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Symmetry
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Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves
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Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves

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

Deadline for manuscript submissions: closed (15 February 2024) | Viewed by 5893

Special Issue Editors

Prof. Dr. Sheng Zhang

E-Mail Website
Guest Editor
School of Mathematical Sciences, Bohai University, Jinzhou 121013, China
Interests: soliton and integrable system; mathematical mechanization and mathematical physics; fractional differential equation and its applications
Dr. Bo Xu

E-Mail Website
Guest Editor
School of Educational Sciences, Bohai University, Jinzhou 121013, China
Interests: soliton and integrable system; fractional calculus

Special Issue Information

Dear Colleagues,

Complex nonlinear phenomena such as solitons and rogue waves are a feature of countless systems worldwide. Nonlinear waves, including solitons, rogue waves, fractals and chaos, often exhibit symmetrical and asymmetrical characteristics, and are always associated with some solvable nonlinear mathematical and physical equations. Developing strategies to solve such equations and obtain exact solutions to explain nonlinear phenomena is very important, and has been a subject of concern for mathematics and physicists for many years.

Solvable equations, especially those with soliton solutions, are often related to the integrable properties of nonlinear evolution equations, such as Lax integrability, Liouville integrability, Painlevé integrability, Calogero integrability, inverse scattering integrability, etc. In addition, many analytical methods for constructing exact solutions of nonlinear evolution equations have been developed in the research field, such as Bäcklund transformation, Darboux transformation, inverse scattering method, Riemann–Hilbert approach, Dbar dressing method, Lie symmetry method, Painlevé analysis, Hirota bilinear method, exp-function method, sub-equation method, negative power expansion method, dynamical system method, traveling wave method, etc. Some of these use symmetry theory and symmetry properties. The use of these methods to obtain the exact solutions of nonlinear evolution equations and study the symmetric/asymmetric structures of solutions has research significance and scientific value.

The purpose of this Special Issue is to provide academic and industrial communities with a platform to discuss the symmetry/asymmetry issues related to solitons, nonlinear waves and integrable systems, and to share their research results. We will address the latest developments in nonlinear mathematical physics. Only high-quality articles describing previously unpublished, original, state-of-the-art research, which are not currently under review by a conference or journal, will be considered.

Prof. Dr. Sheng Zhang
Dr. Bo Xu
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

  • integrable system
  • soliton
  • nonlinear wave
  • Bäcklund and Darboux transformation
  • inverse scattering method
  • Lie symmetry method
  • Painlevé analysis
  • Hirota bilinear method
  • exp-function method
  • sub-equation method

Published Papers (6 papers)

