The Bi-Hamiltonian Structure of the Perturbation Equations of KdV Hierarchy

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压缩原子托马斯-费米方程的变分解(英文)1. IntroductionThepressed atomic Thomas-Fermi (CATF) model is a widely used and efficient approach for describing the electronic structure of atoms and molecules. However, the standard Thomas-Fermi (TF) equation can be further improved by employing the variational principle to account for the electron density distribution in a more accurate and systematic manner. In this article, we will explore the variational approach to develop apressed version of the Thomas-Fermi equation, focusing on the mathematical formulation and physical significance of the variational solution.2. Thomas-Fermi ModelThe Thomas-Fermi model is a theoretical framework based on the density functional theory (DFT), which provides a powerful andputationally efficient method for predicting the electronic properties of atoms and molecules. The model 本人ms to determine the ground-state electron density by minimizing the total energy functional, which consists of the kinetic energy and the Coulomb potential energy of the electrons within a given external potential. The standard Thomas-Fermi equation represents the balance between the kinetic and potentialenergies of the electrons, leading to a simplified expression for the electron density distribution.3. Variational PrincipleTo enhance the accuracy and flexibility of the Thomas-Fermi model, we can employ the variational principle, which states that the true ground-state energy of a system is always greater than or equal to the expectation value of the Hamiltonian in any trial wave function. By introducing a trial electron density function, we can minimize the total energy functional with respect to the variational parameters, leading to a more refined description of the electronic structure. The variational approach allows us to incorporate additional physical effects and interactions, thus improving the overall predictive power of the model.4. Compressed Thomas-Fermi EquationBuilding upon the variational principle, we can derive apressed version of the Thomas-Fermi equation that accounts for the electron density distribution in a more rigorous manner. By introducing the variational parameters, we can systematically optimize the electron density function to minimize the total energy functional, leading to a more accurate representation ofthe electronic structure. Thepressed Thomas-Fermi equation provides a refined balance between the kinetic and potential energies, capturing the intricate interplay of electron correlations and exchange effects.5. Mathematical FormulationThe variational solution of thepressed Thomas-Fermi equation involves the systematic optimization of the electron density function with respect to the variational parameters, leading to a set of coupled nonlinear differential equations. By solving these equations numerically, we can obt本人n the ground-state electron density and the corresponding total energy. The mathematical formulation of thepressed Thomas-Fermi equation offers aprehensive framework for studying the electronic properties of diverse atomic and molecular systems, enabling