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Four types of bounded wave solutions of CH-■ equation

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science china Mathematics 2007年第1期50卷 132-152页

作者： Min-ying TANG & Wen-ling ZHANG school of mathematical sciences and center for nonlinear science studies, south china university of technology, guangzhou 510640, china ( Department of Mathematics and Physics, National Natural science Foundation of china, Beijing 100085, china

Recently, many authors have studied the following CH-γequation: ut + c0ux + 3uux -α2(wxxt + uuxxx + 2uxuxx) 4-γuxxx = 0,whereα2, c0 andγare paramters. Its bounded wave solutions have been investigated mainly for the caseα2 > 0. For the caseα2 < 0, the existence of three bounded waves (regular solitary waves, compactons, periodic peakons) was pointed out by Dullin et al. But the proof has not been given. In this paper, not only the existence of four types of bounded waves: periodic waves, compacton-like waves, kink-like waves, regular solitary waves, is shown, but also their explicit expressions or implicit expressions are given for the caseα2 < 0. Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.

关键词： CH-γequation bounded waves explicit expressions implicit expressions.

Invariant tori for asymptotically linear impact oscillators

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science china Mathematics 2006年第5期49卷 669-687页

作者： QIAN Dingbian & SUN Xiying school of mathematical sciences, Suzhou university, Suzhou 215006, china Laboratory of Mathematics for nonlinear sciences, Fudan university, Shanghai 200433, china school of mathematical sciences Suzhou University Suzhou 215006 China Laboratory of Mathematics for nonlinear sciences Fudan University Shanghai 200433 China

The existence of invariant tori and quasi-periodic solutions for asymptotically linear impact oscillators is proved by using the successor map and some generalized versions of the Moser's twist theorem.

关键词： invariant tori, boundedness of solutions, quasi-periodic solution, impact oscillator.

Deterministic learning from control of nonlinear systems with disturbances

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Progress in Natural science:Materials International 2009年第8期19卷 1011-1019页

作者： Guopeng Zhou a,b, Cong Wang a, a school of Automation and center for Control and Optimization, south china university of technology, guangzhou 510640, china b College of Mathematics and Statistics, Xianning university, Xianning 437005, china [a]school of Automation and center for Control and Optimization South China University of Technology Guangzhou 510640 China [b]College of Mathematics and Statistics Xianning University Xianning 437005 China

In this paper, we investigate deterministic learning in environments with disturbances. We will show that for a class of uncertain nonlinear systems with bounded disturbances, by using an appropriately designed adaptive neural controller, the disturbances are attenuated and the system output tracks a periodic orbit in finite time. As radial basis function (RBF) neural networks (NN) are employed, this leads to the satisfaction of a partial persistence of the excitation (PE) condition. By using the uniform complete observability (UCO) technique, it is analyzed that partial estimated NN weights will converge to a neighborhood of zero, with the size of the neighborhood depending on the amplitude of disturbances as well as on the control gains. Locally-accurate approximation of unknown system dynamics can still be achieved in the stable NN control process. The approximation error level is influenced by the amplitude of disturbances. The obtained knowledge of system dynamics can be reused in another control process towards stability and improved performance. Simulation studies are included to demonstrate the effectiveness of the approach.

关键词： Deterministic learning Adaptive NN control Disturbance Partial PE condition Uniform complete observability

Entropy number and non-linear approximations on multivariate Besov space by manifolds of finite pseudo-dimension

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Progress in Natural science:Materials International 2006年第3期16卷 260-265页

作者： LONG Jingfan and FANG Gensun (school of mathematical sciences, Beijing Normal university, Beijing 100875, china Department of Computer science, Information technology Institute, Beijing 100101, china) [a]school of mathematical sciences Beijing Normal University Beijing China [b]Department of Computer science Information Technology Institute Beijing China

We determine the asymptotic order of entropy number and optimal non -linear approximations of anisotropic periodic Besov class of Bpθr(Td') (1≤p≤∞,1≤0≤∞) by manifolds of finite pseudo-dimension in the metric Lq... 详细信息

关键词： Pseudo-dimension anisotropic Besov class of functions entropy number nonlinear width.

Power functions with low uniformity on odd characteristic finite fields

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science china Mathematics 2010年第8期53卷 1931-1940页

作者： ZHA ZhengBang 1,3 & WANG XueLi 2 1 school of mathematical sciences,Luoyang Normal university,Luoyang 471022,china 2 school of mathematical sciences,south china Normal university,guangzhou 510631,china 3 State Key Laboratory of Information Security,Graduate university of Chinese Academy of sciences,Beijing 100049,china 1. school of mathematical sciences Luoyang Normal University Luoyang 471022 China3. State Key Laboratory of Information Security Graduate University of Chinese Academy of Sciences Beijing 100049 China2. School of Mathematical Sciences South China Normal University Guangzhou 510631 China

In this paper,we give some new low differential uniformity of some power functions defined on finite fields with odd *** corollaries of the uniformity,we obtain two families of almost perfect nonlinear functions in GF... 详细信息

关键词： almost perfect nonlinear function differential uniform function finite field

