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Symmetry
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Exact Solutions with Symmetry Reduction and Long Time Behaviors of...
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Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations

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

Deadline for manuscript submissions: closed (30 September 2023) | Viewed by 8850

Special Issue Editors

Prof. Dr. Manwai Yuen

E-Mail Website
Guest Editor
Department of Mathematics and Information Technology, The Education University of Hong Kong, 10 Lo Ping Road, Tai Po, New Territories, Hong Kong
Interests: partial differential equations; symmetry reduction; blowup; Euler-Poisson equations; Euler equations with or without Coriolis Force; Camassa-Holm equations; Navier-Stokes equations; Magnetohydrodynamics (MHD); analytical and exact solutions; mathematical methods in fluids; classical cosmology
Special Issues, Collections and Topics in MDPI journals
Dr. Imre Ferenc Barna

E-Mail Website
Guest Editor
Wigner Research Centre for Physics, 1121 Budapest, Hungary
Interests: analytic solutions of non-linear partial differential equations; laser-atom interaction
Prof. Dr. Baofeng Feng

E-Mail Website
Guest Editor
Department of Mathematics, The University of Texas-RGV, Edinburg, TX 78539, USA
Interests: continuous and discrete integrable systems; nonlinear waves with applications; scientific computing and numerical analysis; nonlinear optics and optical fiber communications
Prof. Dr. Biao Li

E-Mail Website
Guest Editor
Ningbo University, System of Ocean and Atmosphere and Dept. of Mathematics, Ningbo 315000, China
Interests: nonlinear mathematical physics; applications of lie group to differential equations
Prof. Dr. Li Jun Zhang

E-Mail Website
Guest Editor
Department of Mathematics, Shandong University of Science and Technology, Qingdao 266000, China
Interests: partial differential equations

Special Issue Information

Dear Colleagues,

It is well-known that most nonlinear partial differential equations do not have a general solution in closed form. However, by using symmetry reductions we can construct their special exact solutions, which can reflect the properties or long time behaviors of the nonlinear partial differential systems. In other words, symmetry is especially useful in the analysis of some particular cases of complex systems.

Similarly, it is often possible to write down some special exact solutions explicitly in terms of elementary functions, and those elementary functions often appear in a highly symmetric form.

In this Special Issue, we are expecting theoretical or numerical analyses of some special solutions with symmetry assumptions of some nonlinear partial differential systems, such as those arising in fluid mechanics (original or special cases of the Navier–Stokes equations, Euler equations, Euler–Poisson equations, etc.), general relativity (original or special cases of the Einstein field equations, etc.), and other nontrivial and nonlinear partial differential equations.

Submit your paper and select the Journal “Symmetry” and the Special Issue “Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations” via: MDPI submission system. Our papers will be published on a rolling basis and we will be pleased to receive your submission once you have finished it.

Prof. Dr. Manwai Yuen
Dr. Imre Ferenc Barna
Prof. Dr. Baofeng Feng
Prof. Dr. Biao Li
Prof. Dr. Li Jun Zhang
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

  • nonlinear partial differential equations
  • symmetry reduction
  • exact solutions
  • fluid dynamics
  • solution in closed form
  • qualitative properties

Published Papers (6 papers)