accurate predictions and in-depth analysis.6. Physical SignificanceFrom a physical perspective, the variational solution of thepressed Thomas-Fermi equation captures the intricate interplay of electron-electron correlations, exchange effects, and external potentials, providing a more realistic description of the electronic structure. The refined balance between the kineticand potential energies allows for a more accurate determination of the electron density distribution, which is crucial for understanding chemical bonding, reactivity, and other fundamental properties of atoms and molecules. The physical significance of thepressed Thomas-Fermi equation lies in its ability to capture the essential electronic interactions within aputationally efficient framework.7. ConclusionIn conclusion, the variational approach to developing apressed version of the Thomas-Fermi equation offers a systematic and rigorous framework for studying the electronic structure of atoms and molecules. By systematically optimizing the electron density function with respect to the variational parameters, we can enhance the accuracy and predictive power of the standard Thomas-Fermi model, capturing the intricate interplay of electron correlations and exchange effects. The mathematical formulation and physical significance of thepressed Thomas-Fermi equation provide a solid foundation for further advancements in the field of density functional theory andputational chemistry.。

257ISSN: 1749-3889 ( print ) 1749-3897 (online)BimonthlyVol.7 (2009) No.3JuneEngland, UK ***************************.uk International Journal ofN o n l i n e a r S c i e n c e Edited by International Committee for Nonlinear Science, WAUPublished by World Academic Union (World Academic Press)CONTENTS259.New Exact Travelling Wave Solutions for Some Nonlinear Evolution EquationsA. Hendi268.A New Hierarchy of Generalized Fisher Equations and Its Bi- Hamiltonian StructuresLu Sun274.New Exact Solutions of Nonlinear Variants of the RLW, the PHI-four and Boussinesq Equations Based on Modified Extended Direct Algebraic MethodA. A. Soliman, H. A. Abdo283. Monotone Methods in Nonlinear Elliptic Boundary Value ProblemG.A.Afrouzi, Z.Naghizadeh, S.Mahdavi290. Influence of Solvents Polarity on NLO Properties of Fluorone Dye Ahmad Y. Nooraldeen1, M. Palanichant, P. K. Palanisamy301.Projective Synchronization of Chaotic Systems with Different Dimensions via Backstepping DesignXuerong Shi, Zhongxian Wang307.Adaptive Control and Synchronization of a Four-Dimensional Energy Resources System of JiangSu ProvinceLin Jia , Huanchao Tang312.Optimal Control of the Viscous KdV-Burgers Equation Using an Equivalent Index MethodAnna Gao, Chunyu Shen，Xinghua Fan319.Adaptive Control of Generalized Viscous Burgers’ Equation Xiaoyan Deng, Wenxia Chen, Jianmei Zhang327. Wavelet Density Degree of a Class of Wiener Processes Xuewen Xia, Ting Dai332.Niches’ Similarity Degree Based on Type-2 Fuzzy Niches’ Model Jing Hua, Yimin Li340.Full Process Nonlinear Analysis the Fatigue Behavior of the Crane Beam Strengthened with CFRPHuaming Zhu, Peigang Gu, Jinlong Wang, Qiyin Shi345.The Infinite Propagation Speed and the Limit Behavior for the B-family Equation with