Erratum: Solutions of the nonlinear 3‐wave equations in three spatial dimensions [J. Math. Phys. 20, 1653 (1979)]

1981 ◽  
Vol 22 (1) ◽  
pp. 219-219
Author(s):  
H. Cornille
Keyword(s):  
Wave Equations ◽  
Spatial Dimensions ◽  
Math Phys

Short wave limit of the Novikov equation and its integrable semi-discretizations

2021 ◽  
Author(s):  
Ruyun Ma ◽  
Yujuan Zhang ◽  
Na Xiong ◽  
Bao-Feng Feng
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Short Wave ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Special Cases ◽  
Two Component ◽  
Novikov Equation ◽  
Wave Limit ◽  
Math Phys

Abstract In this paper, we are concerned with one of the generalized short wave equations proposed by Hone et al. (Lett. Math. Phys 108 927 (2018)). We show that the derivative form of this equation can be viewed as a short wave limit of the Novikov (sw-Novikov) equation. Furthermore, this generalized short wave equation and its derivative form are found to be connected to period 3 reduction of two-dimensional CKP(BKP)-Toda hierarchy, same as the short wave limit of the Depasperis-Procesi (sw-DP) equation. We propose a two-component short wave equation which contain the sw-Novikov equation and sw-DP equation as two special cases. As a main result, we construct two types of integrable semi-discretizations via Hirota’s bilinear method and provide multi-soliton solution to the semi-discrete sw-Novikov equation.

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.

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