Optical pulse compression using a nonlinear optical loop mirror constructed from dispersion decreasing fiber

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science china(Technological sciences) 2004年第1期47卷 33-50页

作者： CAO Wenhua & LIU Songhaoschool of Information, Wuyi university, Jiangmen 529020, china school for Information and Optoelectronic science and Engineering, south china Normal university, guangzhou 510631, china 1. school of Information Wuyi University 529020 Jiangmen China 2. school for Information and Optoelectronic science and Engineering South China Normal University 510631 Guangzhou China

A novel scheme to compress optical pulses is proposed and demonstrated numerically, which is based on a nonlinear optical loop mirror constructed from dispersion decreasing fiber (DDF). We show that, in contrast to the conventional soliton-effect pulse compression in which compressed pulses are always accompanied by pedestals and frequency chirps owning to nonlinear effects, the proposed scheme can completely suppress pulse pedestals and frequency chirps. Unlike the adiabatic compression technique in which DDF length must increase exponentially with input pulsewidth, the proposed scheme does not require adiabatic condition and therefore can be used to compress long pulses by using reasonable fiber lengths. For input pulses with peak powers higher than a threshold value, the compressed pulses can propagate like fundamental solitons. Furthermore, the scheme is fairly insensitive to small variations in the loop length and is more robust to higher-order nonlinear effects and initial frequency chirps than the adiabatic compression technique.

关键词： dispersion decreasing fiber (DDF), optical pulse compression, optical solitons.

Sharpness on the Lower Bound of the Lifespan of Solutions to nonlinear Wave Equations

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Chinese Annals of Mathematics,Series B 2011年第4期32卷 521-526页

作者： Yi ZHOU 1 Wei HAN 2 1 nonlinear mathematical Modeling and Methods Laboratory Shanghai Key Laboratory for Contem- porary Applied Mathematics school of mathematical sciences, Fudan university, Shanghai 200433, china. 2 school of mathematical sciences, Fudan university, Shanghai 200433, china Department of Mathematics, North university of china, Taiyuan 030051, china. nonlinear mathematical Modeling and Methods Laboratory Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai 200433 China

This paper is devoted to proving the sharpness on the lower bound of the lifespan of classical solutions to general nonlinear wave equations with small initial data in the case n = 2 and cubic nonlinearity （see the results of T. T. Li and Y. M. Chen in 1992）. For this purpose, the authors consider the following Cauchy problem： { where □=δt^2-∑i=1n δx^^2 is the wave operator, g（x）（≡/） 0 is a smooth non-negative function with compact support, and ε 〉 0 is a small parameter. It is shown that the solution blows up in a finite time, and the lifespan T（ε） of solutions has an upper bound T（ε） ≤ exp（Aε-2） with a positive constant A independent of ε, which belongs to the same kind of the lower bound of the lifespan.

关键词： nonlinear wave equation Cauchy problem Lifespan

The Real-gas Effect in Single-bubble Sonoluminescence

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Communications in nonlinear science and Numerical Simulation 1999年第1期4卷 29-34页

作者： Li YUAN (LSEC, Institute of Computational Mathematics and Scientific-Engineering Computing, Chinese Academy of sciences, Beijing 100080, china) Wei WEI (center for nonlinear sciences, university of science and technology of china, Hefei 230026,china) LSEC Inst. Compl. Math./Sci.-Eng. Comp. Chinese Academy of Sciences Beijing 100080 China center for nonlinear sciences Univ. of Sci. and Technol. of China Hefei 230026 China Department of Physics Chinese University of Hong Kong Shatin Hong Kong

The full set of hydrodynamic equations governing the oscillation of a bubble is solved numerically by using a modified equation of state with ionization. We show that the hydro-thermal process in the form of compression waves and low-level ionization are dominant inside a sonoluminescing bubble for a wide range of driving pressure. The resulting thermal bremsstrahlung radiation has pulse heights and widths that agree with experimental data for cases considered in this paper.

关键词： single-bubble sonoluminescence ionization bremsstrahlung

Global Classical Solutions to Partially Dissipative Quasilinear Hyperbolic Systems

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Chinese Annals of Mathematics,Series B 2011年第5期32卷 771-780页

作者： Yi ZHOU 11 Key Laboratory of Mathematics for nonlinear sciences,Ministry of Education,china Shanghai Key Laboratory for Contemporary Applied Mathematics school of mathematical sciences,Fudan university,Shanghai 200433,china. Key Laboratory of Mathematics for nonlinear sciences Ministry of Education China Shanghai Key Laboratory for Contemporary Applied Mathematics School of Mathematical Sciences Fudan University Shanghai 200433 China

The author considers the Cauchy problem for quasilinear inhomogeneous hyperbolic *** the assumption that the system is weakly dissipative,Hanouzet and Natalini established the global existence of smooth solutions for small initial data (in *** ***.,Vol.169,2003,pp.89-117).The aim of this paper is to give a completely different proof of this result with slightly different assumptions.

关键词： Cauchy problem Global classical solution Partially dissipativequasilinear hyperbolic system