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Axioms
Special Issues
Differential Equations and Its Application
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Differential Equations and Its Application

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 June 2025 | Viewed by 6918

Special Issue Editors

Dr. Yuting Ding

E-Mail Website
Guest Editor
College of Science, Northeast Forestry University, Harbin, China
Interests: functional differential equations (bifurcation theory and numerical analysis); reaction diffusion equation (bifurcation theory of and its application); mathematical biology
Prof. Dr. Feng Li

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Linyi University, Linyi, China
Interests: bifurcation theory of and its application; mathematical biology
Dr. Patricia J. Y. Wong

E-Mail Website
Guest Editor
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
Interests: differential equations; difference equations; integral equations; numerical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Axioms is an international, open access journal that provides an advanced forum for mathematics, mathematical logic, and mathematical physics studies. This Special Issue focuses on Differential Equations and Applications, particularly the mathematical aspects of differential equations and on applications of the mathematical theory to issues arising in the physical, engineering, financial, and life sciences.  In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following: dynamic stability, local and global methods, bifurcations, chaos, and deterministic and random vibrations.

We look forward to receiving your contributions.

Dr. Yuting Ding
Prof. Dr. Feng Li
Dr. Patricia J. Y. Wong
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. Axioms 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

  • mathematical model
  • dynamic system
  • bifurcation
  • chaos
  • normal form

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Published Papers (7 papers)

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Research

37 pages, 2252 KiB  
Article
Rogue Waves in the Nonlinear Schrödinger, Kadomtsev–Petviashvili, Lakshmanan–Porsezian–Daniel and Hirota Equations
by Pierre Gaillard
Axioms 2025, 14(2), 94; https://doi.org/10.3390/axioms14020094 - 27 Jan 2025
Cited by 1 | Viewed by 843
Abstract
We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS [...] Read more.
We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order 2N. These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appear as deformations of the Peregrine breathers PN as the last one can be obtained when all parameters are equal to zero. At order N, these solutions are the product of a ratio of two polynomials of degree N(N+1) in x and t by an exponential depending on time t and depending on 2N−2 real parameters: they are called quasi-rational solutions. For the KPI equation, we explicitly obtain solutions at order N depending on 2N−2 real parameters. We present different examples of rogue waves for the LPD and Hirota equations. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
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16 pages, 276 KiB  
Article
Existence and Uniqueness of Second-Order Impulsive Delay Differential Systems
by Yingxia Zhou and Mengmeng Li
Axioms 2024, 13(12), 834; https://doi.org/10.3390/axioms13120834 - 27 Nov 2024
Viewed by 584
Abstract
In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and [...] Read more.
In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
15 pages, 294 KiB  
Article
Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs
by Dandan Yang, Zhenyu Bai and Chuanzhi Bai
Axioms 2024, 13(11), 762; https://doi.org/10.3390/axioms13110762 - 3 Nov 2024
Cited by 1 | Viewed by 880
Abstract
In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N [...] Read more.
In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
16 pages, 299 KiB  
Article
Morse Thoery of Saddle Point Reduction with Applications
by Ran Yang and Qin Xing
Axioms 2024, 13(9), 603; https://doi.org/10.3390/axioms13090603 - 4 Sep 2024
Viewed by 995
Abstract
In this paper, we demonstrate that when saddle point reduction is applicable, there is a clear relationship between the Morse index and the critical groups before and after the reduction. As an application of this result, we use saddle point reduction along with [...] Read more.
In this paper, we demonstrate that when saddle point reduction is applicable, there is a clear relationship between the Morse index and the critical groups before and after the reduction. As an application of this result, we use saddle point reduction along with the critical point theorem to show the existence of periodic solutions in second-order Hamiltonian systems. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
10 pages, 256 KiB  
Article
Coexistence of Algebraic Limit Cycles and Small Limit Cycles of Two Classes of Near-Hamiltonian Systems with a Nilpotent Singular Point
by Huimei Liu, Meilan Cai and Feng Li
Axioms 2024, 13(9), 593; https://doi.org/10.3390/axioms13090593 - 30 Aug 2024
Viewed by 684
Abstract
In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist 2n+1 limit cycles, which include an algebraic limit [...] Read more.
In this paper, two classes of near-Hamiltonian systems with a nilpotent center are considered: the coexistence of algebraic limit cycles and small limit cycles. For the first class of systems, there exist 2n+1 limit cycles, which include an algebraic limit cycle and 2n small limit cycles. For the second class of systems, there exist n2+3n+22 limit cycles, including an algebraic limit cycle and n2+3n2 small limit cycles. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
14 pages, 270 KiB  
Article
Perturbed Dirac Operators and Boundary Value Problems
by Xiaopeng Liu and Yuanyuan Liu
Axioms 2024, 13(6), 363; https://doi.org/10.3390/axioms13060363 - 29 May 2024
Cited by 1 | Viewed by 930
Abstract
In this paper, the time-independent Klein-Gordon equation in R3 is treated with a decomposition of the operator Δ−γ2I by the Clifford algebra Cl(V3,3). Some properties of integral operators associated the [...] Read more.
In this paper, the time-independent Klein-Gordon equation in R3 is treated with a decomposition of the operator Δ−γ2I by the Clifford algebra Cl(V3,3). Some properties of integral operators associated the kind of equations and some Riemann-Hilbert boundary value problems for perturbed Dirac operators are investigated. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
16 pages, 281 KiB  
Article
Solving Nonlinear Second-Order ODEs via the Eisenhart Lift and Linearization
by Andronikos Paliathanasis
Axioms 2024, 13(5), 331; https://doi.org/10.3390/axioms13050331 - 16 May 2024
Cited by 3 | Viewed by 1053
Abstract
The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The [...] Read more.
The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis. Full article
(This article belongs to the Special Issue Differential Equations and Its Application)
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