Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media

Topic Information

Dear Colleagues,

Crystalline features have a wide range of uses in semiconductors, superconductors, optical nonlinear and laser crystals, solid states, engineering, industrial applications, and technological advancements. Mechanical and optical characteristics may cause some critical phenomena that need modern theoretical methods for physics-based growth descriptions such as crystal growth models. This nonlinear effect may modulate the dynamical behaviors in crystal models that relate to nonlinear equations in crystal physics.

Random applications in physical systems have been a topic of extensive theoretical and analytical research around the world in recent decades. Furthermore, various complex nonlinear wave propagation models have been developed in several fields, such as superfluids, chemical physics, plasma physics, semiconductor, solid state, biophysics, optical fibers, plasma fluids, solid-state physics, quantum mechanics and super-conductivity.

The purpose of this Research Topic is to bring together top researchers and academicians from a variety of fields concerned with theoretical physics, experimental solid-state physics, engineering and applied mathematics. For the submitted papers, we are especially interested in experimental, theoretical, analytical, and numerical modeling as well as new progresses in random differential equations in mathematical physics.

Dr. Mahmoud A.E. Abdelrahman
Prof. Dr. Emad El-Shewy
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Crystals crystals	2.4	4.2	2011	11.1 Days	CHF 2100
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700
Axioms axioms	1.9	-	2012	22.8 Days	CHF 2400

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Published Papers (8 papers)

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All Crystals Mathematics Symmetry Fractal Fract Axioms

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11 pages, 884 KiB  

Open AccessEditor’s ChoiceArticle

One-Dimensional Gap Soliton Molecules and Clusters in Optical Lattice-Trapped Coherently Atomic Ensembles via Electromagnetically Induced Transparency

by Zhiming Chen, Hongqiang Xie, Qi Zhou and Jianhua Zeng

Crystals 2024, 14(1), 36; https://doi.org/10.3390/cryst14010036 - 27 Dec 2023

Viewed by 1390

Abstract

In past years, optical lattices have been demonstrated as an excellent platform for making, understanding, and controlling quantum matters at nonlinear and fundamental quantum levels. Shrinking experimental observations include matter-wave gap solitons created in ultracold quantum degenerate gases, such as Bose–Einstein condensates with [...] Read more.

In past years, optical lattices have been demonstrated as an excellent platform for making, understanding, and controlling quantum matters at nonlinear and fundamental quantum levels. Shrinking experimental observations include matter-wave gap solitons created in ultracold quantum degenerate gases, such as Bose–Einstein condensates with repulsive interaction. In this paper, we theoretically and numerically study the formation of one-dimensional gap soliton molecules and clusters in ultracold coherent atom ensembles under electromagnetically induced transparency conditions and trapped by an optical lattice. In numerics, both linear stability analysis and direct perturbed simulations are combined to identify the stability and instability of the localized gap modes, stressing the wide stability region within the first finite gap. The results predicted here may be confirmed in ultracold atom experiments, providing detailed insight into the higher-order localized gap modes of ultracold bosonic atoms under the quantum coherent effect called electromagnetically induced transparency. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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16 pages, 1668 KiB  

Open AccessArticle

A Study of Traveling Wave Structures and Numerical Investigations into the Coupled Nonlinear Schrödinger Equation Using Advanced Mathematical Techniques

by Taghread Ghannam Alharbi and Abdulghani Alharbi

Mathematics 2023, 11(22), 4597; https://doi.org/10.3390/math11224597 - 9 Nov 2023

Cited by 3 | Viewed by 1329

Abstract

This article explores adapted mathematical methods to solve the coupled nonlinear Schrödinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrödinger equation into real and [...] Read more.

This article explores adapted mathematical methods to solve the coupled nonlinear Schrödinger (C-NLS) equation through analytical and numerical methods. To obtain exact solutions for the (C-NLS) equation, we utilize the improved modified, extended tanh-function method. By separating the Schrödinger equation into real and imaginary parts, we can obtain four coupled equations, which we then analyze using the generalized tanh method to extract exact solutions. This system of equations is essential for understanding the behavior of quantum systems and has various applications in quantum mechanics. We obtain an analytical solution and demonstrate numerical solutions using implicit finite difference. Studies have shown that this scheme is second-order in space and time, and the von Neumann stability analysis confirms its unconditional stability. We introduce the comparison between numerical and exact solutions. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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12 pages, 957 KiB  

Open AccessArticle

Modulations of Collapsing Stochastic Modified NLSE Structures

by Mahmoud A. E. Abdelrahman, Emad K. El-Shewy, Y. Omar and N. F. Abdo

Mathematics 2023, 11(20), 4330; https://doi.org/10.3390/math11204330 - 18 Oct 2023

Cited by 3 | Viewed by 1023

Abstract

The exact solutions of the nonlinear Schrödinger equation (NLSE) predict consistent novel applicable existences such as solitonic localized structures, rouge forms, and shocks that rely on physical phenomena to propagate. Theoretical explanations of randomly nonlinear new extension NLSE structure solutions have undergone stochastic [...] Read more.

