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Symmetry
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New Trends on the Mathematical Models and Solitons Arising in...
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New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition

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

Deadline for manuscript submissions: 30 April 2026 | Viewed by 3046

Special Issue Editors

Prof. Dr. Haci Mehmet Baskonus

E-Mail Website
Guest Editor
Faculty of Education, Harran University, 63050 Sanliurfa, Turkey
Interests: applied science; partial differential equations; soliton theory; analytical methods; numerical methods
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Shuli Mei

E-Mail Website
Guest Editor
College of Information and Electrical Engineering, China Agricultural University, Beijing 100083, China
Interests: artificial intelligence and image processing; deep learning based on sparse representation; wavelet analysis
Dr. Md Nurul Raihen

E-Mail Website
Guest Editor Assistant
Department of Mathematics and Statistics, University of Toledo, Toledo, OH, USA
Interests: mathematical modelling

Special Issue Information

Dear Colleagues, 

The essence of mathematical tools for exemplifying practical problems in daily life is as old humanity itself. Mathematical models in science and technology have recently attracted increasing research attention, with the aim of understanding, describing, and predicting the future behavior of natural phenomena. Recent studies on fractional calculus have been particularly popular among researchers due to their favorable properties when analyzing real-world models associated with properties such as anomalous diffusion, non-Markovian processes, random walk, long range, and, most importantly, heterogeneous behaviors. The development of local differential operators, along with power law settings and non-local differential operators, has been suggested to more accurately replicate these processes. The complexities of nature have driven mathematicians and physicists to derive increasingly sophisticated and scientific mathematical operators to accurately replicate and capture pragmatic realities.

Mathematical physics plays a vital role in studying the determinants and distribution of solitons. It enables the identification of wave distributions across many fields of nonlinear science, and many experts have recently focused their efforts on this area. Such studies may also provide foundations for developing public policy, informing regulatory decisions on engineering problems, and evaluating both existing and emerging perspectives. Major areas of mathematical physics research using mathematical models include symmetry, transmission, outbreak investigation, and epidemiological problems.

This Special Issue is devoted to collecting new results, spanning from theory to practice, with the aim of developing innovative technological tools. Topics of interest include, but are not limited to, the following topics:

  • Theoretical, computational, and experimental aspects of mathematical physics models;
  • Performance evaluation of mathematical models with fractional differential and integral equations;
  • Assessment of models involving different types of fractional operators;
  • Validation of models with fractal–fractional differential and integral operators;
  • Effects of new fractal differential and integral operators in modeling applications, such as epidemiological diseases, mathematical physics, soliton theory, etc.

Prof. Dr. Haci Mehmet Baskonus
Dr. Shuli Mei
Guest Editors

Dr. Md Nurul Raihen
Guest Editor Assistant

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 250 words) can be sent to the Editorial Office for assessment.

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

  • mathematical physics
  • partial differential equations
  • epidemic models
  • basic reproduction number
  • fractional differential equations
  • dynamical systems
  • stability analysis
  • bifurcation
  • optimal control

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

Related Special Issue

Published Papers (5 papers)