Dispersive TermXiuming Li353.The Classification of all Single Traveling Wave Solutions to Fornberg-Whitham EquationChunxiang Feng, Changxing Wu360.New Jacobi Elliptic Functions Solutions for the Higher-orderNonlinear Schrodinger EquationBaojian Hong, Dianchen Lu368.A Counterexample on Gap Property of Bi-Lipschitz Constants Ying Xiong, Lifeng Xi371.A Method for Recovering the Shape for Inverse Scattering Problem of Acoustic WavesLihua Cheng, Tieyan Lian, Ping Li379.An Approach of Image Hiding and Encryption Based on a New Hyper-chaotic SystemHongxing Yao, Meng Li258International Journal of Nonlinear Science (IJNS)BibliographicISSN: 1749-3889 (print), 1749-3897 (online), BimonthlyEdited by International Editorial Committee of Nonlinear Science, WAUPublished by World Academic Union (World Academic Press)Publisher Contact:Academic House, 113 Mill LaneWavertree Technology Park, Liverpool L13 4AH, England, UKEmail:******************.uk,*********************************URL: www. 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关于科学的英语演讲稿关于科学的英语演讲稿the presence of students, ladies and teachers, everyone! i was prepared intervals of cloud today, in honor here entitled “technology and future” speech, i am very proud of both, but some unease. in recent years, we have seen our great motherland, the cause of the rapid development of technology, which allow me to a chinese i feel very proud. remember that long ago, cell phone use almost the only one, which is called, but a few years ago, cell phones has undergone great changes, not only look more beautiful, but also use more, you can use the phones to take pictures, meetings, internet, text messages, etc. a series of things that i their life more convenient, so i am more aware of the strength of the technology, but i am just a fledgling s students, “technology” as the word also aware of the limited, i am unable to use some very difficult theory to elaborate technology xuanji, no right to work on their elders i can promise of the technology blueprint. but i am willing to use a student’s perspective to the imagine technology and the future.from genetic engineering “is a live princes” dream, nano-technology “- not washing your clothing,” promises; fromartificial intelligence “will give you a cute robot dog” warm, transgenic technology “let people grow mouse ears” wonders. the new technology in the birth of a new technology that will let people wild with joy, because these new technologies is gradually improving our lives, let us learn more about ourselves. in the near term, china completed its first sars virus genome sequencing, sars is now the world’s largest recognized the dangers of the disease, but why not other countries completed first, and we just completed the countries? very si-mp-le, this illustrates that our country backward than others, worse than others, we look at the past, had just started a country’s reform and opening up to the current level of science and technology has lead a large country, our motherland experienced a number of ups and downs, how many difficulties and bumpy however, we still back all of the motherland, the motherland because we firmly believe that - not only technological change destiny, can