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Mathematics
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Differential Geometry and Related Integrable Systems
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Differential Geometry and Related Integrable Systems

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

Deadline for manuscript submissions: closed (20 August 2022) | Viewed by 7827

Special Issue Editor

Prof. Dr. Magdalena Toda

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA
Interests: differential geometry; geometric PDE and integrable systems with applications to fluid flows; geometry of proteins; Riemannian and Lorentzian geometry (relativistic math physics)

Special Issue Information

Dear Colleagues, 

This volume is dedicated to the study of geometric objects that represent critical points of certain curvature functionals. Many interesting classes of surfaces appear from the calculus of variations, as solutions of Euler–Lagrange equations, thus bringing close together the fields of geometric integrable systems and mathematical physics. Common examples include Willmore surfaces and constant-mean-curvature surfaces. The tools used to study these surfaces involve the calculus of variations and PDE, Lie group theory, harmonic map theory, gauge theory, along with real and complex manifolds. The different techniques and fields involved unify research interests, rather than dividing them. The key idea of this work is to present groundbreaking and recent research in differential geometry and integrable systems, and to stimulate further collaborations.

Prof. Dr. Magdalena Toda
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

  • minimal surfaces
  • constant-mean-curvature surfaces
  • Willmore surfaces
  • harmonic maps
  • integrable systems
  • elastic energies

Published Papers (5 papers)

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Research

8 pages, 470 KiB  
Article
The Geometry of the Kiepert Trefoil
by Vladimir I. Pulov, Magdalena D. Toda, Vassil M. Vassilev and Ivaïlo M. Mladenov
Mathematics 2022, 10(18), 3357; https://doi.org/10.3390/math10183357 - 15 Sep 2022
Viewed by 1142
Abstract
This article presents a comparative study of Kiepert’s trefoil and its related curves, combining a variety of tools from differential and algebraic geometry, integrable systems, elastica theory, and special functions. While this curve was classically known and well studied in the literature, some [...] Read more.
This article presents a comparative study of Kiepert’s trefoil and its related curves, combining a variety of tools from differential and algebraic geometry, integrable systems, elastica theory, and special functions. While this curve was classically known and well studied in the literature, some related open problems were recently solved, and the goal of this paper is to present and characterize the general solution of the equation that governs this trefoil’s family of curves by involving elliptic functions and elastica theory in the mechanics. Full article
(This article belongs to the Special Issue Differential Geometry and Related Integrable Systems)
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10 pages, 5028 KiB  
Article
λ-Spheres as a New Reference Model for Geoid: Explicit Solutions of the Direct and Inverse Problems for Loxodromes (Rhumb Lines)
by Vasyl Kovalchuk and Ivaïlo M. Mladenov
Mathematics 2022, 10(18), 3356; https://doi.org/10.3390/math10183356 - 15 Sep 2022
Cited by 3 | Viewed by 1081
Abstract
In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter λ. These surfaces, which are called λ-spheres, were introduced in another setting [...] Read more.
In this paper, we present a new reference model that approximates the actual shape of the Earth, based on the concept of the deformed spheres with the deformation parameter λ. These surfaces, which are called λ-spheres, were introduced in another setting by Faridi and Schucking as an alternative to the spheroids (i.e., ellipsoids of revolution). Using their explicit parametrizations that we have derived in our previous papers, here we have defined the corresponding isothermal (conformal) coordinates as well as obtained and solved the differential equation describing the loxodromes (or rhumb lines) on such surfaces. Next, the direct and inverse problems for loxodromes have been formulated and the explicit solutions for azimuths and arc lengths have been presented. Using these explicit solutions, we have assessed the value of the deformation parameter λ for our reference model on the basis of the values for the semi-major axis of the Earth a and the quarter-meridian mp (i.e., the distance between the Equator and the North or South Pole) for the current best ellipsoidal reference model for the geoid, i.e., WGS 84 (World Geodetic System 1984). The latter is designed for use as the reference system for the GPS (Global Positioning System). Finally, we have compared the results obtained with the use of the newly proposed reference model for the geoid with the corresponding results for the ellipsoidal (WGS 84) and spherical reference models used in the literature. Full article
(This article belongs to the Special Issue Differential Geometry and Related Integrable Systems)
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24 pages, 35586 KiB  
Article
Periodic Solutions and Stability Analysis for Two-Coupled-Oscillator Structure in Optics of Chiral Molecules
by Jing Li, Yuying Chen and Shaotao Zhu
Mathematics 2022, 10(11), 1908; https://doi.org/10.3390/math10111908 - 02 Jun 2022
Cited by 1 | Viewed by 1389
Abstract
Chirality is an indispensable geometric property in the world that has become invariably interlocked with life. The main goal of this paper is to study the nonlinear dynamic behavior and periodic vibration characteristic of a two-coupled-oscillator model in the optics of chiral molecules. [...] Read more.
Chirality is an indispensable geometric property in the world that has become invariably interlocked with life. The main goal of this paper is to study the nonlinear dynamic behavior and periodic vibration characteristic of a two-coupled-oscillator model in the optics of chiral molecules. We systematically discuss the stability and local dynamic behavior of the system with two pairs of identical conjugate complex eigenvalues. In particular, the existence and number of periodic solutions are investigated by establishing the curvilinear coordinate and constructing a Poincaré map to improve the Melnikov function. Then, we verify the accuracy of the theoretical analysis by numerical simulations, and take a comprehensive look at the nonlinear response of multiple periodic motion under certain conditions. The results might be of important significance for the vibration control, safety stability and design optimization for chiral molecules. Full article
(This article belongs to the Special Issue Differential Geometry and Related Integrable Systems)
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5 pages, 394 KiB  
Article
The Trefoil Soliton
by David A. Singer
Mathematics 2022, 10(9), 1512; https://doi.org/10.3390/math10091512 - 01 May 2022
Cited by 1 | Viewed by 1204
Abstract
The Kiepert trefoil is an algebraic curve with remarkable geometric and number theoretic properties. Ludwig Kiepert, generalizing ideas due to Serret and Liouville, determined that it could be parametrized by arc length in terms of elliptic functions. In this note, we observe some [...] Read more.
The Kiepert trefoil is an algebraic curve with remarkable geometric and number theoretic properties. Ludwig Kiepert, generalizing ideas due to Serret and Liouville, determined that it could be parametrized by arc length in terms of elliptic functions. In this note, we observe some other properties of the curve. In particular, the curve is a special example of a buckled ring, and thus a solitary wave solution to the planar filament equation, evolving by rotation. It is also a solitary wave solution to a flow in the (three-dimensional) filament hierarchy, evolving by translation. Full article
(This article belongs to the Special Issue Differential Geometry and Related Integrable Systems)
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20 pages, 351 KiB  
Article
Negative Order KdV Equation with No Solitary Traveling Waves
by Miguel Rodriguez, Jing Li and Zhijun Qiao
Mathematics 2022, 10(1), 48; https://doi.org/10.3390/math10010048 - 24 Dec 2021
Cited by 4 | Viewed by 2080
Abstract
We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve [...] Read more.
We consider the negative order KdV (NKdV) hierarchy which generates nonlinear integrable equations. Selecting different seed functions produces different evolution equations. We apply the traveling wave setting to study one of these equations. Assuming a particular type of solution leads us to solve a cubic equation. New solutions are found, but none of these are classical solitary traveling wave solutions. Full article
(This article belongs to the Special Issue Differential Geometry and Related Integrable Systems)
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