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Research

14 pages, 1105 KB  
Article
Exact Soliton Structures and Modulation Instability in Extended Kadomtsev–Petviashvili–Boussinesq Equation
by Nadiyah Hussain Alharthi, Rubayyi T. Alqahtani and Melike Kaplan
Symmetry 2026, 18(4), 626; https://doi.org/10.3390/sym18040626 - 8 Apr 2026
Viewed by 308
Abstract
In this study, we consider an extended form of the Kadomtsev–Petviashvili–Boussinesq equation motivated by wave propagation phenomena in dissipative media. The primary aim of this work is to construct exact analytical solutions and clarify the types of nonlinear wave structure admitted by the [...] Read more.
In this study, we consider an extended form of the Kadomtsev–Petviashvili–Boussinesq equation motivated by wave propagation phenomena in dissipative media. The primary aim of this work is to construct exact analytical solutions and clarify the types of nonlinear wave structure admitted by the considered model. For this purpose, the Riccati equation expansion method is applied for the first time within this framework. This method allows us to obtain several distinct families of solitary wave solutions whose qualitative behaviors and physical characteristics are illustrated through graphical representations. In addition, modulation instability analysis is carried out to assess the stability of continuous wave solutions and further elucidate the underlying nonlinear dynamics of the system. Full article
(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition)
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19 pages, 845 KB  
Article
Delay-Induced Stability Transitions and Hopf Bifurcation in a Model of Cumulative Hot Pollutant Concentration
by Dipesh, Pankaj Kumar, Hacer Ozden Ayna and Ismail Naci Cangul
Symmetry 2026, 18(3), 404; https://doi.org/10.3390/sym18030404 - 25 Feb 2026
Viewed by 320
Abstract
The present work provides a detailed discussion of the dynamical behavior of the delay-induced model of cumulative concentration of hot pollutants, including the contribution of the time-delay parameter to the system’s stability. Analytical results indicate that time delay is a bifurcation mechanism that [...] Read more.
The present work provides a detailed discussion of the dynamical behavior of the delay-induced model of cumulative concentration of hot pollutants, including the contribution of the time-delay parameter to the system’s stability. Analytical results indicate that time delay is a bifurcation mechanism that leads to a critical threshold, at which a steady state loses asymptotic stability and a Hopf bifurcation occurs. The directional analysis is carried out to further explain the behavior of the system in the neighborhood of this transition, and this offers some understanding of the nature and stability of the resulting periodic solutions, as well as the qualitative evolution. Numerical simulations are done on representative parameter values to support the theoretical results. Comprehensively, the findings reveal the strong dependence of the accumulation processes of pollutants on the effects of time delays and the significance of considering the temporal lags in environmental modeling. The study provides a viable analytical and numerical system of interpreting transitions caused by delays in pollutant concentration systems. Full article
(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition)
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35 pages, 901 KB  
Article
Modeling the Dynamic of Herpes Simplex Virus II Incorporating Voluntary Laboratory Test and Medical Treatment
by Abdulfatai Atte Momoh, Salaudeen Yusuf, Goni Umar Modu, Ali Inalegwu Michael, Idris Ahmed and Jessada Tariboon
Symmetry 2026, 18(1), 86; https://doi.org/10.3390/sym18010086 - 3 Jan 2026
Viewed by 578
Abstract
This study develops a mathematical model to investigate the transmission dynamics of HSV-II within the framework of symmetry in dynamical systems. The basic reproduction number (R0HSV<1) of the model was determined using the next generation [...] Read more.
This study develops a mathematical model to investigate the transmission dynamics of HSV-II within the framework of symmetry in dynamical systems. The basic reproduction number (R0HSV<1) of the model was determined using the next generation method (NGM). The stability of the disease-free equilibrium point was also investigated using the Routh–Hurwitz Criterion and was found to be locally asymptotically stable (LAS) when R0HSV<1 but not globally asymptotically stable (GAS). To help ensure that the control variables were included correctly, sensitivity analysis was performed on the fundamental reproduction number parameters. Four control variables were applied for the model: HSV-II vaccination, effective condom use, laboratory test, and treatment. The optimality system was solved using Pontryagin’s maximum principle (PMP) to establish the optimal control strategy for combating the spread of the disease. Numerical solution was obtained by using the forward-backward Runge–Kutta fourth-order approach. The most effective approach to help eradicate HSV-II disease in the system is to combine the HSV-II vaccine, effective condom use, laboratory testing, and HSV therapy (strategy D). Full article
