Periodic wave solutions of nonlinear equations by Hirota's bilinear method

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时滞脉冲周期细胞神经网络指数稳定性的平均准则

时滞脉冲周期细胞神经网络指数稳定性的平均准则宋学力;赵盼;王小伟【摘要】本文致力于研究具有周期系数的脉冲时滞非自治细胞神经网络的全局指数稳定性。

具体地，利用非线性测度方法和周期Halanay微分不等式，我们获得了该类神经网络的积分平均意义下的稳定性准则。

我们的方法不要求激活函数的有界性和单调性，这意味着我们的稳定性准则和现有的某些结果相比受到的约束限制少，可以被应用到更一般的实际问题。

此外，我们获得的稳定性准则是现有某些研究结果的推广和改进。

最后，文中的例子说明了我们的方法是有效的，获得的结果是正确的。

%This paper investigates the global exponential stability of non-autonomous impul-sive and delayed cellular neural networks with periodic coeﬃcients. Particularly, by means of the nonlinear measure method and periodic Halanay differential inequal-ity, we obtain an integral average criterion for global exponential stability of this class of neural networks. Our method does not require assumptions on bounded-ness and monotonicity of activation functions, which demonstrates that our derived cri-terion is less restrictive than some existing ones and can be applied to more gen-eral practical problems. Moreover, our stability criterion is the generalization and improvement of some existing ones. Finally, an example illustrates the effectiveness of our method and the correctness of our results.【期刊名称】《工程数学学报》【年(卷),期】2015(000)004【总页数】15页(P608-622)【关键词】全局指数稳定性;细胞神经网络;周期Halanay微分不等式;非线性测度【作者】宋学力;赵盼;王小伟【作者单位】长安大学理学院，西安 710064;长安大学理学院，西安 710064;长安大学理学院，西安 710064【正文语种】中文【中图分类】O175.131 IntroductionIn the viewpoint of mathematics,cellular neural networks(CNNs)introduced by Chua and Yang[1,2]can be characterized by an array of locally interconnected identical nonlinear dynamical systems(calledcells)[3].During the last nearly three decades,the dynamic behaviors of CNNs have been intensively investigated because they are important and in fact,necessary for applications of CNNs.As we all know,in electronic implementation of neural networks,time delays are inevitable due to axonal conduction times and f i nite switching speeds of amplif i ers[4]and the abrupt changes in the voltagesproduced by faulty circuit elements are exemplary of impulse phenomena,which can af f ect the dynamic behaviors of the networks[5].Consequently,a great deal of attention has been devoted to stability analysis of autonomous CNNs with time delays and/or impulses[4-17].Although the stability is one of the major problems encountered in design and applications of neural networks,periodic oscillation is an important dynamical behavior for non-autonomous neural networks[18]because one has found that some networks(such as human brain)are often periodic oscillatory[19]and periodic oscillatory solutions can represent various storage or memory patterns.Moreover,an equilibrium point can be deemed as a special periodic solution with arbitrary period.Therefore,the dynamic study of periodic oscillatory solutions of non-autonomous neural networks is more general and practical than that of the equilibrium points.Accordingly,the properties of periodic oscillatory solutions of non-autonomous CNNs are recently of great interest and there come forth many excellent results[19-27].In order to characterize the exponential stability of periodic oscillatory solutions of the non-autonomous CNNs,in fact,most of existing methods have to verify the Halanay dif f erential inequalitywhere V(t)is a positive function,a(t)and b(t)are related to the coefficients of the non-autonomous CNNs in question.The paper[28]has provided an integral average criterion such that the Halanay dif f erential inequality holds.It has been illustrated that this criterion is less conservative and ef f ective for describing global exponential stability of delayed CNNs with periodic coefficients in[20].Motivated by this,we employ this integral average criterion and nonlinear measure method introduced in[29]to invastigate global exponential stability of periodic solution of delayed andimpulsive CNNs with periodic coefficients.In fact,our stability criterion will be the generalization and improvement of Theorem 3.1 and Theorem 3.4 