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Symmetry
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Symmetry in Integrable Systems: Theory and Application
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Symmetry in Integrable Systems: Theory and Application

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

Deadline for manuscript submissions: closed (28 February 2023) | Viewed by 9269

Special Issue Editor

Prof. Dr. Bo Ren

E-Mail Website
Guest Editor
Department of Science, Zhejiang University of Technology, Hangzhou 310014, China
Interests: nonlinear integrable systems; lie symmetry method; nonlocal symmetry; hirota bilinear method; solitons; lumps; rogue waves; breathers

Special Issue Information

Dear Colleagues,

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science. Finding exact solutions for nonlinear differential equations has always been one of the central themes of perpetual interest in mathematics and physics. The use of symmetry reduction can be carried out using the classical Lie group approach, the nonclassical Lie group approach, and the Clarkson–Kruskal direct method, which are effective methods for obtaining exact solutions of nonlinear differential equations. The interactions between solitons and other different nonlinear excitation are obtained by symmetry reduction based on nonlocal symmetry.

This Special Issue will cover the symmetries of nonlinear partial differential equations, discrete dynamical systems, and nonlinear integrable systems. Potential topics include but are not limited to the following:

  • Nonlinear partial differential equations;
  • Symmetries of nonlinear equations;
  • Conservation laws;
  • Orthogonal polynomials;
  • Hirota bilinear method;
  • Darboux method;
  • Lump wave;
  • Rogue wave;
  • Riemann-Hilbert problems;
  • Soliton molecules;
  • Other miscellaneous applications of nonlinear integrable systems.

Prof. Dr. Bo Ren
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.

Published Papers (8 papers)

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Editorial

Jump to: Research

2 pages, 168 KiB  
Editorial
Review of Contributions to the Special Edition: Symmetry in Integrable Systems: Theory and Application
by Bo Ren
Symmetry 2023, 15(4), 934; https://doi.org/10.3390/sym15040934 - 19 Apr 2023
Viewed by 770
Abstract
Nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of science [...] Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)