change the future.for our generation, the general feeling of the community is a strong sense of competition, learning a foot down. science knowledge is the focus of our attention, albert einstein, and stephen hawking, bill gates is the star we have in mind, computer science, physics and chemistry of modern dynamic is constantly affect us. we have to understand theimportance of technology, know that the technology universal.although technology to create a new life prospects soliciting thoughts, inspiring. however, the final analysis is that we rely on our common efforts to realize. , as the future construction of the backbone of our generation of young people shoulders the burden is not light, new opportunities are always accompanied risks and challenges, but we will not give up that easily, we use our youth to predecessors vowed: never live up to their predecessors of our hope.looking back at the history of civilization, anderson is the history of mankind against the darkness of ignorance, is the scientific lit a fire in the raging human soul of hope; technology support civilization, science and technology to create the future, but the future is in our hands. let us become knowledge explorers, let us unknown roaming on the road, let me use our creativity to the world we live a better place.参考翻译：在场的学生们，女士们，老师，大家好！我准备间隔云今天，为了纪念在这里题为“科技与未来”的讲话，我感到非常自豪的两个，但有些不安。

a r X i v :n l i n /0212002v 2 [n l i n .S I ] 21 M a r 2003Nonlinear superposition formula for N =1supersymmetric KdV Equation Q.P.Liu and Y.F.Xie Department of Mathematics,China University of Mining and Technology,Beijing 100083,China.Abstract In this paper,we derive a B¨a cklund transformation for the supersymmetric Korteweg-de Vries equation.We also construct a nonlinear superposition formula,which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea,Ramani and Grammaticos.The celebrated Korteweg-de Vries (KdV)equation was extended into super frame-work by Kupershmidt [3]in 1984.Shortly afterwards,Manin and Radul [7]proposed another super KdV system which is a particular reduction of their general supersymmet-ric Kadomstev-Petviashvili hierarchy.In [8],Mathieu pointed out that the super version of Manin and Radul for the KdV equation is indeed invariant under a space supersymmetric transformation,while Kupershmidt’s version does not.Thus,the Manin-Radul’s super KdV is referred to the supersymmetric KdV equation.We notice that the supersymmetric KdV equation has been studied extensively in lit-erature and a number of interesting properties has been established.We mention here the infinite conservation laws [8],bi-Hamiltonian structures [10],bilinear form [9][2],Darboux transformation [6].By the constructed Darboux transformation,Ma˜n as and one of us calculated the soli-ton solutions for the supersymmetric KdV system.This sort of solutions was also obtained by Carstea in the framework of bilinear formalism [1].However,these solutions are char-acterized by the presentation of some constraint on soliton parameters.Recently,using super-bilinear operators,Carstea,Ramani and Grammaticos [2]constructed explicitly new two-and three-solitons for the supersymmetric KdV equation.These soliton solutions are interesting