presented in[20].The remainder of this article is organized as follows.In section 2,we provide a description and related assumptions of the non-autonomous delayed and impulsive CNNs with periodic coefficients.Section 3 is devoted to preliminaries,where we introduce periodic Halanay dif f erential inequality and prove an integral average criterion for global exponential stability of a non-autonomous impulsive functional dif f erential equation with periodic coefficient operators.In section 4,an integral average criterion is obtained for global exponential stability of periodic solutions of the non-autonomous delayed and impulsive CNNs with periodic coefficients by the results derived in section 3.Moreover,an example is presented to illustrate that our method is valid and that our derived results are new and correct.Conclusions are given in section 5.2 Model descriptionIn this paper,we plan to discuss the model of delayed and impulsive CNNs with periodic coefficients described by the following delayed and impulsive dif f erential equationswhere n denotes the number of the cells and i=1,2,···,n;τ>0 is the time delayed constant;xi(t)corresponds to the state of the ith cell;fjand gjdenote the activation functions of the jth cell;ai(t)is a positive continuous T-periodic function and represents the rate with which the ith cell will resetits potential to the resting state in isolation when disconnected from the network and external input at time t;coefficients bij(t),cij(t)and Ii(t)are continuous and T-periodic functions;is the impulse at the moment tk;0=t1<t2< ···is a strictly increasing sequence such thatand there exists q∈such thatϕibelongs to the space of all continuous functions from[−τ,0]to,i.e.,.In order to investigate the global exponential stability of periodic solutions of the model(1),we only suppose:(H1): The time delayed constant τ is a multiple of the period T of the coefficients functions;(H2): Activation functions fjand gjare globally Lipschitz continuous forj=1,2,···,n;(H3): for all i=1,2,···,n and k=1,2,···,q.3 PreliminariesLet n-dimensional real vector spacebe endowed with 1-norm def i ned bywhere the superscript “T” denotes the transpose.Let 〈·,·〉denote the inner product inand sign(x)=(sign(x1),sign(x2),···,sign(xn))Tthe sign vector of x ∈,where sign()represents the sign function of ∈.Obviously,the relationshold for all x,y∈.In order to discuss the global exponential stability of periodic solutions of the neural networks(1),we f i rstly consider one of the following non-autonomous impulsive functional dif f erential equationwhere τ>0 is time delayed constant;C([−τ,0],)denotes the space of all continuous functions from[−τ,0]into;the coefficients F(t),G(t):are nonlinear globally Lipschitz continuous and T-periodic;0=t1<t2< ···is a strictly increasing sequence such that=+∞ and there exists q∈Nsuch thatDefinition 1[30] For any t≥0,a nonlinear operator T(t):n→nis called be globally Lipschitz continuous if there exists a nonnegative constant Mtdepending on t such thatwhere Mtis called the Lipschitz constant of T(t).The constantis called the minimal Lipschitz constant of T(t).Definition 2[29]Assume that F(t)is a nonlinear operator fromintofor t≥0.The constant related to tis called the nonlinear measure of F(t).Definition 3 The dif f erential equation(3)is globally exponentially stable if there exist two positive constants σ and M such that the followinginequalityholds for t≥0,where y(t)and x(t)are the periodic oscillatory solutions of the dif f erential equation(3)initiated from the function ϕ,ψ ∈c([−τ,0],),respectively.To characterize the exponential stability of the dif f erentialequation(3),one has to verify the Halanay dif f erential inequalitywhere V(t)is a positive function,a(t)and b(t)are continuous and T-periodic functions and b(t)≥0.If some assumptions are imposed to the coefficients of the model(3)such that the inequalityholds for all t∈[0,T],then from the Halanay dif f erential inequality(4)one can conclude the fundamental