The exact solutions of the nonlinear Schrödinger equation (NLSE) predict consistent novel applicable existences such as solitonic localized structures, rouge forms, and shocks that rely on physical phenomena to propagate. Theoretical explanations of randomly nonlinear new extension NLSE structure solutions have undergone stochastic mode examination. This equation enables accurate and efficient solutions capable of simulating developed solitonic structures with dynamic features. The generated random waves are a dynamically regulated system that are influenced by random water currents behaviour. It has been noticed that the stochastic parameter modulates the wave force and supplies the wave collapsing energy with related medium turbulence. It has been observed that noise effects can alter wave characteristics, which may lead to innovative astrophysics, physical density, and ocean waves. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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12 pages, 751 KiB  

Open AccessArticle

Langmuir Forcing and Collapsing Subsonic Density Cavitons via Random Modulations

by Maged A. Azzam, H. G. Abdelwahed, Emad K. El-Shewy and Mahmoud A. E. Abdelrahman

Symmetry 2023, 15(8), 1558; https://doi.org/10.3390/sym15081558 - 9 Aug 2023

Cited by 4 | Viewed by 1340

Abstract

Electrostatic nonlinear random Langmuir structures have been propagated in stochastic magnetospheres, clouds and solar wind. A theoretical description of Langmuir waves can be modeled by Schrödinger and Zakharov models with stochastic terms. It was explained that the stochastic parameter affects the forcing, collapsing [...] Read more.

Electrostatic nonlinear random Langmuir structures have been propagated in stochastic magnetospheres, clouds and solar wind. A theoretical description of Langmuir waves can be modeled by Schrödinger and Zakharov models with stochastic terms. It was explained that the stochastic parameter affects the forcing, collapsing in strongly density turbulence and density crystalline structures. The unified method has been implemented to provide new stochastic solutions for a Zakharov system in subsonic limit with noises via the Itô sense. This unified approach provides a variety of advantages, such as avoiding difficult calculations and explicitly providing pivotal solutions. It is easy to use, efficient, and precise. The induced generated energy during the collapsing of solar Langmuir wave bursts and clouds is determined by the solitonic formations. In addition, the collapsing strong turbulence or forcing density crystalline structures depend mainly on stochastic processes. Furthermore, electrostatic waves in clouds that may collapse are represented sometimes as dissipative shapes. So, the results of this investigation could be applicable to observations of energy seeding and collapsing in clouds. This energy is based on the electrostatic field and its related densities’ perturbation in subsonic limits. Finally, it has been explored how noise parameters in the Itô sense affect the solar wind Langmuir waves’ properties. So, the findings of this discussion may be applicable to real observations of energy collapsing and seeding in clouds. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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18 pages, 7921 KiB  

Open AccessArticle

Negative Linear Compressibility of Formate Crystals from the Viewpoint of Quantum Electronic Pressure

by Yury V. Matveychuk, Sergey A. Sobalev, Polina I. Borisova, Ekaterina V. Bartashevich and Vladimir G. Tsirelson

Crystals 2023, 13(7), 1147; https://doi.org/10.3390/cryst13071147 - 23 Jul 2023

Cited by 3 | Viewed by 1724

Abstract

In order to understand the phenomenon of negative linear compressibility (NLC) in organic crystals, it is necessary to investigate not only the structural features but also the electronic changes taking place under external hydrostatic pressure. It is also necessary to clarify which electronic [...] Read more.

In order to understand the phenomenon of negative linear compressibility (NLC) in organic crystals, it is necessary to investigate not only the structural features but also the electronic changes taking place under external hydrostatic pressure. It is also necessary to clarify which electronic properties allow the quantification and comparison of the compressibility of crystals. In our study, the crystal structures of sodium and cadmium formates under hydrostatic compression were modeled, as well as the α and β-phases of calcium formate. The changes in cell parameters and spatial dependences of the linear compressibility were analyzed, and the ranges of external pressure, which must be applied for NLC onset, were predicted for the sodium and α-calcium formates. Although the behavior of chemical bonds is not predicted by the sign or absolute value of the quantum electronic pressure, its relative change under external pressure clearly distinguishes the soft and rigid regions in a crystal. The relationship between the NLC values and the changes in quantum electronic pressure in the cavities of formate crystals was established. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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23 pages, 6620 KiB  

Open AccessArticle

Study of Thermoelectric Responses of a Conductive Semi-Solid Surface to Variable Thermal Shock in the Context of the Moore–Gibson–Thompson Thermoelasticity

by Sami F. Megahid, Ahmed E. Abouelregal, Sameh S. Askar and Marin Marin

Axioms 2023, 12(7), 659; https://doi.org/10.3390/axioms12070659 - 3 Jul 2023

Cited by 9 | Viewed by 1759

Abstract

In this study, the Moore–Gibson–Thompson (MGT) concept of thermal conductivity is applied to a two-dimensional elastic solid in the form of a half-space. This model was constructed using Green and Naghdi’s thermoelastic model to address the infinite velocity problem of heat waves. It [...] Read more.

