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Symmetry
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Geometric Analysis of Nonlinear Partial Differential Equations II
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Geometric Analysis of Nonlinear Partial Differential Equations II

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

Deadline for manuscript submissions: closed (30 May 2022) | Viewed by 13423

Special Issue Editors

Prof. Dr. Valentin Lychagin

E-Mail Website
Guest Editor
1. Department of Mathematics and Statistics, The Arctic University of Norway, N-9037 Tromso, Norway
2. V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 117997 Moscow, Russia
Interests: differential equations; symmetries; conservation laws; differential invariants; Integrability; singularities solutions; shock waves and phase transitions
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Joseph Krasilshchik

E-Mail Website
Guest Editor
V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 117997 Moscow, Russia
Interests: differential equations; nonlocal symmetries; conservation laws
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is the continuation of the previous one recently published in Symmetry with the same title (https://www.mdpi.com/journal/symmetry/special_issues/Geometric_Analysis_Nonlinear_PDEs).

The Special Issue is devoted to the observation of various geometrical structures associated with nonlinear partial differential equations as well as their symmetry and applications to integrability of the equations.

The main topics that we plan to discuss should be concentrated on different notions of symmetry and related to its invariants, conservation laws, and integrability. Its applications will also be of interest.

Prof. Dr. Valentin Lychagin
Prof. Dr. Joseph Krasilshchik
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. 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

  • differential equations
  • symmetries
  • conservation laws
  • differential invariants
  • integrability

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Published Papers (6 papers)

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Research

34 pages, 327 KiB  
Article
Integrable Kuralay Equations: Geometry, Solutions and Generalizations
by Zhanna Sagidullayeva, Gulgassyl Nugmanova, Ratbay Myrzakulov and Nurzhan Serikbayev
Symmetry 2022, 14(7), 1374; https://doi.org/10.3390/sym14071374 - 4 Jul 2022
Cited by 36 | Viewed by 2260
Abstract
In this paper, we study the Kuralay equations, namely the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these two equations is established. With the help of [...] Read more.
In this paper, we study the Kuralay equations, namely the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these two equations is established. With the help of the Hirota bilinear method, the simplest soliton solutions are also presented. The nonlocal and dispersionless versions of the Kuralay equations are considered. Some integrable generalizations and other related nonlinear differential equations are presented. Full article
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations II)
28 pages, 410 KiB  
Article
Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds
by Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone and Alessandro Zampini
Symmetry 2022, 14(1), 70; https://doi.org/10.3390/sym14010070 - 4 Jan 2022
Cited by 8 | Viewed by 1717
Abstract
As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a [...] Read more.
As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a symplectic regularization based on an application of Gotay’s coisotropic embedding theorem. An analysis of electrodynamics and of the Klein–Gordon theory illustrate the main results of the theory as well as the emergence of the energy–momentum tensor algebra of conserved currents. Full article
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations II)
39 pages, 494 KiB  
Article
ReLie: A Reduce Program for Lie Group Analysis of Differential Equations
by Francesco Oliveri
Symmetry 2021, 13(10), 1826; https://doi.org/10.3390/sym13101826 - 30 Sep 2021
Cited by 10 | Viewed by 2783
Abstract
Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the [...] Read more.
Lie symmetry analysis provides a general theoretical framework for investigating ordinary and partial differential equations. The theory is completely algorithmic even if it usually involves lengthy computations. For this reason, along the years many computer algebra packages have been developed to automate the computation. In this paper, we describe the program ReLie, written in the Computer Algebra System Reduce, since 2008 an open source program for all platforms. ReLie is able to perform almost automatically the needed computations for Lie symmetry analysis of differential equations. Its source code is freely available too. The use of the program is illustrated by means of some examples; nevertheless, it is to be underlined that it proves effective also for more complex computations where one has to deal with very large expressions. Full article
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations II)
27 pages, 490 KiB  
Article
Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter
by Mahmood R. Tarayrah and Alexei F. Cheviakov
Symmetry 2021, 13(9), 1612; https://doi.org/10.3390/sym13091612 - 2 Sep 2021
Cited by 3 | Viewed by 1937
Abstract
The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed [...] Read more.
The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. For algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE and can be systematically computed. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq equation reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided. Full article
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations II)
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12 pages, 289 KiB  
Article
Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations
by Yarema A. Prykarpatskyy
Symmetry 2021, 13(6), 1077; https://doi.org/10.3390/sym13061077 - 16 Jun 2021
Cited by 4 | Viewed by 1762
Abstract
Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s [...] Read more.
Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems. Full article
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations II)
8 pages, 424 KiB  
Article
On Galilean Invariant and Energy Preserving BBM-Type Equations
by Alexei Cheviakov, Denys Dutykh and Aidar Assylbekuly
Symmetry 2021, 13(5), 878; https://doi.org/10.3390/sym13050878 - 14 May 2021
Cited by 1 | Viewed by 1668
Abstract
We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis [...] Read more.
We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability. Full article
(This article belongs to the Special Issue Geometric Analysis of Nonlinear Partial Differential Equations II)
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