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19 pages, 560 KiB

Open AccessArticle

Eigenvalue Spectra of Rabi Models with Infinite Matrix Representations

by Hongbin Liang, Shucan Xia, Yixiang Chen, Yuguo Su and Jie Chen

Axioms 2025, 14(4), 263; https://doi.org/10.3390/axioms14040263 - 30 Mar 2025

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We investigate the relationship between confluent Heun functions and the eigenvalue spectra of infinite matrices related to the semi-classical and quantum Rabi models, revealing distinct connections in each case. In the semi-classical model, the eigenvalues are explicitly expressed through confluent Heun functions, whereas [...] Read more.

We investigate the relationship between confluent Heun functions and the eigenvalue spectra of infinite matrices related to the semi-classical and quantum Rabi models, revealing distinct connections in each case. In the semi-classical model, the eigenvalues are explicitly expressed through confluent Heun functions, whereas in the quantum Rabi model, they are determined by zeros of a condition involving confluent Heun functions. Our findings establish a unified framework for solving the eigenvalue problem of infinite-dimensional unbounded matrices related to the Rabi models. We derive some new identities for confluent Heun functions, enabling simplifications and broader applications in mathematics and physics. The explicit eigenvalue expressions in the semi-classical case align with approximate results from earlier studies, while the derived conditions for the quantum model provide a concise and unified form, encompassing special cases that are typically treated as exceptions. We also discuss the energy spectrum of the quantum Rabi model, uncovering intriguing phenomena and patterns. Our results deepen the understanding of Rabi models and extend their potential applications in quantum optics and quantum information. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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11 pages, 501 KiB

Open AccessArticle

Relativistic Scalar Particle Systems in a Spacetime with a Spiral-like Dislocation

by Ricardo L. L. Vitória

Axioms 2025, 14(3), 227; https://doi.org/10.3390/axioms14030227 - 19 Mar 2025

Viewed by 247

Abstract

We have analyzed solutions of bound states of a scalar particle in spacetime with torsion. In the first analysis, we investigate the confinement of a scalar particle in a cylindrical shell. In the second step, we investigate the Klein–Gordon oscillator. Then, we finish [...] Read more.

We have analyzed solutions of bound states of a scalar particle in spacetime with torsion. In the first analysis, we investigate the confinement of a scalar particle in a cylindrical shell. In the second step, we investigate the Klein–Gordon oscillator. Then, we finish our analysis by searching for solutions of bound states of the Klein–Gordon oscillator by interacting with a hard-wall potential. In all these systems, we determine the relativistic energy profile in the background characterized by the presence of torsion in spacetime represented by a spiral-like dislocation. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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13 pages, 3104 KiB

Open AccessArticle

Interaction of a Four-Level Atom with a Deformed Quantum Field: Mathematical Model and Quantum Resources

by Mariam Algarni, Sayed Abdel-Khalek and Kamal Berrada

Axioms 2025, 14(3), 211; https://doi.org/10.3390/axioms14030211 - 13 Mar 2025

Viewed by 273

Abstract

We introduce a framework presenting the interaction between a four-level atom (F-LA) and a field mode that begins in a coherent state within the para-Bose field (P-BF). The F-LA is considered in a cascade configuration and initially prepared in the upper level. We [...] Read more.

We introduce a framework presenting the interaction between a four-level atom (F-LA) and a field mode that begins in a coherent state within the para-Bose field (P-BF). The F-LA is considered in a cascade configuration and initially prepared in the upper level. We display the system dynamics by solving the motion equation. We discuss various dynamical behaviors of fundamental quantum resources used in quantum optics and information tasks, including atomic population inversion, quantum entanglement (QE), and the statistical properties of the P-BF based on the parameters of the quantum model. In this context, we demonstrate the impact of various system parameters on these quantum resources. Finally, we illustrate the dynamic relationships among the quantum resources within the model. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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14 pages, 264 KiB

Open AccessArticle

Strong Stability of the Thermoelastic Bresse System with Second Sound and Fractional Delay

by Khaled Zennir and Loay Alkhalifa

Axioms 2025, 14(3), 176; https://doi.org/10.3390/axioms14030176 - 27 Feb 2025

Viewed by 285

Abstract

The thermoelastic Bresse system is a mathematical model that describes the dynamic behavior of elastic beams accounting for both mechanical deformations and thermal effects. Incorporating concepts such as second sound and fractional delay into this system enhances its ability to model complex physical [...] Read more.