since they are free of any constraint on soliton parameters.Furthermore,the fermionic part of these solutions is dressed through the interactions.In addition to the bilinear form approach,B¨a cklund transformation (BT)is also a powerful method to construct solutions.Therefore,it is interesting to see if the soliton solutions of Carstea-Ramani-Grammaticos can be constructed by BT approach.In this paper,we first construct a BT for the supersymmetric KdV equation.Then,we derive a nonlinear superposition formula.In this way,the soliton solutions can be producedsystematically.We explicitly show that the two-soliton solution of Carstea,Ramani and Grammaticos appears naturally in the framework of BT.To introduce the supersymmetric extension for the KdV equation,we recall some terminology and notations.The classical spacetime is(x,t)and we extend it to a super-spacetime(x,t,θ),whereθis a Grassmann odd variable.The dependent variable u(x,t) in the KdV equation is replaced by a fermionic variableΦ=Φ(x,t,θ).Now the super-symmetric KdV equation reads asΦt−3(ΦDΦ)x+Φxxx=0,(1)where D=∂∂x is the superderivative.Mathieu found the following supersymmetricversion of Gardner type mapΦ=χ+ǫχx+ǫ2χ(Dχ),(2) whereǫis an ordinary(bosonic)parameter.It is easy to show thatχsatisfies the following supersymmetric Gardner equationχt−3(χDχ)x−3ǫ2(Dχ)(χDχ)x+χxxx=0.(3) This map was used in[8]to prove that there exists an infinite number of conservation laws for the supersymmetric KdV equation(1).In the classical case,Gardner type of map was studied extensively by Kupershmidt[4].It is well known that such map may be used to construct interesting BT.We will show that it is also the case for the supersymmetric KdV equation.We notice that the supersymmetric Gardner equation(3)is invariant underǫ→−ǫ. The new solution of the supersymmetric KdV equation corresponding to with−ǫis denoted as˜Φ.Thus we have˜Φ=χ−ǫχx+ǫ2χ(Dχ).(4) From above relations(3-4),wefindΦ−˜Φ=2ǫχx,(5)Φ+˜Φ=2χ+2ǫχ(Dχ).(6) Let us introduce the potentials as followsΦ=Ψx,˜Φ=˜Ψx,thus,the equation(5)provides usχ=12(Ψ−˜Ψ)(DΨ−D˜Ψ),(8)whereλ=1/ǫis the B¨a cklund parameter.The transformation(8)is in fact the spatial part of BT.Its temporal counterpart can be easily worked out.We also remark here that the BT is reduced to the well known BT for the classical KdV equation ifη=0,as it should be.A BT can be used to generate special solutions.If we start with the trivial solution ˜Ψ=0,we obtain2a(ζ+θλ)eλx−λ3tΨ=−(Ψ1−Ψ0)(DΨ1−DΨ0),(10)2and1(Ψ0+Ψ2)x=λ2(Ψ2−Ψ0)+(Ψ3−Ψ1)(DΨ3−DΨ1),(12)2and1(Ψ2+Ψ3)x=λ1(Ψ3−Ψ2)+Subtraction (10)form (11),we have(Ψ2−Ψ1)x =λ2Ψ2−λ1Ψ1+(λ1−λ2)Ψ0+12Ψ2(D Ψ0)−12Ψ1(D Ψ1)+12Ψ0(D Ψ1),(14)similarly,from (12)and (13)we have(Ψ2−Ψ1)x = λ1−λ2+12(D Ψ2) Ψ3+12Ψ2(D Ψ2)−12(D Ψ1)−12(Ψ1−Ψ2)[(D Ψ3)+2λ1+2λ2−(D Ψ0)]=0.