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Symmetry
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Partial Differential Equations and Their Applications in Nonlinear Optics
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Partial Differential Equations and Their Applications in Nonlinear Optics

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

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 10817

Special Issue Editors

Dr. Mohamed S. Osman

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
Interests: solitons; mathematical photonics; computational and applied mathematics; mathematical physics
Prof. Dr. Abdul-Majid Wazwaz

E-Mail Website
Guest Editor
Department of Mathematics, Saint Xavier University, Chicago, IL 60655, USA
Interests: solitons and compactons; N-solitons; ODE and PDE; numerical analysis; mathematical physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Optical solitons are commonly known in all-optical ultrafast switching systems and the protracted communication after being conceived and validated practically. As a result, it has piqued the interest of a majority of nonlinear optics researchers. In diverse sectors, such as applied sciences, mathematical photonics, nonlinear wave propagation, and plasma physics, partial differential equations (PDEs) can be employed to quantify a plethora of dynamical systems. The quest for their numerical and analytical solutions provides the most insightful discussion about these equations and the nonlinear physical phenomena they are linked to.

In interacting systems, symmetry is frequently utilized to establish conservation principles and to create forbidden/allowed transitions. Symmetries are commonly employed in the field of nonlinear optics to identify whether a specific nonlinear process is permitted or prohibited based on the point group of the medium.

Our primary driving force behind this Special Issue is to look for various wave shapes for the PDEs' achieved solutions. To accomplish our objective, we use different analytical or numerical techniques to identify several analytical (or numerical) solutions, such as solitary, kink-soliton, anti-kink soliton, shock, dark-soliton, bright-soliton, and elliptic wave solutions. Topics of interest include (but are not limited to): nonlinear optics, wave transmission and propagation in homogeneous and inhomogeneous materials, and optical properties of materials.

Dr. Mohamed S. Osman
Prof. Dr. Abdul-Majid Wazwaz
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

  • nonlinear optics
  • analytical and numerical techniques
  • wave propagation
  • soliton theory
  • PDEs

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Published Papers (5 papers)