The thermoelastic Bresse system is a mathematical model that describes the dynamic behavior of elastic beams accounting for both mechanical deformations and thermal effects. Incorporating concepts such as second sound and fractional delay into this system enhances its ability to model complex physical phenomena. The paper studies a Bresse thermoelastic system with fractional delay and second sound. Firstly, we prove the existence and uniqueness of the solution for our system using semi-group theory. Additionally, we derive an exponential decay estimate for the associated semi-group utilizing suitable multiplier techniques. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

18 pages, 2240 KiB

Open AccessArticle

The Characteristic Relation in Two-Dimensional Type I Intermittency

by Juan Colman and Sergio Elaskar

Axioms 2025, 14(1), 24; https://doi.org/10.3390/axioms14010024 - 31 Dec 2024

Viewed by 477

Abstract

To explore intermittency in discrete systems with two or more degrees of freedom, we analyze the general characteristics of type I intermittency within a two-dimensional map. This investigation is carried out numerically, concentrating on the system’s attractors, bifurcation diagrams, and the characteristic relation [...] Read more.

To explore intermittency in discrete systems with two or more degrees of freedom, we analyze the general characteristics of type I intermittency within a two-dimensional map. This investigation is carried out numerically, concentrating on the system’s attractors, bifurcation diagrams, and the characteristic relation associated with type I intermittency. We present two methods for determining the laminar interval and the channel structure. Our computations yield numerical results for the average laminar length as a function of the control parameter, which we then compare with findings from intermittency in one-dimensional maps. We observe a strong agreement between the numerical data and the theoretical predictions. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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37 pages, 424 KiB

Open AccessArticle

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

by Po-Chun Huang and Bo-Yu Pan

Axioms 2024, 13(10), 673; https://doi.org/10.3390/axioms13100673 - 29 Sep 2024

Viewed by 751

Abstract

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of [...] Read more.

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

24 pages, 6028 KiB

Open AccessArticle

Combined Compact Symplectic Schemes for the Solution of Good Boussinesq Equations

by Zhenyu Lang, Xiuling Yin, Yanqin Liu, Zhiguo Chen and Shuxia Kong

Axioms 2024, 13(9), 574; https://doi.org/10.3390/axioms13090574 - 23 Aug 2024

Viewed by 637

Abstract

Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic [...] Read more.

Good Boussinesq equations are considered in this work. First, we apply three combined compact schemes to approximate spatial derivatives of good Boussinesq equations. Then, three fully discrete schemes are developed based on a symplectic scheme in the time direction, which preserves the symplectic structure. Meanwhile, the convergence and conservation of the fully discrete schemes are analyzed. Finally, we present numerical experiments to confirm our theoretical analysis. Both our analysis and numerical tests indicate that the fully discrete schemes are efficient in solving the spatial derivative mixed equation. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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39 pages, 514 KiB

Open AccessArticle

Well-Posedness of the Schrödinger–Korteweg–de Vries System with Robin Boundary Conditions on the Half-Line

by Po-Chun Huang and Bo-Yu Pan

Axioms 2024, 13(8), 508; https://doi.org/10.3390/axioms13080508 - 28 Jul 2024

Cited by 1 | Viewed by 807

Abstract

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, [...] Read more.

The Schrödinger–Korteweg–de Vries (SKdV) system can describe the nonlinear dynamics of phenomena such as Langmuir and ion acoustic waves, which are highly valuable for studying wave behavior and interactions. The SKdV system has wide-ranging applications in physics and applied mathematics. In this article, we investigate the local well-posedness of the SKdV system with Robin boundary conditions and polynomial terms in the Sobolev space. We want to enhance the applicability of this type of SKdV system. Our verification process is as follows: We estimate Fokas solutions for the Robin problem with external forces. Next, we define an iteration map in suitable solution space and prove the iteration map is a contraction mapping and onto some closed ball

B (0, r)

. Finally, by the contraction mapping theorem, we obtain the uniqueness solution. Moreover, we show that the data-to-solution map is locally Lipschitz continuous and conclude with the well-posedness of the SKdV system. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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12 pages, 275 KiB

Open AccessArticle

Special Geometric Objects in Generalized Riemannian Spaces

by Marko Stefanović, Nenad Vesić, Dušan Simjanović and Branislav Randjelović

Axioms 2024, 13(7), 463; https://doi.org/10.3390/axioms13070463 - 9 Jul 2024

Viewed by 682

Abstract

In this paper, we obtained the geometrical objects that are common in different definitions of the generalized Riemannian spaces. These objects are analogies to the Thomas projective parameter and the Weyl projective tensor. After that, we obtained some geometrical objects important for applications [...] Read more.

