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Symmetry
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Symmetry in Nonlinear Partial Differential Equations and Rogue Waves
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Symmetry in Nonlinear Partial Differential Equations and Rogue Waves

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

Deadline for manuscript submissions: 31 March 2025 | Viewed by 5411

Special Issue Editor

Dr. Natanael Karjanto

E-Mail Website
Guest Editor
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Gyeonggi-do, Korea
Interests: nonlinear wave phenomena; partial differential equations; soliton theory; mathematics education
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Study of rogue waves has assisted many people not only to better understand natural phenomena but also to progress in the knowledge of nonlinear waves in general. The nonlinear Schrödinger (NLS) equation and its exact analytical solutions have been used as a mathematical model and for prototypes of rogue waves, also known as freak or extreme waves. Although the family of solitons on nonvanishing backgrounds was discovered in the 1970s and 1980s, it was not until the 2010s that experimental observations confirmed those theoretical predictions. In this Special Issue of Symmetry, we seek contributions from researchers on the topic of Symmetry in Rogue Waves. All types of contribution are welcome, including modeling, mathematical, physical, numerical, statistical, and experimental.

Dr. Natanael Karjanto
Guest Editor

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 waves
  • coherent structures
  • the nonlinear Schrödinger (NLS) equation
  • soliton on constant background
  • breathers
  • freak waves
  • rogue waves
  • extreme waves
  • wave packets
  • modulational instability
  • phase singularity
  • wavefront dislocation
  • variational formulation
  • maximum temporal amplitude
  • amplification factor
  • dissipation
  • surface gravity waves
  • nonlinear optics
  • superconductivity
  • plasma physics
  • Bose–Einstein condensates

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Published Papers (4 papers)

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Research

31 pages, 12862 KiB  
Article
Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques
by Miguel Vivas-Cortez, Maham Nageen, Muhammad Abbas and Moataz Alosaimi
Symmetry 2024, 16(8), 1085; https://doi.org/10.3390/sym16081085 - 21 Aug 2024
Viewed by 871
Abstract
Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to [...] Read more.
Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to create an ordinary differential equation form for the nonlinear partial differential equation. This work presents the construction of novel trigonometric, hyperbolic and Jacobi elliptic functions to the nonlinear Klein–Gordon equation using the mapping and generalized Riccati equation mapping methods. In the fields of fluid motion, plasma science, and classical physics the nonlinear Klein–Gordon equation is frequently used to identify of a wide range of interesting physical occurrences. It is considered that the obtained results have not been established in prior study via these methods. To fully evaluate the wave character of the solutions, a number of typical wave profiles are presented, including bell-shaped wave, anti-bell shaped wave, W-shaped wave, continuous periodic wave, while kink wave, smooth kink wave, anti-peakon wave, V-shaped wave and flat wave solitons. Several 2D, 3D and contour plots are produced by taking precise values of parameters in order to improve the physical description of solutions. It is noteworthy that the suggested techniques for solving nonlinear partial differential equations are capable, reliable, and captivating analytical instruments. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Partial Differential Equations and Rogue Waves)
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14 pages, 279 KiB  
Article
An Improvement of the Upper Bound for the Number of Halving Lines of Planar Sets
by Estrella Alonso, Mariló López and Javier Rodrigo
Symmetry 2024, 16(7), 936; https://doi.org/10.3390/sym16070936 - 22 Jul 2024
Viewed by 1491
Abstract
In this paper, we provide improvements in the additive constant of the current best asymptotic upper bound for the maximum number of halving lines for planar sets of n points, where n is an even number. We also improve this current best upper [...] Read more.
In this paper, we provide improvements in the additive constant of the current best asymptotic upper bound for the maximum number of halving lines for planar sets of n points, where n is an even number. We also improve this current best upper bound for small values of n, namely, 106≤n≤336. To obtain this enhancements, we provide lower bounds for the sum of the squares of the degrees of the vertices of a graph related to the halving lines. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Partial Differential Equations and Rogue Waves)
17 pages, 2059 KiB  
Article
Lump, Breather, Ma-Breather, Kuznetsov–Ma-Breather, Periodic Cross-Kink and Multi-Waves Soliton Solutions for Benney–Luke Equation
by Miguel Vivas-Cortez, Sajawal Abbas Baloch, Muhammad Abbas, Moataz Alosaimi and Guo Wei
Symmetry 2024, 16(6), 747; https://doi.org/10.3390/sym16060747 - 15 Jun 2024
Viewed by 944
Abstract
The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface [...] Read more.
The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Partial Differential Equations and Rogue Waves)
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18 pages, 340 KiB  
Article
New Hermite–Hadamard Type Inequalities in Connection with Interval-Valued Generalized Harmonically (h1,h2)-Godunova–Levin Functions
by Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Donal O’Regan, Muhammad Tariq and Kamsing Nonlaopon
Symmetry 2022, 14(10), 1964; https://doi.org/10.3390/sym14101964 - 20 Sep 2022
Cited by 2 | Viewed by 1131
Abstract
As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically m,h1,h2-Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we [...] Read more.
As is known, integral inequalities related to convexity have a close relationship with symmetry. In this paper, we introduce a new notion of interval-valued harmonically m,h1,h2-Godunova–Levin functions, and we establish some new Hermite–Hadamard inequalities. Moreover, we show how this new notion of interval-valued convexity has a close relationship with many existing definitions in the literature. As a result, our theory generalizes many published results. Several interesting examples are provided to illustrate our results. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Partial Differential Equations and Rogue Waves)
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