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Research

16 pages, 972 KiB  
Article
Dynamics and Soliton Propagation in a Modified Oskolkov Equation: Phase Plot Insights
by Muhammad Bilal Riaz, Adil Jhangeer, Jan Martinovic and Syeda Sarwat Kazmi
Symmetry 2023, 15(12), 2171; https://doi.org/10.3390/sym15122171 - 7 Dec 2023
Cited by 7 | Viewed by 1495
Abstract
This study explores the modified Oskolkov equation, which depicts the behavior of the incompressible viscoelastic Kelvin–Voigt fluid. The primary focus of this research lies in several key areas. Firstly, the Lie symmetries of the considered equation are identified. These symmetries are utilized to [...] Read more.
This study explores the modified Oskolkov equation, which depicts the behavior of the incompressible viscoelastic Kelvin–Voigt fluid. The primary focus of this research lies in several key areas. Firstly, the Lie symmetries of the considered equation are identified. These symmetries are utilized to transform the discussed model into an ordinary differential equation. Analytical solutions are subsequently derived using the new auxiliary equation technique. Next, a comprehensive analysis of the equation’s dynamic nature is undertaken from multiple aspects. Bifurcation is carried out at fixed points within the system, and chaotic behavior is unveiled by introducing an external force to the dynamic system. Various tools, including 3D and 2D phase plots, time series, Poincaré maps, and multistability analysis, are employed to identify the chaotic nature of the system. Furthermore, the sensitivity of the model is explored across diverse initial conditions. In general, comprehending the dynamic characteristics of systems holds immense significance in forecasting outcomes and innovating new technologies. Full article
(This article belongs to the Special Issue Partial Differential Equations and Their Applications in Nonlinear Optics)
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19 pages, 1216 KiB  
Article
The Analysis of Bifurcation, Quasi-Periodic and Solitons Patterns to the New Form of the Generalized q-Deformed Sinh-Gordon Equation
by Syeda Sarwat Kazmi, Adil Jhangeer, Nauman Raza, Haifa I. Alrebdi, Abdel-Haleem Abdel-Aty and Hichem Eleuch
Symmetry 2023, 15(7), 1324; https://doi.org/10.3390/sym15071324 - 28 Jun 2023
Cited by 52 | Viewed by 2201
Abstract
In this manuscript, a new form of the generalized q-deformed Sinh-Gordon equation is investigated which could model physical systems with broken symmetries and to incorporate phenomena involving amplification or dissipation. The proposed model is explored based on the Lie symmetry approach. Using [...] Read more.
In this manuscript, a new form of the generalized q-deformed Sinh-Gordon equation is investigated which could model physical systems with broken symmetries and to incorporate phenomena involving amplification or dissipation. The proposed model is explored based on the Lie symmetry approach. Using similarity reduction, the partial differential equation is transformed into an ordinary differential equation. By employing the generalized auxiliary equation approach, precise results for the derived equation are obtained. The solutions are graphically depicted as 3D, 2D, and contour plots. Furthermore, the qualitative analysis of the considered model is investigated by employing the concepts of bifurcation and chaos. The phase profiles are displayed for different sets of the parameters. Additionally, by applying an external periodic strength, quasi-periodic and chaotic behaviors are documented. Various tools for detecting chaos are discussed, including 3D and 2D phase patterns, time series, and Poincaré maps. Additionally, a sensitivity analysis is conducted for various initial conditions. The obtained findings are unique and indicate the viability and efficacy of the suggested strategies for evaluating soliton solutions and phase illustrations for various nonlinear models. Full article
(This article belongs to the Special Issue Partial Differential Equations and Their Applications in Nonlinear Optics)
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10 pages, 1428 KiB  
Article
The Solitary Solutions for the Stochastic Jimbo–Miwa Equation Perturbed by White Noise
by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed
Symmetry 2023, 15(6), 1153; https://doi.org/10.3390/sym15061153 - 26 May 2023
Cited by 16 | Viewed by 1705
Abstract
We study the (3+1)-dimensional stochastic Jimbo–Miwa (SJM) equation induced by multiplicative white noise in the Itô sense. We employ the Riccati equation mapping and He’s semi-inverse techniques to provide trigonometric, hyperbolic, and rational function solutions of SJME. Due to the applications of the [...] Read more.
We study the (3+1)-dimensional stochastic Jimbo–Miwa (SJM) equation induced by multiplicative white noise in the Itô sense. We employ the Riccati equation mapping and He’s semi-inverse techniques to provide trigonometric, hyperbolic, and rational function solutions of SJME. Due to the applications of the Jimbo–Miwa equation in ocean studies and other disciplines, the acquired solutions may explain numerous fascinating physical phenomena. Using a variety of 2D and 3D diagrams, we illustrate how white noise influences the analytical solutions of SJM equation. We deduce that the noise destroys the symmetry of the solutions of SJM equation and stabilizes them at zero. Full article
(This article belongs to the Special Issue Partial Differential Equations and Their Applications in Nonlinear Optics)
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17 pages, 1664 KiB  
Article
Nonlinear Wave Propagation for a Strain Wave Equation of a Flexible Rod with Finite Deformation
by Aqilah Aljuaidan, Mamdouh Elbrolosy and Adel Elmandouh
Symmetry 2023, 15(3), 650; https://doi.org/10.3390/sym15030650 - 5 Mar 2023
Cited by 11 | Viewed by 2387
Abstract
The present work is attentive to studying the qualitative analysis for a nonlinear strain wave equation describing the finite deformation elastic rod taking into account transverse inertia, and shearing strain. The strain wave equation is rewritten as a dynamic system by applying a [...] Read more.
The present work is attentive to studying the qualitative analysis for a nonlinear strain wave equation describing the finite deformation elastic rod taking into account transverse inertia, and shearing strain. The strain wave equation is rewritten as a dynamic system by applying a particular transformation. The bifurcation of the solutions is examined, and the phase portrait is depicted. Based on the bifurcation constraints, the integration of the first integral of the dynamic system along specified intervals leads to real wave solutions. We prove the strain wave equation has periodic, solitary wave solutions and does not possess kink (or anti-kink) solutions. In addition, the set of discovered solutions contains Jacobi-elliptic, trigonometric, and hyperbolic functions. This model contains many kinds of solutions, which are always symmetric or anti-symmetric in space. We study how the change in the physical parameters impacts the solutions that are found. Numerically, the behavior of the strain wave for the elastic rod is examined when particular periodic forces act on it, and moreover, we clarify the existence of quasi-periodic motion. To clarify these solutions, we present a 3D representation of them and the corresponding phase orbit. Full article
(This article belongs to the Special Issue Partial Differential Equations and Their Applications in Nonlinear Optics)
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21 pages, 8799 KiB  
Article
Stochastic Solitons in Birefringent Fibers for Biswas–Arshed Equation with Multiplicative White Noise via Itô Calculus by Modified Extended Mapping Method
by Yazid Alhojilan, Hamdy M. Ahmed and Wafaa B. Rabie
Symmetry 2023, 15(1), 207; https://doi.org/10.3390/sym15010207 - 10 Jan 2023
Cited by 45 | Viewed by 1961
Abstract
Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons and other exact stochastic solutions through birefringent fibers for the Biswas–Arshed equation with multiplicative white noise using the modified extended mapping method. [...] Read more.
Stochastic partial differential equations have wide applications in various fields of science and engineering. This paper addresses the optical stochastic solitons and other exact stochastic solutions through birefringent fibers for the Biswas–Arshed equation with multiplicative white noise using the modified extended mapping method. This model contains many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Stochastic bright soliton solutions, stochastic dark soliton solutions, stochastic combo bright–dark soliton solutions, stochastic combo singular-bright soliton solutions, stochastic singular soliton solutions, stochastic periodic solutions, stochastic rational solutions, stochastic Weierstrass elliptic doubly periodic solutions, and stochastic Jacobi elliptic function solutions are extracted. The constraints on the parameters are considered to guarantee the existence of these stochastic solutions. Furthermore, some of the selected solutions are described graphically to demonstrate the physical nature of the obtained solutions. Full article
(This article belongs to the Special Issue Partial Differential Equations and Their Applications in Nonlinear Optics)
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