In this paper, we obtained the geometrical objects that are common in different definitions of the generalized Riemannian spaces. These objects are analogies to the Thomas projective parameter and the Weyl projective tensor. After that, we obtained some geometrical objects important for applications in physics. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

22 pages, 693 KiB

Open AccessArticle

Biequivalent Planar Graphs

by Bernard Piette

Axioms 2024, 13(7), 437; https://doi.org/10.3390/axioms13070437 - 28 Jun 2024

Cited by 2 | Viewed by 745

Abstract

We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an [...] Read more.

We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an algebraic method, which is followed by an algorithmic method, to determine all such graphs with up to 300 nodes each with valencies ranging between three and six. We also present a graphic representation of every graph found. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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14 pages, 1633 KiB

Open AccessArticle

An Optimization Problem for Computing Predictive Potential of General Sum/Product-Connectivity Topological Indices of Physicochemical Properties of Benzenoid Hydrocarbons

by Sakander Hayat, Azri Arfan, Asad Khan, Haziq Jamil and Mohammed J. F. Alenazi

Axioms 2024, 13(6), 342; https://doi.org/10.3390/axioms13060342 - 22 May 2024

Cited by 1 | Viewed by 840

Abstract

For a graph

G = (V_{G}, E_{G})

, a degree-based graphical index

G I_{d}

takes the general form

G I_{d} = \sum_{x y \in E_{G}} ϕ (d_{x}, d_{y})

, [...] Read more.

For a graph

G = (V_{G}, E_{G})

, a degree-based graphical index

G I_{d}

takes the general form

G I_{d} = \sum_{x y \in E_{G}} ϕ (d_{x}, d_{y})

, where

ϕ

is a symmetric map and

d_{i}

is the degree of

i \in V_{G}

. For

α \in R

, if

ϕ = {(d_{x} d_{y})}^{α}

(resp.

ϕ = {(d_{x} + d_{y})}^{α}

), the index is called the general product-connectivity

R_{α}

(resp. general sum-connectivity

S C I_{α}

) index. In this paper, by formulating an optimization problem, we determine the value(s) of

α

, for which the linear/multiple correlation coefficient of

R_{α}

and

S C I_{α}

with physicochemical properties of benzenoid hydrocarbons is the strongest. This, in turn, fills some research gaps left by similar studies in this area. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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24 pages, 20089 KiB

Open AccessArticle

Basic Computational Algorithms for Representing an Aircraft Flight (Calculation of 3D Displacement and Displaying)

by Adan Ramirez-Lopez

Axioms 2024, 13(5), 313; https://doi.org/10.3390/axioms13050313 - 10 May 2024

Cited by 2 | Viewed by 1838

Abstract

This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. [...] Read more.

This manuscript describes the computational process to calculate an airplane path and display it in a 2D and 3D coordinate system on a computer screen. The airplane movement is calculated as a function of its dynamic’s conditions according to physical and logical theory. Here, the flight is divided into maneuvers and the aircraft conditions are defined as boundary conditions. Then the aircraft position is calculated using nested loops, which execute the calculation procedure at every step time (Δt). The calculation of the aircraft displacement is obtained as a function of the aircraft speed and heading angles. The simulator was created using the C++ programming language, and each part of the algorithm was compiled independently to reduce the source code, allow easy modification, and improve the programming efficiency. Aerial navigation involves very complex phenomena to be considered for an appropriate representation; moreover, in this manuscript, the influence of the mathematical approach to properly represent the aircraft flight is described in detail. The flight simulator was successfully tested by simulating some basic theoretical flights with different maneuvers, which include stationary position, running along the way, take off, and some movements in the airspace. The maximum aircraft speed tested was 120 km/h, the maximum maneuver time was 12 min, and the space for simulation was assumed to be without obstacles. Here, the geometrical description of path and speed is analyzed according to the symmetric and asymmetric results. Finally, an analysis was conducted to evaluate the approach of the numerical methods used; after that, it was possible to confirm that precision increased as the step time was reduced. According to this analysis, no more than 500 steps are required for a good approach in the calculation of the aircraft displacement. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)

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