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Symmetry
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Symmetry in the Soliton Theory
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Symmetry in the Soliton Theory

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

Deadline for manuscript submissions: 31 March 2025 | Viewed by 7345

Special Issue Editors

Dr. Baojian Hong

E-Mail Website
Guest Editor
Faculty of Mathematical Physics, Nanjing Institute of Technology, Nanjing 211167, China
Interests: soliton theory; bifurcation theory; numerical analysis; fractional model; exact solution; approximate solution; nonlinear partial differential equations
Prof. Dr. Dianchen Lu

E-Mail Website
Guest Editor
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
Interests: soliton theory; nonlinear system; bifurcation analysis; homotopy analysis method; numerical analysis; mathematical physics; partial differential equations; fractional calculus
Special Issues, Collections and Topics in MDPI journals
Dr. Yusuf Gürefe

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Mersin University, 33110 Mersin, Turkey
Interests: numerical analysis; mathematical physics; partial differential equations; fractional calculus
Special Issues, Collections and Topics in MDPI journals
Dr. Naila Nasreen

E-Mail Website
Guest Editor
School of Mathematical Sciences, Jiangsu University, Zhenjiang 212013, China
Interests: mathematical analysis; fractional differential equations

Special Issue Information

Dear Colleagues,

Soliton is a nonlinear wave which was first discovered by the Scottish scientist Russell in 1834. Solitons appear in almost all branches of mathematical and physical science, such as nonlinear optics, hydrodynamics, chaotic oscillations, ecological and economic systems, plasma physics, chemistry and biochemistry, etc. Until now, it was proven that a large class of nonlinear fractional partial differential equations (NLFPDEs) have the soliton solutions through numerical calculations and theoretical analysis.

It is well known that there is a tight connection between symmetry and soliton solutions. Most of the existing techniques to manage the NLFPDEs and find the exact or approximate soliton solutions are, in essence, a case of symmetry reduction, including nonclassical symmetry and Lie symmetries, etc. Numerous methods have been developed in terms of obtaining the exact, approximate solutions of NLFPDEs, such as Darboux transformation, Bäcklund transformation method, Hirota bilinear method, Jacobi method, homotopy analysis method, variation iteration method, Adomian decomposition method, etc. In addition, many researchers have successfully achieved significant results by using these methods, which are useful in studying nonlinear phenomena including soliton waves.

The aim of this Special Issue is to construct a platform to collect new results about solitary waves, soliton solutions, and achievements related to soliton theory in mathematics, physics, and science fields. This Special Issue will also focus on studying the behavior and properties of the obtained solutions. Among others, papers on the above topics are welcome. This Special Issue will be focused on but not limited to:

Topics:

  • Applications of soliton theory in mathematical and physical differential equations;
  • Applications of fractional calculus in science and engineering;
  • Review performance of mathematical models with fractional differential and integral equations;
  • Numerical and analytical methods for fractional nonlinear differential equations;
  • Recent advances in fractional calculus;
  • Some new definitions and properties about the fractional operators;
  • New numerical schemes for fractional partial differential equations;
  • Exact solutions to some nonlinear mathematical physics problems.

Dr. Baojian Hong
Prof. Dr. Dianchen Lu
Dr. Yusuf Gürefe
Dr. Naila Nasreen
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

  • soliton
  • solitary waves
  • dynamical systems theory
  • bifurcation analysis
  • exact solutions
  • approximate solutions
  • numerical methods
  • fractional partial differential equations
  • fractional calculus
  • Darboux transformation
  • Bäcklund transformation
  • homotopy analysis method
  • homotopy perturbation method
  • lie symmetry
  • laplace transform

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Published Papers (4 papers)

