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Symmetry
Special Issues
Symmetry in Integrable Systems and Soliton Theories
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Symmetry in Integrable Systems and Soliton Theories

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

Deadline for manuscript submissions: closed (30 September 2024) | Viewed by 2171

Special Issue Editor

Dr. Yufeng Zhang

E-Mail Website
Guest Editor
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
Interests: integrate system; inverse scattering method; Riemann-Hilbert problem; soliton solution
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to provide an overview of recent developments in integrable systems and soliton theories. Papers that present various solutions to partial differential and integral equations, Hamiltonian structures, Darboux transformations, solitons, Lie symmetry theory (including local symmetry and nonlocal symmetry) and Riemann–Hilbert problems, among others, are welcome.

Dr. Yufeng Zhang
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

  • solutions for partial differential and integral equations
  • Hamiltonian structures
  • Darboux transformations
  • solitons
  • Lie symmetry theory
  • Riemann–Hilbert problems

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Published Papers (2 papers)

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Research

24 pages, 347 KiB  
Article
The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line
by Jiawei Hu and Ning Zhang
Symmetry 2024, 16(10), 1258; https://doi.org/10.3390/sym16101258 - 25 Sep 2024
Viewed by 832
Abstract
The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and [...] Read more.
The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)
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19 pages, 310 KiB  
Article
Onthe Reducibility of a Class Nonlinear Almost Periodic Hamiltonian Systems
by Nina Xue and Yanmei Sun
Symmetry 2024, 16(6), 656; https://doi.org/10.3390/sym16060656 - 26 May 2024
Viewed by 801
Abstract
Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic [...] Read more.
Inthis paper, we consider the reducibility of a class of nonlinear almost periodic Hamiltonian systems. Under suitable hypothesis of analyticity, non-resonant conditions and non-degeneracy conditions, by using KAM iteration, it is shown that the considered Hamiltonian system is reducible to an almost periodic Hamiltonian system with zero equilibrium points for most small enough parameters. As an example, we discuss the reducibility and stability of an almost periodic Hill’s equation. Full article
(This article belongs to the Special Issue Symmetry in Integrable Systems and Soliton Theories)
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