(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition)
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23 pages, 1798 KB  
Article
New Insights into Delay-Impulsive Interactions and Stability in Almost Periodic Cohen–Grossberg Neural Networks
by Münevver Tuz and Gülden Altay Suroğlu
Symmetry 2025, 17(12), 2063; https://doi.org/10.3390/sym17122063 - 2 Dec 2025
Viewed by 476
Abstract
This paper investigates the existence and global exponential stability of almost periodic solutions in a class of impulsive Cohen–Grossberg-type bidirectional associative memory (BAM) neural networks with time-varying delays. Real neural systems often experience sudden perturbations and nonuniform temporal interactions, leading to complex oscillatory [...] Read more.
This paper investigates the existence and global exponential stability of almost periodic solutions in a class of impulsive Cohen–Grossberg-type bidirectional associative memory (BAM) neural networks with time-varying delays. Real neural systems often experience sudden perturbations and nonuniform temporal interactions, leading to complex oscillatory behaviors. To capture these effects, a new impulsive Cohen–Grossberg BAM model is developed that integrates both delays and impulsive influences within a unified framework. Using the theory of almost periodic functions, fixed point methods, and impulsive differential inequalities, new sufficient conditions are derived for the existence and stability of almost periodic solutions. A Lyapunov functional combined with a generalized Gronwall-type inequality provides rigorous global exponential stability criteria. Numerical simulations confirm the theoretical analysis. The results extend existing studies and offer new insights into how delay and impulsive factors jointly shape the stability and dynamics of hybrid neural systems, contributing to the design of robust and delay-tolerant neural architectures. Full article
(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition)
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17 pages, 680 KB  
Article
Stochastic SO(3) Lie Method for Correlation Flow
by Yasemen Ucan and Melike Bildirici
Symmetry 2025, 17(10), 1778; https://doi.org/10.3390/sym17101778 - 21 Oct 2025
Viewed by 617
Abstract
It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. [...] Read more.
It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to solve mathematical models. This paper introduces a novel methodological framework based on the SO(3) Lie method to estimate time-dependent correlation matrices (correlation flows) among three variables that have chaotic, entropy, and fractal characteristics, from 11 April 2011 to 31 December 2024 for daily data; from 10 April 2011 to 29 December 2024 for weekly data; and from April 2011 to December 2024 for monthly data. So, it develops the stochastic SO(2) Lie method into the SO(3) Lie method that aims to obtain the correlation flow for three variables with chaotic, entropy, and fractal structure. The results were obtained at three stages. Firstly, we applied entropy (Shannon, Rényi, Tsallis, Higuchi) measures, Kolmogorov–Sinai complexity, Hurst exponents, rescaled range tests, and Lyapunov exponent methods. The results of the Lyapunov exponents (Wolf, Rosenstein’s Method, Kantz’s Method) and entropy methods, and KSC found evidence of chaos, entropy, and complexity. Secondly, the stochastic differential equations which depend on S2 (SO(3) Lie group) and Lie algebra to obtain the correlation flows are explained. The resulting equation was numerically solved. The correlation flows were obtained by using the defined covariance flow transformation. Finally, we ran the robustness check. Accordingly, our robustness check results showed the SO(3) Lie method produced more effective results than the standard and Spearman correlation and covariance matrix. And, this method found lower RMSE and MAPE values, greater stability, and better forecast accuracy. For daily data, the Lie method found RMSE = 0.63, MAE = 0.43, and MAPE = 5.04, RMSE = 0.78, MAE = 0.56, and MAPE = 70.28 for weekly data, and RMSE = 0.081, MAE = 0.06, and MAPE = 7.39 for monthly data. These findings indicate that the SO(3) framework provides greater robustness, lower errors, and improved forecasting performance, as well as higher sensitivity to nonlinear transitions compared to standard correlation measures. By embedding time-dependent correlation matrix into a Lie group framework inspired by physics, this paper highlights the deep structural parallels between financial markets and complex physical systems. Full article
(This article belongs to the Special Issue New Trends on the Mathematical Models and Solitons Arising in Real-World Problems, 3rd Edition)
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