Integration of the matrix KdV equation with self-consistent source

2013 ◽  
Vol 49 ◽  
pp. 21-27 ◽  
Author(s):  
N. Bondarenko ◽  
G. Freiling ◽  
G. Urazboev
Keyword(s):  
Kdv Equation ◽  
Self Consistent ◽  
The Matrix

scholarly journals On the Integration of the matrix modified Korteweg-de Vries equation with a self-consistent source

2019 ◽  
Vol 50 (3) ◽  
pp. 281-291 ◽  
Author(s):  
G. U. Urazboev ◽  
A. K. Babadjanova
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Scattering Data ◽  
Self Consistent ◽  
De Vries ◽  
The Matrix

In this work we deduce laws of the evolution of the scattering  data for the matrix Zakharov Shabat system with the potential that is the solution of the matrix modied KdV equation with a self consistent source.

scholarly journals Solutions and Painlevé Property for the KdV Equation with Self-Consistent Source

2018 ◽  
Vol 2018 ◽  
pp. 1-16
Author(s):  
Yali Shen ◽  
Ruoxia Yao
Keyword(s):  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Hyperbolic Function ◽  
Group Invariant ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
Self Consistent ◽  
Group Invariant Solutions

In this paper, the polynomial solutions in terms of Jacobi’s elliptic functions of the KdV equation with a self-consistent source (KdV-SCS) are presented. The extended (G′/G)-expansion method is utilized to obtain exact traveling wave solutions of the KdV-SCS, which finally are expressed in terms of the hyperbolic function, the trigonometric function, and the rational function. Meanwhile we find the Lie point symmetry and Lie symmetry group and give several group-invariant solutions for the KdV-SCS. Finally, we supplement the results of the Painlevé property in our previous work and get the Bäcklund transformations of the KdV-SCS.

Self-Consistent Determination of Time-Dependent Behavior of Metals

1981 ◽  
Vol 48 (1) ◽  
pp. 41-46 ◽  
Author(s):  
G. J. Weng
Keyword(s):  
Driving Forces ◽  
Pure Shear ◽  
Time Dependent ◽  
Hardening Law ◽  
Dependent Behavior ◽  
Consistent Model ◽  
Self Consistent ◽  
The Matrix ◽  
Latent Hardening

Though Kro¨ner’s self-consistent model is not fully consistent in the elastic-plastic deformation of polycrystals, it is found to be perfectly consistent in the time-dependent deformation of such materials. Hill’s model, on the other hand, should be used with a modified constraint tensor containing the elastic moduli of the matrix in that case. Kro¨ner’s model is supplemented with a physically consistent constitutive equation for the slip system; these, together with Weng’s inverse method, form the basis of a self-consistent determination of time-dependent behavior of metals. The kinematic component of the latent hardening law and the residual stress introduced in more favorably oriented grains are the two major driving forces for recovery and the Bauschinger effect in creep. The proposed method was applied to predict the creep and recovery strains of a 2618-T61 Aluminum alloy under pure shear, step and nonradial loading. The predicted results are seen to be in generally good agreement with the test data.

Analysis of Mixed Mode Cracks in Two-Dimensional Magnetoelectroelastic Media

2006 ◽  
Vol 324-325 ◽  
pp. 403-406 ◽  
Author(s):  
Han Wang ◽  
Xian Hui Ke ◽  
Ming Hao Zhao
Keyword(s):  
Strain Energy Density ◽  
Mixed Mode ◽  
Strain Energy ◽  
The Self ◽  
Two Dimensional ◽  
Elliptical Cavity ◽  
Self Consistent ◽  
The Matrix ◽  
Consistent Method ◽  
Magnetoelectroelastic Medium

Based on the analytical solution for an elliptical cavity and the self-consistent method, the exact solutions for a crack in a two-dimensional magnetoelectroelastic medium is derived. The strain energy density factors are calculated for mixed mode cracks in a composite made of BaTiO3 as the inclusion and CoFe2O4 as the matrix.

Mechanical Properties and Reinforcement Mechanism of Amorphous Particles Reinforced Aluminum Matrix Composites

2011 ◽  
Vol 87 ◽  
pp. 38-42
Author(s):  
Jin Xiang Wang ◽  
Rui Jin Zhao ◽  
Xiao Li Zhang
Keyword(s):  
Mechanical Properties ◽  
Metallic Glass ◽  
Aluminum Matrix ◽  
Aluminum Matrix Composites ◽  
Matrix Composites ◽  
Consistent Theory ◽  
Reinforcement Mechanism ◽  
Self Consistent ◽  
The Matrix ◽  
Amorphous Particles

The metallic glass particles reinforced aluminum matrix composites without obvious defects were obtained successfully by explosive compaction of mixed powders. The quasi-static compressive mechanical properties of the composites with the reinforcement matrix mass fraction 10%, 15% and 20% respectively were researched recur to universal testing machine of Instron 3367 and self-consistent theory. Finally, reinforcement mechanism of the metallic glass particles on the matrix was analyzed by numerical simulation recur to LS-DYNA program. The results show that the mechanical properties obtained by self-consistent theory are well accord with the experimental results; compared with pure aluminum, the yield stress of the composites with metallic glass particles reinforcement of the mass fraction 20% enhances 46.8 percent; the main reinforcement mechanism is the amorphous particles can undertake higher loading, the combined quality of the reinforcement particles with the matrix and the distribute uniformity of the reinforcement particles are important factors which will affect the reinforcement effects.

KdV Equation with Self-consistent Sources in Non-uniform Media

2009 ◽  
Vol 51 (6) ◽  
pp. 989-999 ◽  
Author(s):  
Hao Hong-Hai ◽  
Wang Guang-Sheng ◽  
Zhang Da-Jun
Keyword(s):  
Kdv Equation ◽  
Self Consistent

scholarly journals Integration of the Matrix Nonlinear Schr¨odinger Equation with a Source

2021 ◽  
Vol 37 ◽  
pp. 63-76
Author(s):  
G.U. Urazboev ◽  
◽  
A.A. Reyimberganov ◽  
A.K. Babadjanova ◽  
◽  
...  
Keyword(s):  
Inverse Scattering Method ◽  
The Self ◽  
Scattering Data ◽  
Scattering Method ◽  
Self Consistent ◽  
The Matrix ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Step Algorithm ◽  
Decreasing Functions

This paper is concerned with studying the matrix nonlinear Schr¨odinger equation with a self-consistent source. The source consists of the combination of the eigenfunctions of the corresponding spectral problem for the matrix Zakharov-Shabat system which has not spectral singularities. The theorem about the evolution of the scattering data of a non-self-adjoint matrix Zakharov-Shabat system which potential is a solution of the matrix nonlinear Schr¨odinger equation with the self-consistent source is proved. The obtained results allow us to solve the Cauchy problem for the matrix nonlinear Schr¨odinger equation with a self-consistent source in the class of the rapidly decreasing functions via the inverse scattering method. A one-to-one correspondence between the potential of the matrix Zakharov-Shabat system and scattering data provide the uniqueness of the solution of the considering problem. A step-by-step algorithm for finding a solution to the problem under consideration is presented.

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