Solutions of the nonlinear 3‐wave equations in three spatial dimensions

1979 ◽  
Vol 20 (8) ◽  
pp. 1653-1666 ◽  
Author(s):  
H. Cornille
Keyword(s):  
Wave Equations ◽  
Spatial Dimensions

Nanostructure of semiconductor quantum dots in a borosilicate glass matrix by complementary use of HREM and AEM

1990 ◽  
Vol 48 (4) ◽  
pp. 728-729
Author(s):  
M.J. Kim ◽  
L.C. Liu ◽  
S.H. Risbud ◽  
R.W. Carpenter
Keyword(s):  
Quantum Dots ◽  
Borosilicate Glass ◽  
Glass Matrix ◽  
Optical Switching ◽  
Response Times ◽  
Fast Response ◽  
Processing Technique ◽  
Internal Stresses ◽  
Electron Hole ◽  
Spatial Dimensions

When the size of a semiconductor is reduced by an appropriate materials processing technique to a dimension less than about twice the radius of an exciton in the bulk crystal, the band like structure of the semiconductor gives way to discrete molecular orbital electronic states. Clusters of semiconductors in a size regime lower than 2R {where R is the exciton Bohr radius; e.g. 3 nm for CdS and 7.3 nm for CdTe) are called Quantum Dots (QD) because they confine optically excited electron- hole pairs (excitons) in all three spatial dimensions. Structures based on QD are of great interest because of fast response times and non-linearity in optical switching applications.In this paper we report the first HREM analysis of the size and structure of CdTe and CdS QD formed by precipitation from a modified borosilicate glass matrix. The glass melts were quenched by pouring on brass plates, and then annealed to relieve internal stresses. QD precipitate particles were formed during subsequent "striking" heat treatments above the glass crystallization temperature, which was determined by differential thermal analysis.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

Information theory and dimensions: Enhancing Quantum Mechanics

2015 ◽  
Vol 9 (3) ◽  
pp. 2470-2475
Author(s):  
Bheku Khumalo
Keyword(s):  
Information Theory ◽  
Quantum Tunneling ◽  
Particle Density ◽  
Human Mind ◽  
Particle Theory ◽  
Knowledge Theory ◽  
Spatial Dimensions ◽  
Double Slit Experiment ◽  
Slit Experiment ◽  
Double Slit

This paper seeks to discuss why information theory is so important. What is information, knowledge is interaction of human mind and information, but there is a difference between information theory and knowledge theory. Look into information and particle theory and see how information must have its roots in particle theory. This leads to the concept of spatial dimensions, information density, complexity, particle density, can there be particle complexity, and re-looking at the double slit experiment and quantum tunneling. Information functions/ relations are discussed.

Instability and Collapse in Discrete Wave Equations

2005 ◽  
Vol 5 (3) ◽  
pp. 223-241
Author(s):  
A. Carpio ◽  
G. Duro
Keyword(s):  
Continuum Limit ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Blow Up ◽  
Theoretical Predictions ◽  
Initial States ◽  
The Impact ◽  
The Continuum

AbstractUnstable growth phenomena in spatially discrete wave equations are studied. We characterize sets of initial states leading to instability and collapse and obtain analytical predictions for the blow-up time. The theoretical predictions are con- trasted with the numerical solutions computed by a variety of schemes. The behavior of the systems in the continuum limit and the impact of discreteness and friction are discussed.

History of Tour de France from the Geographical Point of View

2013 ◽  
Vol 22 (3-4) ◽  
pp. 255-277 ◽  
Author(s):  
Vladimír Bačík ◽  
Michal Klobučník
Keyword(s):  
Point Of View ◽  
The Past ◽  
Advanced Technologies ◽  
The World ◽  
Spatial Dimensions ◽  
Tour De France ◽  
Wide Circle ◽  
History Of ◽  
Expert Community ◽  
Geographical Point

Abstract The Tour de France, a three week bicycle race has a unique place in the world of sports. The 100th edition of the event took place in 2013. In the past of 110 years of its history, people noticed unique stories and duels in particular periods, celebrities that became legends that the world of sports will never forget. Also many places where the races unfolded made history in the Tour de France. In this article we tried to point out the spatial context of this event using advanced technologies for distribution of historical facts over the Internet. The Introduction briefly displays the attendance of a particular stage based on a regional point of view. The main topic deals with selected historical aspects of difficult ascents which every year decide the winner of Tour de France, and also attract fans from all over the world. In the final stage of the research, the distribution of results on the website available to a wide circle of fans of this sports event played a very significant part (www.tdfrance.eu). Using advanced methods and procedures we have tried to capture the historical and spatial dimensions of Tour de France in its general form and thus offering a new view of this unique sports event not only to the expert community, but for the general public as well.

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