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VORTEX DYNAMICS OF THE ANISOTROPIC GINZBURG-LANDAU EQUATION

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Acta Mathematica Scientia 2010年第3期30卷 949-962页

作者：温焕尧丁时进 School of Mathematical Sciences South China Normal University

In this article, using coordinate transformation and gronwall inequality, we study the vortex motion law of the anisotropic Cinzburg-Landau equation in a smooth bounded domain Ω （R^2,that is ,Эtuε=j,k=1∑2（ajkЭxkuε）xj＋ε^2^-b（x）（1-｜uε｜^2）uε,x∈Ω,and conclude that each vortex,bj（t）（j=1,2,…,N）satisfies dt^-dbj（t）=-（a（bj（t））^-a1k（bj（t））Эxka（bj（t））,a（aj（t））^-a2k（bj（t））Эxka（bj（t）））,where a（x）=√a11a22-a12^2. We prove that all the vortices are pinned together to the critical points of a（x）. Furthermore, we prove that these critical points can not be the maximum points.

关键词： Anisotropic Ginzburg-Landau equation gronwall inequality vortex dynamics

Contributions to Hom-Schunck optical flow equations-part I： Stability and rate of convergence of classical algorithm

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Journal of Central South University 2013年第7期20卷 1909-1918页

作者： DONG Guo-hua AN Xiang-jing FANG Yu-qiang HU De-wen College of Mechatronics and Automation National University of Defense Technology Changsha 410073 China

Globally exponential stability （which implies convergence and uniqueness） of their classical iterative algorithm is established using methods of heat equations and energy integral after embedding the discrete iteration into a continuous flow. The stability condition depends explicitly on smoothness of the image sequence, size of image domain, value of the regularization parameter, and finally discretization step. Specifically, as the discretization step approaches to zero, stability holds unconditionally. The analysis also clarifies relations among the iterative algorithm, the original variation formulation and the PDE system. The proper regularity of solution and natural images is briefly surveyed and discussed. Experimental results validate the theoretical claims both on convergence and exponential stability.

关键词： optical flow Hom-Schunck equations globally exponential stability convergence convergence rate heat equations energy integral and estimate gronwall inequality natural images regularity

Local Nonautonomous Schrodinger Flows on Kahler Manifolds

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Acta Mathematica Sinica,English Series 2019年第8期35卷 1251-1299页

作者： Zong Lin JIA You De WANG Institute of Applied Physics and Computational Mathematics China Academy of Engineering PhysicsBeijing 100088 P.R.China Academy of Mathematics and Systems Sciences Chinese Academy of SciencesBeijing 100190P.R.China

In this paper, we prove that the nonautonomous Schrodinger flow from a compact Riemannian manifold into a Kahler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher r... 详细信息

关键词： Nonautonomous Schrodinger flow gronwall inequality interpolation

Optimal Error Estimates of Compact Finite Difference Discretizations for the Schrodinger-Poisson System

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Communications in Computational Physics 2013年第5期13卷 1357-1388页

作者： Yong Zhang Department of Mathematical Sciences Tsinghua UniversityBeijing100084P.R.China

We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related *** some useful inequalities are established in this ***,extensive numerical results are reported to support our error estimates of the numerical methods.

关键词： Schrodinger-Poisson system Crank-Nicolson scheme semi-implicit scheme compact finite difference method gronwall inequality the maximum principle

Finite-Time Stability of Fractional-Order Neural Networks with Delay

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Communications in Theoretical Physics 2013年第8期60卷 189-193页

作者：吴然超黑鑫东陈立平 School of Mathematics Anhui University School of Automation Chongqing University

Finite-time stability of a class of fractional-order neural networks is investigated in this paper. By Laplace transform, the generalized Gronwa11 inequality and estimates of Mittag-Leffier functions, sufficient conditions are pre- sented to ensure the finite-time stability of such neural models with the Caputo fractionM derivatives. Furthermore, results about asymptotical stability of fractional-order neural models are also obtained.

关键词： neural networks fractional-order finite-time stability gronwall inequality

Robust Exponential Stability of a Class of Fractional Order Hopfield Neural Networks

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Journal of Mathematical Research with Applications 2015年第6期35卷 653-658页

作者： Xiaolei LIU Mingjiu GAI Shiwei CUI Institute of System Science and Mathematics Naval Aeronautical and Astronautical University

In this paper, we investigate the robust exponential stability of a class of fractional order Hopfield neural network with Caputo derivative, and we get some sufficient conditions to guarantee its robust exponential stability. Finally, we use one numerical simulation example to illustrate the correctness and effectiveness of our results.

关键词： fractional order neural networks gronwall inequality robust exponential stability

EXPONENTIAL STABILITY OF FIRST ORDER DYNAMIC EQUATION ON TIME SCALES

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Annals of Differential Equations 2011年第3期27卷 372-378页

作者： Zongyi Wang(Dept. of Math.,Huizhou University,Huizhou 516007,Guangdong)

This paper is concerned with the stability of a first order dynamic equation on time scales. In particular,using the properties of the generalized exponential function with time scales,different kinds of sufficient co... 详细信息

关键词： exponential stability generalized exponential function regressive gronwall inequality time scales

A NOTE ON ＂SMALL AMPLITUDE SOLUTIONS OF THE GENERALIZED IMBQ EQUATION＂

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Journal of Partial Differential Equations 2006年第4期19卷 377-383页

作者： Zhu Youbin Graduate School of Chinese Academy of Engineering Physics P. O. Box 2101 Beijing 100088 China

Global existence of small amplitude solution and nonlinear scattering result for the Canchy problem of the generalized IMBq equation were considered in the paper titled ＂Small amplitude solutions of the generalized IMBq equation＂ [1]. It is a pity that the authors overlooked the bad behavior of low frequency part of S（t）ψ which causes troubles in L^∞ and H^＊ estimates. In this note, we will present a new proof of global existence under same conditions as in [1] but for space dimension n ≥ 3.

关键词： IMBq equation Duhamel's principle Hoelder inequality gronwall inequality Hausdorff-Young inequality.

Impulsive Synchronization of Typical Hopfield Neural Networks

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Impulsive Synchronization of Typical Hopfield Neural Network...