
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Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and...
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Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods

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

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 3594

Special Issue Editors

Dr. Kamyar Hosseini

E-Mail Website
Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
Interests: analytical methods; numerical methods; fractional differential equations; wave propagation; mathematical physics; nonlinear partial differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Evren Hınçal

E-Mail Website
Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Nicosia 99138, Turkey
Interests: applied mathematics; statistics; epidemiology
Special Issues, Collections and Topics in MDPI journals
Dr. Mohammad Mirzazadeh

E-Mail Website
Guest Editor
Department of Engineering Science, Faculty of Technology and Engineering, East of Guilan, University of Guilan, Rasht, ‎Guilan‎, ‎Iran
Interests: theory of solitons; partial differential equations; nonlinear optics; conservation laws; lie groups

Special Issue Information

Dear Colleagues,

As is well known, a wide range of nonlinear phenomena in the real world can be described by nonlinear Schrödinger equations. More precisely, nonlinear Schrödinger equations are capable tools to model a lot of nonlinear phenomena from plasma physics to nonlinear optics. There are different families of nonlinear Schrödinger equations, such as the Sasa–Satsuma equation, Ginzburg–Landau equation, Biswas–Milovic equation, and Gerdjikov–Ivanov equation, which have been the subject of many studies. In recent decades, with the developments of symbolic computation packages, many effective methods such as the Lie symmetry method, the exponential method, and the Kudryashov method have been used to deal with nonlinear Schrödinger equations and their families. The main purpose of the present Special Issue is to address the latest research on new analytical and numerical solutions of nonlinear Schrödinger equations and their families. Original research and review articles are welcome. Potential topics include but are not limited to the following:

  • Lie symmetry analysis of nonlinear Schrödinger equations and their families;
  • Exact soliton solutions of nonlinear Schrödinger equations and their families;
  • Numerical soliton solutions of nonlinear Schrödinger equations and their families;
  • Conservation law of nonlinear Schrödinger equations and their families;
  • Stability analysis of nonlinear Schrödinger equations and their families.

Prof. Dr. Kamyar Hosseini
Prof. Dr. Evren Hınçal
Dr. Mohammad Mirzazadeh
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.

Published Papers (4 papers)

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Research

29 pages, 1528 KiB  
Article
Exact Solutions of M-Fractional Kuralay Equation via Three Analytical Schemes
by Asim Zafar, Muhammad Raheel, Mohamed R. Ali, Zhaidary Myrzakulova, Ahmet Bekir and Ratbay Myrzakulov
Symmetry 2023, 15(10), 1862; https://doi.org/10.3390/sym15101862 - 04 Oct 2023
Cited by 3 | Viewed by 862
Abstract
This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the [...] Read more.
This article concerns new analytical wave solutions of the Kuralay-II equations (K-IIAE and K-IIBE) with exploration of a new definition of the derivative. This model is used in various fields, like nonlinear optics, ferromagnetic materials and optical fibers. For this purpose, the expa function, the extended sinh-Gordon equation expansion scheme, and the generalized Kudryashov schemes were utilized. The resulting solutions are dark, bright, dark-bright, periodic, singular and other kinds of solitons. These results are obtained and also verified by the Mathematica tool. Some of the solutions are explained with 2-D, 3-D and contour plots using the Mathematica tool. The solutions obtained succede the present solutions in the literature. For the first time, the effect of the fractional derivative on the solutions is also shown graphically for this model. The analytical wave solutions are highly desirable as they offer insights into the underlying physics or mathematics of a system and provide a framework for further analysis. The results obtained can also be fruitful for the development of models in the future. The schemes used in this research are effective, easy to apply, and reliably handle other fractional non-linear partial differential equations. Full article
(This article belongs to the Special Issue Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods)
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16 pages, 2427 KiB  
Article
New Three Wave and Periodic Solutions for the Nonlinear (2+1)-Dimensional Burgers Equations
by Waseem Razzaq, Asim Zafar, Abdulaziz Khalid Alsharidi and Mohammed Ahmed Alomair
Symmetry 2023, 15(8), 1573; https://doi.org/10.3390/sym15081573 - 12 Aug 2023
Cited by 1 | Viewed by 698
Abstract
This research paper is about the new three wave, periodic wave and other analytical wave solutions of (2+1)-Dimensional Burgers equations by utilizing Hirota bilinear and extended sinh-Gordon equation expansion (EShGEE) schemes. Achieved solutions are verified and demonstrated by different plots with the use [...] Read more.
This research paper is about the new three wave, periodic wave and other analytical wave solutions of (2+1)-Dimensional Burgers equations by utilizing Hirota bilinear and extended sinh-Gordon equation expansion (EShGEE) schemes. Achieved solutions are verified and demonstrated by different plots with the use of Mathematica 11.01 software. Some of the achieved solutions are also described graphically by two-dimensional, three-dimensional and contour plots. The gained solutions are helpful for the future study of concerned models. Finally, these two schemes are simple, fruitful and reliable to handle the nonlinear PDEs. Full article
(This article belongs to the Special Issue Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods)
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7 pages, 215 KiB  
Article
A Study on the Centroid of a Class of Solvable Lie Algebras
by Demin Yu, Chan Jiang and Jiejing Ma
Symmetry 2023, 15(7), 1451; https://doi.org/10.3390/sym15071451 - 20 Jul 2023
Viewed by 640
Abstract
The centroid of Lie algebra is a basic concept and a necessary tool for studying the structure of Lie algebraic structure. The extended Heisenberg algebra is an important class of solvable Lie algebras. In any Lie algebra, the anti symmetry of Lie operations [...] Read more.
The centroid of Lie algebra is a basic concept and a necessary tool for studying the structure of Lie algebraic structure. The extended Heisenberg algebra is an important class of solvable Lie algebras. In any Lie algebra, the anti symmetry of Lie operations is an important property of Lie algebra. This article investigates the centroids and structures of 2n+2 dimensional extended Heisenberg algebras, where all invertible elements form a group and all elements form a ring. Then, its main research results are extended to infinite dimensional extended Heisenberg algebras. Full article
(This article belongs to the Special Issue Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods)
19 pages, 250 KiB  
Article
Study on Poisson Algebra and Automorphism of a Special Class of Solvable Lie Algebras
by Demin Yu, Chan Jiang and Jiejing Ma
Symmetry 2023, 15(5), 1115; https://doi.org/10.3390/sym15051115 - 19 May 2023
Cited by 1 | Viewed by 826
Abstract
We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀x,y∈g, [...] Read more.
We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀x,y∈g,[x,y]=−[y,x], that is, the operation [,] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by Aut(g)=G3G1G2G3G4G7G8G5, or Aut(g)=G3G1G2G3G4G7G8G5G6, or Aut(g)=G3G1G2G3G4G7G8G5G3, where Gi is a commutative subgroup of Aut(g). We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups. Full article
(This article belongs to the Special Issue Analytic/Numeric Solutions of Schrödinger-Type Equations: Applications of Lie Symmetry and Other Methods)
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