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Mathematics
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Symmetries of Integrable Systems, 2nd Edition
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Symmetries of Integrable Systems, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: 31 August 2025 | Viewed by 2430

Special Issue Editor

Prof. Dr. Chuanzhong Li

E-Mail Website
Guest Editor
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
Interests: mathematical physics; algebra nonlinear systems; differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue builds on the previous edition of "Symmetries of Integrable Systems" in the MDPI journal Mathematics. The study of symmetries of integrable systems, such as Virasoro symmetries, is crucial in the field of integrable systems. Additional symmetries play a significant and intriguing role in these systems. Notably, the Kadomtsev–Petviashvili (KP) hierarchy, a fundamental integrable system in mathematical physics, has garnered increasing attention. Orlov and Shulman introduced additional symmetries to the KP hierarchy, including Virasoro symmetries, which impose Virasoro constraints on the partition functions of matrix models of string theory under additional non-isospectral symmetries. The BKP hierarchy and the CKP hierarchy, two pivotal sub-hierarchies of the KP hierarchy, exhibit similar cases in the Toda hierarchy.

We invite authors to submit their papers to our Special Issue.

Prof. Dr. Chuanzhong Li
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • KP hierarchy
  • toda hierarchy
  • symmetry

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Published Papers (5 papers)