Research

Jump to: Editorial

17 pages, 727 KiB  
Article
Cubic-Quartic Optical Solitons in Fiber Bragg Gratings with Dispersive Reflectivity Having Parabolic Law of Nonlinear Refractive Index by Lie Symmetry
by Sandeep Malik, Sachin Kumar, Anjan Biswas, Yakup Yıldırım, Luminita Moraru, Simona Moldovanu, Catalina Iticescu and Hashim M. Alshehri
Symmetry 2022, 14(11), 2370; https://doi.org/10.3390/sym14112370 - 10 Nov 2022
Cited by 14 | Viewed by 1295
Abstract
This work recovers cubic-quartic optical solitons with dispersive reflectivity in fiber Bragg gratings and parabolic law of nonlinearity. The Lie symmetry analysis first reduces the governing partial differential equations to the corresponding ordinary differential equations which are subsequently integrated. This integration is conducted [...] Read more.
This work recovers cubic-quartic optical solitons with dispersive reflectivity in fiber Bragg gratings and parabolic law of nonlinearity. The Lie symmetry analysis first reduces the governing partial differential equations to the corresponding ordinary differential equations which are subsequently integrated. This integration is conducted using two approaches which are the modified Kudryashov’s approach as well as the generalized Arnous’ scheme. These collectively yielded a full spectrum of cubic-quartic optical solitons that have been proposed to control the depletion of the much-needed chromatic dispersion. They range from bright, dark, singular to combo solitons. These solitons are considered with dispersive reflectivity, which maintains the necessary balance between chromatic dispersion and nonlinear refractive index structure for an uninterrupted transmission of solitons along intercontinental distances. Their respective surface and contour plots are also exhibited. A few closing words are included with some prospective future avenues of research to extend this topic further. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
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18 pages, 341 KiB  
Article
On New Hamiltonian Structures of Two Integrable Couplings
by Yu Liu, Jin Liu and Da-jun Zhang
Symmetry 2022, 14(11), 2259; https://doi.org/10.3390/sym14112259 - 27 Oct 2022
Cited by 2 | Viewed by 823
Abstract
In this paper, we present new Hamiltonian operators for the integrable couplings of the Ablowitz–Kaup–Newell–Segur hierarchy and the Kaup–Newell hierarchy. The corresponding Hamiltonians allow nontrivial degeneration. Multi-Hamiltonian structures are investigated. The involutive property is proven for the new and known Hamiltonians with respect [...] Read more.
In this paper, we present new Hamiltonian operators for the integrable couplings of the Ablowitz–Kaup–Newell–Segur hierarchy and the Kaup–Newell hierarchy. The corresponding Hamiltonians allow nontrivial degeneration. Multi-Hamiltonian structures are investigated. The involutive property is proven for the new and known Hamiltonians with respect to the two Poisson brackets defined by the new and known Hamiltonian operators. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
24 pages, 2094 KiB  
Article
Anti-Symmetric Medium Chirality Leading to Symmetric Field Helicity in Response to a Pair of Circularly Polarized Plane Waves in Counter-Propagating Configuration
by Hyoung-In Lee
Symmetry 2022, 14(9), 1895; https://doi.org/10.3390/sym14091895 - 10 Sep 2022
Cited by 4 | Viewed by 1418
Abstract
We examine how a chiral medium responds to a pair of plane waves of circular polarizations. To this goal, we assume the chiral medium to be spatially homogeneous for simplicity. By assuming the medium to be a lossless, we provide analytic formulas of [...] Read more.
We examine how a chiral medium responds to a pair of plane waves of circular polarizations. To this goal, we assume the chiral medium to be spatially homogeneous for simplicity. By assuming the medium to be a lossless, we provide analytic formulas of key bilinear parameters such as the pair of electromagnetic and reactive Poynting vectors in addition to the pair of electromagnetic and reactive helicities. By examining two obliquely colliding plane waves, we learned that most of those key parameters are asymmetric with respect to the medium chirality. Only for a counter-propagating pair, some of those key parameters are found to exhibit symmetry with respect to the medium chirality. We will discuss the implications of those asymmetries and symmetries from the viewpoints of typical applications in optics and physics. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
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10 pages, 1435 KiB  
Article
Symmetry Reductions, Cte Method and Interaction Solutions for Sharma-Tasso-Olver-Burgers Equation
by Jun Yu, Bo Ren and Wan-Li Wang
Symmetry 2022, 14(8), 1690; https://doi.org/10.3390/sym14081690 - 15 Aug 2022
Cited by 3 | Viewed by 1156
Abstract
In this paper, the Sharma-Tasso-Olver-Burgers (STOB) system is analyzed by the Lie point symmetry method. The hypergeometric wave solution of the STOB equation is derived by symmetry reductions. In the meantime, the consistent tanh expansion (CTE) method is applied to the STOB equation. [...] Read more.
In this paper, the Sharma-Tasso-Olver-Burgers (STOB) system is analyzed by the Lie point symmetry method. The hypergeometric wave solution of the STOB equation is derived by symmetry reductions. In the meantime, the consistent tanh expansion (CTE) method is applied to the STOB equation. An nonauto-Bäcklund (BT) theorem that includes the over-determined equations and the consistent condition is obtained by the CTE method. By using the nonauto-BT theorem, the interactions between one-soliton and the cnoidal wave, and between one-soliton and the multiple resonant soliton solutions, are constructed. The dynamics of these novel interaction solutions are shown both in analytical and graphical forms. The results are potentially useful for explaining ocean phenomena. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
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5 pages, 227 KiB  
Article
Non-Zero Order of an Extended Temme Integral
by Robert Reynolds and Allan Stauffer
Symmetry 2022, 14(8), 1573; https://doi.org/10.3390/sym14081573 - 30 Jul 2022
Cited by 1 | Viewed by 845
Abstract
A new three-dimensional integral containing f(x,y,z)Iv(xα) is derived where Iv(xα) is the Modified Bessel Function of the first kind and the integral is taken over [...] Read more.
A new three-dimensional integral containing f(x,y,z)Iv(xα) is derived where Iv(xα) is the Modified Bessel Function of the first kind and the integral is taken over the infinite cubic space 0<x<∞,0<y<∞,0<z<∞. The integral is not easily evaluated for complex ranges of the parameters. A representation in terms of the Hurwitz–Lerch zeta function, polylogarithm function and Riemann zeta functions are evaluated. This representation yields triple integral representations in terms of fundamental constants that can be derived. Almost all Lerch functions have an asymmetrical zero distribution. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
14 pages, 283 KiB  
Article
Second-Order Approximate Equations of the Large-Scale Atmospheric Motion Equations and Symmetry Analysis for the Basic Equations of Atmospheric Motion
by Ping Liu, Senyue Lou and Lei Peng
Symmetry 2022, 14(8), 1540; https://doi.org/10.3390/sym14081540 - 27 Jul 2022
Cited by 1 | Viewed by 822
Abstract
In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and [...] Read more.
In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and eight one-parameter group invariant solutions are demonstrated. Three types of nontrivial similarity solutions and group invariants are proposed. With the help of perturbation method, we derive the second-order approximate equations for the large-scale atmospheric motion equations, including the non-dimensional equations and the dimensional equations. The second-order approximate equations of the large-scale atmospheric motion equations not only show the characteristics of physical quantities changing with time, but also describe the characteristics of large-scale atmospheric vertical motion. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
20 pages, 321 KiB  
Article
Lax Operator Algebras and Applications to τ-Symmetries for Multilayer Integrable Couplings
by Chun-Xia Li, Wen-Xiu Ma and Shou-Feng Shen
Symmetry 2022, 14(6), 1185; https://doi.org/10.3390/sym14061185 - 09 Jun 2022
Cited by 1 | Viewed by 1281
Abstract
The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As [...] Read more.
The algebraic structures of zero curvature representations are furnished for multilayer integrable couplings associated with matrix spectral problems, both discrete and continuous. The key elements are a class of matrix loop algebras consisting of block matrices with blocks of the same size. As illustrative examples, isospectral and non-isospectral integrable couplings and the corresponding commutator relations of their Lax operators are computed explicitly in the cases of the Volterra lattice hierarchy and the AKNS hierarchy, along with their τ-symmetry algebras. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems: Theory and Application)
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