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Research

14 pages, 2730 KiB  
Article
A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods
by Abdulrahman B. M. Alzahrani, Mohamed A. Abdoon, Mohamed Elbadri, Mohammed Berir and Diaa Eldin Elgezouli
Symmetry 2023, 15(11), 1991; https://doi.org/10.3390/sym15111991 - 28 Oct 2023
Cited by 5 | Viewed by 710
Abstract
This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested [...] Read more.
This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge–Kutta’s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines. Full article
(This article belongs to the Special Issue Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves)
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10 pages, 276 KiB  
Article
Periodic Solution Problems for a Class of Hebbian-Type Networks with Time-Varying Delays
by Mei Xu, Honghui Yin and Bo Du
Symmetry 2023, 15(11), 1985; https://doi.org/10.3390/sym15111985 - 27 Oct 2023
Viewed by 548
Abstract
By using Gronwall’s inequality and coincidence degree theory, the sufficient conditions of the globally exponential stability and existence are given for a Hebbian-type network with time-varying delays. The periodic behavior phenomenon is one of the hot topics in network systems research, from which [...] Read more.
By using Gronwall’s inequality and coincidence degree theory, the sufficient conditions of the globally exponential stability and existence are given for a Hebbian-type network with time-varying delays. The periodic behavior phenomenon is one of the hot topics in network systems research, from which we can discover the symmetric characteristics of certain neurons. The main theorems in the present paper are illustrated using a numerical example. Full article
(This article belongs to the Special Issue Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves)
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66 pages, 866 KiB  
Article
Riemann–Hilbert Problems, Polynomial Lax Pairs, Integrable Equations and Their Soliton Solutions
by Vladimir Stefanov Gerdjikov and Aleksander Aleksiev Stefanov
Symmetry 2023, 15(10), 1933; https://doi.org/10.3390/sym15101933 - 18 Oct 2023
Cited by 2 | Viewed by 1110
Abstract
The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: (1) Lax representation [L,M]=0; (2) construction of fundamental analytic solutions (FAS); (3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem [...] Read more.
The standard approach to integrable nonlinear evolution equations (NLEE) usually uses the following steps: (1) Lax representation [L,M]=0; (2) construction of fundamental analytic solutions (FAS); (3) reducing the inverse scattering problem (ISP) to a Riemann-Hilbert problem (RHP) ξ+(x,t,λ)=ξ−(x,t,λ)G(x,tλ) on a contour Γ with sewing function G(x,t,λ); (4) soliton solutions and possible applications. Step 1 involves several assumptions: the choice of the Lie algebra g underlying L, as well as its dependence on the spectral parameter, typically linear or quadratic in λ. In the present paper, we propose another approach that substantially extends the classes of integrable NLEE. Its first advantage is that one can effectively use any polynomial dependence in both L and M. We use the following steps: (A) Start with canonically normalized RHP with predefined contour Γ; (B) Specify the x and t dependence of the sewing function defined on Γ; (C) Introduce convenient parametrization for the solutions ξ±(x,t,λ) of the RHP and formulate the Lax pair and the nonlinear evolution equations (NLEE); (D) use Zakharov–Shabat dressing method to derive their soliton solutions. This requires correctly taking into account the symmetries of the RHP. (E) Define the resolvent of the Lax operator and use it to analyze its spectral properties. Full article
(This article belongs to the Special Issue Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves)
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11 pages, 2887 KiB  
Article
Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations
by Sheng Zhang, Feng Zhu and Bo Xu
Symmetry 2023, 15(6), 1211; https://doi.org/10.3390/sym15061211 - 06 Jun 2023
Cited by 5 | Viewed by 826
Abstract
The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are [...] Read more.
The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case. Full article
(This article belongs to the Special Issue Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves)
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10 pages, 4179 KiB  
Article
Exact Solutions for Coupled Variable Coefficient KdV Equation via Quadratic Jacobi’s Elliptic Function Expansion
by Xiaohua Zeng, Xiling Wu, Changzhou Liang, Chiping Yuan and Jieping Cai
Symmetry 2023, 15(5), 1021; https://doi.org/10.3390/sym15051021 - 04 May 2023
Cited by 2 | Viewed by 1001
Abstract
The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic [...] Read more.
The exact traveling wave solutions to coupled KdV equations with variable coefficients are obtained via the use of quadratic Jacobi’s elliptic function expansion. The presented coupled KdV equations have a more general form than those studied in the literature. Nine couples of quadratic Jacobi’s elliptic function solutions are found. Each couple of traveling wave solutions is symmetric in mathematical form. In the limit cases m→1, these periodic solutions degenerate as the corresponding soliton solutions. After the simple parameter substitution, the trigonometric function solutions are also obtained. Full article
(This article belongs to the Special Issue Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves)
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20 pages, 433 KiB  
Article
Completeness of Bethe Ansatz for Gaudin Models with 𝔤𝔩(1|1) Symmetry and Diagonal Twists
by Kang Lu
Symmetry 2023, 15(1), 9; https://doi.org/10.3390/sym15010009 - 21 Dec 2022
Viewed by 776
Abstract
We studied the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules. Namely, we gave an explicit [...] Read more.
We studied the Gaudin models with gl(1|1) symmetry that are twisted by a diagonal matrix and defined on tensor products of polynomial evaluation gl(1|1)[t]-modules. Namely, we gave an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation gl(1|1)[t]-modules and showed that a bijection exists between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also gave dimensions of the generalized eigenspaces. Full article
(This article belongs to the Special Issue Symmetries/Asymmetries in Mathematical Physics: Integrable Systems, Solitons and Nonlinear Waves)
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