(16)The equation (16)is a differential equation for Ψ3.Solving it we obtainΨ3=Ψ0−(λ1+λ2)(Ψ1−Ψ2)λ1−λ2+v 1−v 2−(λ1+λ2)(η1−η2)(η1,x −η2,x )(λ1−λ2+v 1−v 2).It is clear that our nonlinear superposition formula reduces to the well-known superposition formula for the KdV as it should be.The advantage to have a superposition formula is that it is an algebraic one and can be used easily to find solutions.Using the solutions (9)as our seeds,we may construct a 2-soliton solution of (1)by means of our superposition formula.Indeed,letΨ0=0,Ψ1=−2a 1(ζ1+θλ1)e λ1x −λ31t1+a 2e λ2x −λ32tthen from our nonlinear superposition formula (17),we obtainΨ3=2 ζ1a 1e δ1−ζ2a 2e δ2+(ζ1−ζ2)a 1a 2e δ1+δ2+θ(λ1a 1e δ1−λ2a 2e δ2+(λ1−λ2)a 1a 2e δ1+δ2)whereδi=λi x−λ3i t+θζi(i=1,2).Now,takingλ1>λ2,a1=−λ1−λ2λ1+λ2we recover the2-soliton solution foundfirst by Carstea,Ramani and Grammaticos[2].As in the classical KdV case[11],we generate this2-soliton from a regular solution and a singular solution.We could continue this process to build the higher soliton solutions and the calculation will be tedious but straightforward.Acknowledgment The work is supported in part by National Natural Scientific Foun-dation of China(grant number10231050)and Ministry of Education of China. References[1]Carstea A S2000Extension of the bilinear formalism to supersymmetric KdV-typeequations Nonlinearity131645.[2]Carstea A S,Ramani A and Grammaticos B2001Constructing the soliton solutionsfor the N=1supersymmetric KdV hierarchy Nonlinearity141419.[3]Kupershmidt B A1984A super Korteweg-de Vries equation Phys.Lett.102A213.[4]Kupershmidt B A1981On the nature of Gardner transformation J.Math.Phys.22449;Kupershmidt B A1983Deformations of integrable systems Proc.R.Soc.Irish 83A45.[5]Liu Q P1995Darboux transformation for the supersymmetric KdV equations Lett.Math.Phys.35115.[6]Liu Q P and Ma˜n as M1997Darboux transformation for the Mann-Radul supersym-metric KdV equation Phys.Lett.B394337;Liu Q P and Ma˜n as M in Supersymmetry and Integrable Systems eds.H.Aratyn et al Lecture Notes in Physics502(Springer).[7]Manin Yu I and Radul A O1985A supersymmetric extension of the Kadomtzev-Petviashvili hierarchy Commun.Math.Phys.9865.[8]Mathieu P1988Supersymmetric extension of the Korteweg-de Vries equation J.Math.Phys.292499.[9]McArthur I N and Yung C M1993Hirota bilinearfform for the super KdV hierarchyMod.Phys.Lett.8A1739.[10]Oevel W and Popowicz Z1991The bi-Hamiltonian structure of fully supersymmetricKorteweg-de Vries systems Commun.Math.Phys.139441;Figueroa-O’Farril J,Mas J and Ramos E1991Integrability and biHamiltonian structure of the even order sKdV hierarchies Rev.Math.Phys.3479.[11]Wahlquist H D and Estabrook F B1974B¨a cklund transformation for solutions of theKorteweg-de Vries equation Phys.Rev.Lett.311386.。

KdV方程的高阶保能量算法蒋朝龙;孙建强;何逊峰;闫静叶【摘要】The KdV equation is transformed into an infinite dimensional Hamiltonian system. The finite dimensional Hamiltonian system of the KdV equation is obtained by the pseudo-spectral method in spacial direction. Then,the finite dimensional Hamiltonian system is discretizated by the fourth order AVF method. Thus,a high order energy-preserving scheme of the KdV equation is derived. The evolution of the solitary wave is simulated by the high order energy-preserving scheme. Numerical results show that the proposed scheme can preserve the discrete energy of the KdV equation exactly.%KdV方程被转化为无穷维Hamilton系统,在空间方向上用拟谱算法离散得到了KdV方程的有限维Hamilton系统.利用四阶平均向量场(AVF)方法离散KdV方程的有限维Hamilton系统,构造了KdV方程的高阶保能量格式.利用构造的高阶保能量格式数值模拟孤立波的演化行为.数值结果表明,高阶保能量格式可以精确保持方程的离散能量守恒.