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Research

22 pages, 1740 KiB  
Article
Analyzing Dynamics: Lie Symmetry Approach to Bifurcation, Chaos, Multistability, and Solitons in Extended (3 + 1)-Dimensional Wave Equation
by Muhammad Bilal Riaz, Adil Jhangeer, Faisal Z. Duraihem and Jan Martinovic
Symmetry 2024, 16(5), 608; https://doi.org/10.3390/sym16050608 - 14 May 2024
Cited by 3 | Viewed by 1081
Abstract
The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries [...] Read more.
The examination of new (3 + 1)-dimensional wave equations is undertaken in this study. Initially, the identification of the Lie symmetries of the model is carried out through the utilization of the Lie symmetry approach. The commutator and adjoint table of the symmetries are presented. Subsequently, the model under discussion is transformed into an ordinary differential equation using these symmetries. The construction of several bright, kink, and dark solitons for the suggested equation is then achieved through the utilization of the new auxiliary equation method. Subsequently, an analysis of the dynamical nature of the model is conducted, encompassing various angles such as bifurcation, chaos, and sensitivity. Bifurcation occurs at critical points within a dynamical system, accompanied by the application of an outward force, which unveils the emergence of chaotic phenomena. Two-dimensional plots, time plots, multistability, and Lyapunov exponents are presented to illustrate these chaotic behaviors. Furthermore, the sensitivity of the investigated model is executed utilizing the Runge–Kutta method. This analysis confirms that the stability of the solution is minimally affected by small changes in initial conditions. The attained outcomes show the effectiveness of the presented methods in evaluating solitons of multiple nonlinear models. Full article
(This article belongs to the Special Issue Symmetry in the Soliton Theory)
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10 pages, 2867 KiB  
Article
Non-Contact Impact Source Localization in Composite Symmetry Panels Based on A0 Mode of Lamb Waves
by Ziping Wang, Jiazhen Zhang, Hangrui Cui, Rahim Gorgin and Yang Zhang
Symmetry 2023, 15(10), 1836; https://doi.org/10.3390/sym15101836 - 28 Sep 2023
Viewed by 1476
Abstract
Traditional methods for detecting damage in engineering structures often use offline static damage detection. To enable the real-time and precise identification of dynamic damage while maintaining symmetry in engineering structures, this study primarily concentrates on isotropic plate structures widely employed in engineering. Moreover, [...] Read more.
Traditional methods for detecting damage in engineering structures often use offline static damage detection. To enable the real-time and precise identification of dynamic damage while maintaining symmetry in engineering structures, this study primarily concentrates on isotropic plate structures widely employed in engineering. Moreover, fiberglass board composite plates were opted as a specific research object. By utilizing the weak S0 mode signals generated by low-frequency ultrasonic Lamb waves, the non-stationary A0 wave signals in the composite symmetry plate structure are collected using the non-contact SLDV (Scanning Laser Doppler Vibrometer) technique. The frequency characteristic parameters in the vibration signals are obtained through HHT (Hilbert–Huang Transform) analysis, followed by filtering and noise reduction. Finally, the circular trajectory intersection method is employed to accurately locate dynamic damage sources in plate structures with different material properties, thereby validating the positioning effect of contact sensors in detecting impacts caused by random impulses. Full article
(This article belongs to the Special Issue Symmetry in the Soliton Theory)
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13 pages, 1366 KiB  
Article
Discovery of New Exact Wave Solutions to the M-Fractional Complex Three Coupled Maccari’s System by Sardar Sub-Equation Scheme
by Abdulaziz Khalid Alsharidi and Ahmet Bekir
Symmetry 2023, 15(8), 1567; https://doi.org/10.3390/sym15081567 - 11 Aug 2023
Cited by 18 | Viewed by 1115
Abstract
In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many [...] Read more.
In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs). Full article
(This article belongs to the Special Issue Symmetry in the Soliton Theory)
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17 pages, 333 KiB  
Article
Inverse Scattering and Soliton Solutions of Nonlocal Complex Reverse-Spacetime Modified Korteweg-de Vries Hierarchies
by Liming Ling and Wen-Xiu Ma
Symmetry 2021, 13(3), 512; https://doi.org/10.3390/sym13030512 - 21 Mar 2021
Cited by 27 | Viewed by 2477
Abstract
This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated [...] Read more.
This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated Riemann-Hilbert problems. A formulation of solutions to specific Riemann-Hilbert problems, with the jump matrix being the identity matrix, is established, where eigenvalues could equal adjoint eigenvalues, and thus N-soliton solutions to the nonlocal complex reverse-spacetime mKdV hierarchies are obtained from the reflectionless transforms. Full article
(This article belongs to the Special Issue Symmetry in the Soliton Theory)
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