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Research

20 pages, 384 KiB  
Article
The Application of Principal Component Analysis and the Wilson Model in Urban Economics
by Yiwei Chen, Congbin Guo and Junhao Fu
Mathematics 2025, 13(10), 1617; https://doi.org/10.3390/math13101617 - 14 May 2025
Viewed by 148
Abstract
This article first selects the “Urban Statistical Yearbook” data of 264 prefecture-level cities in China from 2004 to 2018 as the raw data, and uses principal component analysis and the Wilson model to calculate the spatial information diffusion capacity of each prefecture-level city. [...] Read more.
This article first selects the “Urban Statistical Yearbook” data of 264 prefecture-level cities in China from 2004 to 2018 as the raw data, and uses principal component analysis and the Wilson model to calculate the spatial information diffusion capacity of each prefecture-level city. The correlation analysis between industrial agglomeration, spatial information diffusion capacity, and urban economic resilience is verified, and this article provides reference materials for the specific application of principal component analysis and the Wilson model in urban economics. Full article
(This article belongs to the Special Issue Symmetries of Integrable Systems, 2nd Edition)
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21 pages, 11194 KiB  
Article
A Dynamic Regional-Aggregation-Based Heterogeneous Graph Neural Network for Traffic Prediction
by Xiangting Liu, Chengyuan Qian and Xueyang Zhao
Mathematics 2025, 13(9), 1458; https://doi.org/10.3390/math13091458 - 29 Apr 2025
Viewed by 239
Abstract
Traffic flow prediction, crucial for intelligent transportation systems, has seen advancements with graph neural networks (GNNs), yet existing methods often fail to distinguish between the importance of different intersections. These methods usually model all intersections uniformly, overlooking significant differences in traffic flow characteristics [...] Read more.
Traffic flow prediction, crucial for intelligent transportation systems, has seen advancements with graph neural networks (GNNs), yet existing methods often fail to distinguish between the importance of different intersections. These methods usually model all intersections uniformly, overlooking significant differences in traffic flow characteristics and influence ranges between ordinary and important nodes. To tackle this, this study introduces a dynamic regional-aggregation-based heterogeneous graph neural network (DR-HGNN). This model categorizes intersections into two types—ordinary and important—to apply tailored feature aggregation strategies. Ordinary intersections aggregate features based on local neighborhood information, whereas important intersections utilize deeper neighborhood diffusion and multi-hop dependencies to capture broader traffic influences. The DR-HGNN model also employs a dynamic graph structure to reflect temporal changes in traffic flows, alongside an attention mechanism for adaptive regional feature aggregation, enhancing the identification of critical traffic nodes. Demonstrating its efficacy, the DR-HGNN achieved 19.2% and 15.4% improvements in the RMSE over 50 min predictions in the METR-LA and PEMS-BAY datasets, respectively, offering a more precise prediction method for traffic management. Full article
(This article belongs to the Special Issue Symmetries of Integrable Systems, 2nd Edition)
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15 pages, 238 KiB  
Article
Prolongation Structure of a Development Equation and Its Darboux Transformation Solution
by Lixiu Wang, Jihong Wang and Yangjie Jia
Mathematics 2025, 13(6), 921; https://doi.org/10.3390/math13060921 - 11 Mar 2025
Viewed by 472
Abstract
This paper explores the third-order nonlinear coupled KdV equation utilizing prolongation structure theory and gauge transformation. By applying the prolongation structure method, we obtained an extended version of the equation. Starting from the Lax pairs of the equation, we successfully derived the corresponding [...] Read more.
This paper explores the third-order nonlinear coupled KdV equation utilizing prolongation structure theory and gauge transformation. By applying the prolongation structure method, we obtained an extended version of the equation. Starting from the Lax pairs of the equation, we successfully derived the corresponding Darboux transformation and Bäcklund transformation for this equation, which are fundamental to our solving process. Subsequently, we constructed and calculated the recursive operator for this equation, providing an effective approach to tackling complex problems within this domain. These results are crucial for advancing our understanding of the underlying principles of soliton theory and their implications on related natural phenomena. Our findings not only enrich the theoretical framework but also offer practical tools for further research in nonlinear wave dynamics. Full article
(This article belongs to the Special Issue Symmetries of Integrable Systems, 2nd Edition)
11 pages, 272 KiB  
Article
Symmetries for the Semi-Discrete Lattice Potential Korteweg–de Vries Equation
by Junwei Cheng and Xiang Tian
Mathematics 2025, 13(1), 117; https://doi.org/10.3390/math13010117 - 30 Dec 2024
Viewed by 547
Abstract
In this paper, we prove that the isospectral flows associated with both the x-part and the n-part of the Lax pair of the semi-discrete lattice potential Korteweg–de Vries equation are symmetries of the equation. Furthermore, we show that these two hierarchies [...] Read more.
In this paper, we prove that the isospectral flows associated with both the x-part and the n-part of the Lax pair of the semi-discrete lattice potential Korteweg–de Vries equation are symmetries of the equation. Furthermore, we show that these two hierarchies of symmetries are equivalent. Additionally, we construct the non-isospectral flows associated with the x-part of the Lax pair, which can be interpreted as the master symmetries of the semi-discrete lattice potential Korteweg–de Vries equation. Full article
(This article belongs to the Special Issue Symmetries of Integrable Systems, 2nd Edition)
13 pages, 265 KiB  
Article
Two-Dimensional Dispersionless Toda Lattice Hierarchy: Symmetry, New Extension, Hodograph Solutions, and Reduction
by Hongxia Wu, Jingxin Liu and Haifeng Wang
Mathematics 2024, 12(23), 3706; https://doi.org/10.3390/math12233706 - 26 Nov 2024
Viewed by 549
Abstract
The symmetry for two-dimensional (2D) dispersionless Toda lattice hierarchy (dTLH) is firstly derived, and then the 2D dTLH is extended based on the symmetry constraint. The commutativity of two different flows for this new hierarchy is shown, which leads to the 2D dToda [...] Read more.
The symmetry for two-dimensional (2D) dispersionless Toda lattice hierarchy (dTLH) is firstly derived, and then the 2D dTLH is extended based on the symmetry constraint. The commutativity of two different flows for this new hierarchy is shown, which leads to the 2D dToda lattice equation with self-consistent sources (dTLESCSs) together with its conservation equation. The hodograph solutions to 2D dTLESCSs are also given. One dimensional reduction of extended 2D dTLH is finally investigated by finding the constraint, and a one-dimensional dTLESCS is shown. Full article
(This article belongs to the Special Issue Symmetries of Integrable Systems, 2nd Edition)
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