【期刊名称】《南京师大学报（自然科学版）》【年(卷),期】2017(040)004【总页数】5页(P16-20)【关键词】AVF方法;KdV方程;保能量算法【作者】蒋朝龙;孙建强;何逊峰;闫静叶【作者单位】海南大学信息科学技术学院,海南海口570228;海南大学信息科学技术学院,海南海口570228;海南大学信息科学技术学院,海南海口570228;海南大学信息科学技术学院,海南海口570228【正文语种】中文【中图分类】O241具有能量守恒的Hamilton系统是动力系统的一个重要体系,一切耗散的和耗散忽略不计的物理过程都可以表示为Hamilton系统. 上个世纪80年代,我国著名计算数学家冯康院士及其研究小组提出Hamilton系统辛几何算法[1]. 在辛几何算法的基础上,国内外学者发展了Hamilton系统的多辛几何算法[2-3]. . 然而,向后误差分析表明对于非线性Hamilton系统,辛和多辛算法只能近似保持系统能量守恒[4]. 因此,构造保持Hamilton系统能量守恒的数值法对正确模拟具有能量守恒的Hamilton系统具有重要的意义.最近,Quispel和McLaren给出了如下的保能量平均向量场方法(AVF)[5]=f((1-ξ)zn+ξzn+1)dξ, z∈R2m,(1)式中,f(z)=SH(z),S是反对称常数矩阵,H:R2m→R是Hamilton能量函数.平均向量场方法(1)也被称为平均离散梯度方法[6],可以精确保持Hamilton系统能量守恒,在时间方向具有二阶精度,是一类B级数方法[7]. 基于修正向量场方法的思想,Quispel和McLaren提出了在时间方向上具有四阶精度的高阶平均向量场方法(AVF)[5]=SH((1-ξ)zn+ξzn+1)dξ,(2)式中,Celledoni等人[7]首次将二阶平均向量场方法应用在具有能量守恒的偏微分方程的求解中;龚等人[8]利用平均向量场方法(1)构造了多辛Hamilton系统的局部保能量和保动量格式. 下面我们将利用四阶平均向量场方法(2)求解KdV方程. KdV方程广泛存在于非谐晶体,泡沫液混合物,磁流体动力学,离子声波中. 考虑一般的KdV方程ut+cuux+δ2uxxx=0, t>0,x∈Ω,(3)初始条件为u(x,0)=u0(x), x∈Ω,(4)和周期边界条件u(x+L,t)=u(x,t), x∈Ω,(5)式中,Ω=[a,b],L=b-a,c和δ是实数.KdV方程具有如下的守恒特性动量守恒(6)能量守恒(7)KdV方程的数值算法一直是研究的热点,特别是构造KdV方程的保结构算法的研究. 赵等人[9]最早构造了KdV方程的多辛Preissman 格式;王等人[10-11]研究了数值实现KdV方程的多辛Preissman 格式和它的等价格式并给出了KdV方程的一类显示多辛格式;宋等人[12]研究了KdV方程的多辛拟谱格式;Ascher等人[13]系统地给出KdV方程的多辛box格式. 吕等人[14]给出一类显示多辛拟谱格式;Karasözen等人[15]利用二阶平均向量场,构造了KdV方程的保能量格式. 本文将利用四阶平均向量场方法构造KdV方程的高阶保能量格式.1 KdV方程的高阶保能量格式方程(3)可以被转化为如下的无穷维Hamilton系统(8)式中,∂x是一阶偏导算子,相应的Hamilton函数为(9)将Ω分为N等分,h=L/N为空间步长,N是一个正偶数. xj=a+hj,j=0,…,N-1为空间配置点. 定义SN={gj(x);j=0,1,…,N-1},(10)为插值空间,其中gj(x)是满足的正交三角多项式,并且gj(x)可以被显式地表示为(11)式中,对函数u(x,t)∈C0(Ω),定义插值算IN为(12)插值算子IN在配置点xj 满足(13)令U=(u0,u1,…,uN-1)T,定义称Dk为k阶谱微分矩阵. 通过计算,可以得到INu(x,t)和在配置点xj的值为(15)式中,D1 和 D2分别是如下的一阶和二阶谱微分矩阵令是u(x,t)在网格点(xj,tn)处的近似. 在空间方向利用拟谱算法离散无限维Hamilton系统(8),得到KdV方程的半离散拟谱格式(16)式中,A=D1D2,di,j是矩阵D1第i行第j列元素.方程(16)可以表示为如下的有限维Hamilton系统=f(U)=D1H(U),(17)相应的Hamilton函数为(18)注意到有限维Hamilton系统(17)中D1具有反对称性,所以Hamilton系统(17)具有能量守恒特性. 利用四阶平均向量场方法(2)在时间方向上离散Hamilton系统(17)可以得到KdV方程的高阶保能量格式=D1τ2J2D1H((1-ξ)Un+ξUn+1)dξ,(19)式中,是如下的对角矩阵方程(19)等价于(20)式中,aij和bij分别是矩阵A和B的第i行第j列元素.消去中间变量ξ,方程(20)等价于(21)2 数值试验为了验证KdV方程的高阶保能量格式(21)的有效性,我们利用高阶保能量格式数值模拟KdV方程单孤立波和多孤立波的演化行为和演化时能量和动量误差变化情况. 定义离散能量和动量误差分别为REn=|H(Un)-H(U0)|, RMn=|M(Un)-M(U0)|,(22)离散能量和动量函数分别为式中,H(U0),M(U0)分别是初始能量和动量,REn和RMn分别是t=nτ时刻的能量和动量误差.2.1 单孤立波当c=6,δ=1时,KdV方程具有单孤立波解. 取初始条件(23)和周期边界条件. 取空间步长h=40/120和时间步长τ=0.001,利用构造的高阶保能量格式数值模拟KdV方程单孤立波的演化. 图 1表示KdV方程在t∈[0,20]内的数值解. 从图1中可以看出孤立波以一定的速度向前传播,在传播过程中孤立波的振幅和波形可以被很好地保持. 图2 是孤立波演化过程中的能量和动量误差图. 在图2中我们可以观察到构造的高阶保能量格式可以精确保持方程离散能量守恒,并且动量守恒特性可以被很好地保持.图1 孤立波在t∈[0,20]内的数值解Fig.1 The numerical solution of the solitary at t∈[0,20]图2 孤立波t∈[0,20]内的能量和动量误差的变化Fig.2 The energy and momentum errors of the solitary at t∈[0,20]2.2 多孤立波碰撞取参数c=1,δ=0.022. 考虑初始条件u(x,0)=cos(πx),(24)和周期边界条件. 令空间步长h=2/82和时间步长τ=0.002. 利用高阶保能量格式(21)数值模拟KdV方程的多孤立波的演化行为. 图3 表示多孤立波在t=0,t=1,t=5时刻的数值解. 从图3中可以看出孤立波的振幅和波形被保持得很好. 图4表示孤立波在t∈[0,40]内演化时能量和动量误差变化情况. 从图4中可以看到构造的高阶保能量格式在多孤立波演化过程中精确保持方程离散能量特性,并且能近似保持动量守恒特性.图3 孤立波在t=0,t=1,t=5时刻的数值解Fig.3 The numerical solutions of the solitary at t=0,t=1,t=5图4 孤立波在t∈[0,40]内的能量和动量误差的变化Fig.4 The energy and momentum errors of the solitary at t∈[0,40]3 结论本文基于四阶AVF方法和拟谱方法,构造了KdV方程的高阶保能量格式. 利用构造的新格式数值模拟孤立波的演化并分析孤立波演化中能量误差和动量误差的变化. 数值结果表明,构造的高阶保能量格式是有效的,可以精确地保持KdV方程的离散能量守恒.[参考文献][1] FENG K,QIN M Z. 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