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Mathematics
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Completely Integrable Equations: Algebraic Aspects and Applications
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Special Issue "Completely Integrable Equations: Algebraic Aspects and Applications"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 30 April 2024 | Viewed by 3245

Special Issue Editors

Dr. Andrew Pickering

E-Mail Website
Guest Editor
Área de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, 28933 Móstoles, Spain
Interests: completely integrable systems; Bäcklund and auto-bäcklund transformations; Hamiltonian structures; Painlevé equations and hierarchies; discrete and differential-delay Painlevé equations and hierarchies
Dr. Pilar R. Gordoa

E-Mail Website
Guest Editor
Área de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, Móstoles, Spain
Interests: integrable systems; Bäcklund transformations; Hamiltonian systems; Painlevé equations and Painlevé hierarchies; scattering problems; discrete and differential-delay systems; lie symmetries; solitons

Special Issue Information

Dear Colleagues,

The theory of completely integrable systems that has developed over the last half-century or so is extremely wide-ranging, taking in algebraic, geometric and analytic approaches. In addition to the inherent beauty of much of this theory, with its myriad connections to many other areas of mathematics, physics and other sciences, much of this interest is motivated by the many applications of well-known completely integrable equations.

It is this two-fold interest that we seek to reflect in this Special Issue. On the one hand, we seek to focus on algebraic aspects of integrable equations, in particular on Hamiltonian structures, recursion operators, and generalized symmetries and related properties of integrable PDEs and lattices. We would like to invite papers that explore new techniques and examples in relation to these algebraic aspects. In addition, we are interested in contributions that study the properties of integrable systems arising in applications, as well as such applications themselves.

Dr. Andrew Pickering
Dr. Pilar R. Gordoa
Guest Editors

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

  • completely integrable evolution equations: PDEs and lattices
  • hamiltonian structures
  • recursion operators and generalized symmetries
  • applications of integrable PDEs and lattices

Published Papers (4 papers)

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Article
Integrable Systems: In the Footprints of the Greats
by
Velimir Jurdjevic
Mathematics 2023, 11(4), 1063; https://doi.org/10.3390/math11041063 - 20 Feb 2023
Viewed by 591
Abstract
In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, [...] Read more.
In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal. Full article
(This article belongs to the Special Issue Completely Integrable Equations: Algebraic Aspects and Applications)
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Article
Darboux Transformation and Soliton Solution of the Nonlocal Generalized Sasa–Satsuma Equation
by
Hong-Qian Sun
and
Zuo-Nong Zhu
Mathematics 2023, 11(4), 865; https://doi.org/10.3390/math11040865 - 08 Feb 2023
Cited by 2 | Viewed by 776
Abstract
This paper aims to seek soliton solutions for the nonlocal generalized Sasa–Satsuma (gSS) equation by constructing the Darboux transformation (DT). We obtain soliton solutions for the nonlocal gSS equation, including double-periodic wave, breather-like, KM-breather solution, dark-soliton, W-shaped soliton, M-shaped soliton, W-shaped periodic wave, [...] Read more.
This paper aims to seek soliton solutions for the nonlocal generalized Sasa–Satsuma (gSS) equation by constructing the Darboux transformation (DT). We obtain soliton solutions for the nonlocal gSS equation, including double-periodic wave, breather-like, KM-breather solution, dark-soliton, W-shaped soliton, M-shaped soliton, W-shaped periodic wave, M-shaped periodic wave, double-peak dark-breather, double-peak bright-breather, and M-shaped double-peak breather solutions. Furthermore, interaction of these solitons, as well as their dynamical properties and asymptotic analysis, are analyzed. It will be shown that soliton solutions of the nonlocal gSS equation can be reduced into those of the nonlocal Sasa–Satsuma equation. However, several of these properties for the nonlocal Sasa–Satsuma equation are not found in the literature. Full article
(This article belongs to the Special Issue Completely Integrable Equations: Algebraic Aspects and Applications)
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Article
A Two-Dimensional port-Hamiltonian Model for Coupled Heat Transfer
by
Jens Jäschke
,
Matthias Ehrhardt
,
Michael Günther
and
Birgit Jacob
Mathematics 2022, 10(24), 4635; https://doi.org/10.3390/math10244635 - 07 Dec 2022
Viewed by 719
Abstract
In this paper, we construct a highly simplified mathematical model for studying the problem of conjugate heat transfer in gas turbine blades and their cooling ducts. Our simple model focuses on the relevant coupling structures and aims to reduce the unrelated complexity as [...] Read more.
In this paper, we construct a highly simplified mathematical model for studying the problem of conjugate heat transfer in gas turbine blades and their cooling ducts. Our simple model focuses on the relevant coupling structures and aims to reduce the unrelated complexity as much as possible. Then, we apply the port-Hamiltonian formalism to this model and its subsystems and investigate the interconnections. Finally, we apply a simple spatial discretization to the system to investigate the properties of the resulting finite-dimensional port-Hamiltonian system and to determine whether the order of coupling and discretization affect the resulting semi-discrete system. Full article
(This article belongs to the Special Issue Completely Integrable Equations: Algebraic Aspects and Applications)
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Article
Novel Bäcklund Transformations for Integrable Equations
by
Pilar Ruiz Gordoa
and
Andrew Pickering
Mathematics 2022, 10(19), 3565; https://doi.org/10.3390/math10193565 - 29 Sep 2022
Viewed by 660
Abstract
In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation [...] Read more.
In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation. Full article
(This article belongs to the Special Issue Completely Integrable Equations: Algebraic Aspects and Applications)
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