In this study, the Moore–Gibson–Thompson (MGT) concept of thermal conductivity is applied to a two-dimensional elastic solid in the form of a half-space. This model was constructed using Green and Naghdi’s thermoelastic model to address the infinite velocity problem of heat waves. It has been taken into account that the free surface of the medium is immersed in an electromagnetic field of constant intensity, undergoes thermal shock, and rotates with a uniform angular velocity. The governing equations of a modified version of Ohm’s law account for the impact of temperature gradients and charge densities. By using the method of normal mode analysis, an analytical representation of the studied physical fields was obtained. The effect of rotation and the modulus of modified Ohm’s law on the responses of the field distributions examined is discussed, along with accompanying graphical representations. Other thermoelastic models have been compared with the results of the proposed system when the relaxation time is ignored. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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14 pages, 3887 KiB  

Open AccessArticle

A Novel Strategy for Comprehensive Estimation of Lattice Energy, Bulk Modulus, Chemical Hardness and Electronic Polarizability of A^NB^8-N Binary Inorganic Crystals

by Xinyu Zhao and Xiaoli Wang

Crystals 2023, 13(4), 668; https://doi.org/10.3390/cryst13040668 - 12 Apr 2023

Cited by 5 | Viewed by 1924

Abstract

How to search for a convenient method without a complicated calculation process to predict the physicochemical properties of inorganic crystals through a simple micro-parameter is a greatly important issue in the field of materials science. Herein, this paper presents a new and facile [...] Read more.

How to search for a convenient method without a complicated calculation process to predict the physicochemical properties of inorganic crystals through a simple micro-parameter is a greatly important issue in the field of materials science. Herein, this paper presents a new and facile technique for the comprehensive estimation of lattice energy (U), bulk modulus (B), chemical hardness (ƞ), and electronic polarizability (α), just by using a simple mathematic fitting formula with a few structure parameters, such as the systems of rock salt crystals (group I–VII, II–VI) and tetrahedral coordinated crystals (group II–VI, III–V). For the typical binary A^NB^8-N crystal systems, our present conclusions suggest that a good quantitative correlation between U, B, ƞ, α and chemical bond length (d) is observed, the normal mathematical expression is P = a·d^b (P represents these physicochemical parameters), constants a and b depend on the type of crystals, and the relevant squares of the correlation coefficient (R²) are larger than 0.9. The results indicate that lattice energy, bulk modulus, and chemical hardness decrease with increases in chemical bond length, but electronic polarizability increases with an increase in chemical bond length. Meanwhile, the new data on the lattice energy, bulk modulus, chemical hardness, and electronic polarizability values of binary A^NB^8-N crystal systems considered in the present study are calculated via the obtained curve fitting equations without any complex calculation process. We find that there is a very good linear trend in our calculated results along with the values reported in the literature. The present study will be important in solid-state chemistry, which may give researchers useful guidance in searching for relevant data for predicting the properties of new materials or synthetic routes based on a simple mathematic empirical model. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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19 pages, 2856 KiB  

Open AccessArticle

Studying the Thermoelastic Waves Induced by Pulsed Lasers Due to the Interaction between Electrons and Holes on Semiconductor Materials under the Hall Current Effect

by Nidhal Becheikh, Nejib Ghazouani, Alaa A. El-Bary and Khaled Lotfy

Crystals 2023, 13(4), 665; https://doi.org/10.3390/cryst13040665 - 12 Apr 2023

Cited by 3 | Viewed by 1594

Abstract

In the present work, the interaction between electrons and holes in semiconductor materials is investigated. According to the excitation process, the optical-elastic-thermal-diffusion (OETD) process is considered when the medium is exposed to a strong magnetic field and laser pulses. Photo-elastic and photo-electronics deformations [...] Read more.

In the present work, the interaction between electrons and holes in semiconductor materials is investigated. According to the excitation process, the optical-elastic-thermal-diffusion (OETD) process is considered when the medium is exposed to a strong magnetic field and laser pulses. Photo-elastic and photo-electronics deformations are taken into account when the Hall current impact appears due to the magnetic field pressure on the semiconductor medium. Due to the complexity of the model, the governing equations that describe the system in one dimension (1D) are studied. Mathematical transformations (Laplace transform) were used to simplify the equations to obtain the physical quantities under study which were affected by laser pulses. To obtain complete solutions, some conditions were obtained from the free surface as well as from a mechanical ramp type and pulse heat flux, and then numerical transformations were applied using the inverse Laplace transform. Under the influence of several variables in this question, the results were explained graphically for silicon (Si) material and the results were analyzed in terms of their physical significance. Full article

(This article belongs to the Topic Mathematical Applications of Nonlinear Wave Properties in Crystalline and Dispersive Media)

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