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Research

15 pages, 842 KiB  
Article
Exact Solutions of Population Balance Equation with Aggregation, Nucleation, Growth and Breakage Processes, Using Scaling Group Analysis
by Fubiao Lin, Yang Yang and Xinxia Yang
Symmetry 2024, 16(1), 65; https://0-doi-org.brum.beds.ac.uk/10.3390/sym16010065 - 04 Jan 2024
Viewed by 785
Abstract
Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation [...] Read more.
Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations. Full article
(This article belongs to the Special Issue Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations)
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21 pages, 2716 KiB  
Article
Using Double Integral Transform (Laplace-ARA Transform) in Solving Partial Differential Equations
by Abdelilah Kamal Sedeeg, Zahra. I. Mahamoud and Rania Saadeh
Symmetry 2022, 14(11), 2418; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14112418 - 15 Nov 2022
Cited by 8 | Viewed by 1686
Abstract
The main goal of this research is to present a new approach to double transforms called the double Laplace–ARA transform (DL-ARAT). This new double transform is a novel combination of Laplace and ARA transforms. We present the basic properties of the new approach [...] Read more.
The main goal of this research is to present a new approach to double transforms called the double Laplace–ARA transform (DL-ARAT). This new double transform is a novel combination of Laplace and ARA transforms. We present the basic properties of the new approach including existence, linearity and some results related to partial derivatives and the double convolution theorem. To obtain exact solutions, the new double transform is applied to several partial differential equations such as the Klein–Gordon equation, heat equation, wave equation and telegraph equation; each of these equations has great utility in physical applications. In symmetry to other symmetric transforms, we conclude that our new approach is simpler and needs less calculations. Full article
(This article belongs to the Special Issue Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations)
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15 pages, 280 KiB  
Article
Lie Symmetry Analysis, Particular Solutions and Conservation Laws of a New Extended (3+1)-Dimensional Shallow Water Wave Equation
by Cailing Huo and Lianzhong Li
Symmetry 2022, 14(9), 1855; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091855 - 06 Sep 2022
Cited by 6 | Viewed by 1394
Abstract
In this paper, a new extended (3+1)-dimensional shallow water wave equation is discussed via Lie symmetry analysis. Making use of symmetric nodes, we obtain two kinds of symmetrically reduced ODEs. By means of power series, we obtain the two kinds of exact power [...] Read more.
In this paper, a new extended (3+1)-dimensional shallow water wave equation is discussed via Lie symmetry analysis. Making use of symmetric nodes, we obtain two kinds of symmetrically reduced ODEs. By means of power series, we obtain the two kinds of exact power series solutions. By invoking a new conservation theorem of Ibragimov, the conservation laws are constructed. Full article
(This article belongs to the Special Issue Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations)
11 pages, 249 KiB  
Article
Generalized Nonlocal Symmetries of Two-Component Camassa–Holm and Hunter–Saxton Systems
by Zhenhua Shi and Yan Li
Symmetry 2022, 14(3), 528; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030528 - 04 Mar 2022
Cited by 1 | Viewed by 1378
Abstract
The two-component Camassa–Holm system and two-component Hunter–Saxton system are completely integrable models. In this paper, it is shown that these systems admit nonlocal symmetries by their geometric integrability. As an application, we obtain the recursion operator and conservation laws by using this kind [...] Read more.
The two-component Camassa–Holm system and two-component Hunter–Saxton system are completely integrable models. In this paper, it is shown that these systems admit nonlocal symmetries by their geometric integrability. As an application, we obtain the recursion operator and conservation laws by using this kind of nonlocal symmetries. Full article
(This article belongs to the Special Issue Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations)
10 pages, 256 KiB  
Article
A New Method for Blow-Up to Scale-Invariant Damped Wave Equations with Derivatives and Combined Nonlinear Terms
by Yuanming Chen
Symmetry 2022, 14(2), 198; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020198 - 20 Jan 2022
Cited by 2 | Viewed by 1115
Abstract
The Cauchy problems of scale-invariant damped wave equations with derivative nonlinear terms and with combined nonlinear terms are studied. A new method is provided to show that the solutions will blow up in a finite time, if the nonlinear powers satisfy some conditions. [...] Read more.
The Cauchy problems of scale-invariant damped wave equations with derivative nonlinear terms and with combined nonlinear terms are studied. A new method is provided to show that the solutions will blow up in a finite time, if the nonlinear powers satisfy some conditions. The method is based on constructing appropriate test functions, by using the solution of an ordinary differential equation. It may be useful to prove the nonexistence for global solutions for other nonlinear evolution equations. Full article
(This article belongs to the Special Issue Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations)
27 pages, 2899 KiB  
Article
A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions
by Meng-Li Qin, Xiao-Yong Wen and Manwai Yuen
Symmetry 2021, 13(12), 2315; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122315 - 03 Dec 2021
Cited by 3 | Viewed by 1285
Abstract
This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such [...] Read more.
This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics. Full article
(This article belongs to the Special Issue Exact Solutions with Symmetry Reduction and Long Time Behaviors of Non-linear Partial Differential Equations)
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