Symmetry in Integrable Systems: Topics and Advances
A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".
Deadline for manuscript submissions: 31 May 2024 | Viewed by 1154
Special Issue Editor
Special Issue Information
Dear Colleagues,
Symmetries play a paramount important role in mathematics as well as in physics. Similarity solutions or invariant solutions of a physical problem can be constructed using the Lie group theory. The relationship between symmetries and conservation laws generates the Noether theorem. The related applications of symmetries are to determine higher-order and nonlocal symmetries, conservation laws, nonlocal conservation laws and specific solutions from reductions. The preceding volume would like to offer an overview of the comprehensive treatments of the Lie groups of transformations, the discovery and use of symmetries to construct solutions, the conservation laws and phenomenological applications thereof.
Potential topics include but are not limited to the following:
- Symmetries;
- Conservation laws;
- Solitons;
- Integrable systems;
- Breathers;
- Rogue waves;
- Hirota bilinear method;
- Darboux transformation;
- Other miscellaneous applications of nonlinear integrable systems.
Prof. Dr. Bo Ren
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.
Planned Papers
The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.
Title: Symmetries and exact solutions in the Ginzburg-Landau mean-field theory of the 3D XY model
Authors: Vassil M. Vassilev and Daniel M. Dantchev
Affiliation: Institute of Mechanics, Bulgarian Academy of Sciences Acad. G. Bontchev St., Building 4, 1113
Sofia, Bulgaria
Abstract: The XY model is one of the basic models of statistical mechanics. It is often used for the description of important physical systems like 4He films and liquid crystals. In the present work, we consider the Ginzburg-Landau mean-field approximation of the 3D XY lattice model, which is a continuous analogue of the latter. We study the Lie symmetries of the corresponding Ginzburg-Landau Hamiltonian and the associated Euler-Lagrange equations and establish the variational symmetries among them in the presence of an external ordering field. Then, using the Noether theorem, we find conservation laws which allow us to obtain and present in analytic form the general solution of the regarded problem. It should be remarked that exact solutions are only known in the absence of an external ordering field within the so-called Ψ-theory and some other mean-field-like theories.
Title: Symmetries, Group-Invariant Solutions and Conservation Laws in the Dynamics of Carbon Nanotubes
Authors: Vassil M. Vassilev
Affiliation: Institute of Mechanics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev St., Block 4, 1113 Sofia, Bulgaria
Abstract: The study focuses on the invariance properties of equations of motion of initially straight single- and double-wall carbon nanotubes conveying fluid in an elastic medium. Two different types of beam-like models commonly used to describe the basic features of the dynamic behaviour of such mechanical systems are considered. The first one arises from the classical Bernoulli-Euler beam theory. The second one builds on Timoshenko's beam theory. Both models reflect the influence of the ambient elastic medium, the impact of the fluid flow inside a single or a double-wall carbon nanotube and the effect of the van der Waals interaction, which is assumed to be nonlinear, between the individual nanotubes composing a double-wall one. In both cases, the respective equations of motion are derived as the Euler-Lagrange equations associated with suitable action functionals, i.e., the problems are imbedded in variational statements. In the present work, the point Lie symmetry groups admitted by the equations of motion mentioned above are established. Then, the variational and divergence symmetries among them are identified and used to construct conservation laws inherent to the considered models. Moreover, several group-invariant solutions of the regarded systems of nonlinear partial differential equations are presented.