estimatewith constants M ≥ 1 and σ >0.Obvious ly,the inequality(5)seems to be too restrictive.Fortunately,the paper[20,28]handles the periodic Halanay inequality under the hypotheses weaker than(5),that is,by means of the integral averages conditionrather than the known pointwise criterion(5),whereDenote by λ0(t)the periodic functionwhere is the unique positive solution of the transcendental equationObviously,we enjoyLemma 1[20]Let V(t)be a positive solution of the periodic Halanay inequality(4)with the initial condition V(s)= ϕ(s)≥ 0,s∈ [−τ,0].Assume that the time delay constant τ is a multiple of the period T.There exists M1,M ≥ 1 such that one hasTheorem 1 Assume that the time delay constant τ is a multiple of the period T of coefficients of the dif f erential equation(3)and|1+ γk|≤ 1 fo r k ∈ {1,2,···,q}.If there exists some positive diagonal matrixD=diag(d1,d2,···,dn)such that the inequalityholds,the dif f erential equation(3)is globally exponentially stable,whereT is the period of coefficient operators F(t)and G(t).Particularly,the periodic oscillatory solutions (t)and y(t)of the dif f erential equation(3)initiated from ϕ,ψ ∈c([−τ,0],)satisfy the following relationwhere M ≥ 1 is a constant and σ is the unique positive solution of the transcendental equationProof Let x(t)=(t)−y(t)for all t≥0.From the relations(2),we derive thatholds for all s>0.consequently,the function t 7→ ∥x(t)∥1is absolutely continuous in(0,+∞),which implies that derivatives of∥x(t)∥1exist almost everywhere in(0,+∞).Furthermore,from the equation(3),we can conclude that derivativesof∥x(t)∥1satisfyfor t≥ 0 and t≠=tk.The combination of the condition(10),Lemma 1and(9)implies that there exists a constant M≥1 such thatholds for all t≥ 0 and t≠=tk,where σ is the unique positive solution of the transcendental equationFor t=tk,we enjoyFrom the assumption of|1+ γk|≤ 1 for k ∈ {1,2,···,q},we deriveFinally,we conclude that the inequality(11)holds for all t≥0.Remark 1 Theorem 1 provides a new integral average criterion for global exponential stability of the non-autonomous impulsive functional dif f erential equation with periodic coefficients(3).The equation(3)is essentially dif f erent from the nonautonomous impulsive functional dif f erential equation(15)in[16]because coefficients of the equation(15)in[16]only depend on the exponential function etin the nonautonomoussense.Moreover,Theorem 8 in[16]has only provided a stability criterion for the equilibrium point rather than the generalsolutions.consequently,Theorem 1 is dif f erent from Theorem 8 in[16].4 Global exponential stability of cNNs(1)In this section,we study global exponential stability of cNNs(1).For this,we respectively def i neTheorem 2 Suppose that the assumptions(H1)–(H3)hold and the inequalitieshold for some positive real numbers di(i=1,2,···,n),where T is the period of the functions ai(t),bij(t),cij(t)and Ii(t),i,j=1,2,···,n.Then the model(1)is globally exponentially stable,i.e.,the solutions y(t)and x(t)of the model(1)initiated from ϕ,ψ ∈ c([−τ,0],n)enjoy the following relationwhere M ≥ 1 is a constant and σ is the unique positive solution of the transcendental equationwithProofLet D=diag(d1,d2,···,dn)and c=diag,whereThe inequalities(16)imply that ci>0,i=1,2,···,n.For all x,y ∈ ,And then,we havewhich impliesMoreover,we haveFrom the above inequality,we deriveconsequently,we haveWithout loss of generalization,we assume that the above maximum value is derived at i0,that isFrom the inequalities of(16),(19)and(20),we can deriveAccording to Theorem 1,the solutionsandof the following functional dif f erential equationsatisf i eswhere σ is the unique positive solution of the equation(18).It is obvious that=C−1Dy(t)and=C−1Dx(t)are the solutions of the dif f erential equation(21)if y(t)and x(t)are solutions of the model(1).consequently,the model(1)is globally exponentially stable and the periodic solutions of themodel(1)enjoy the relation(17).Remark 2 Dif f erent stability criteria of non-autonomous cNNs have been provided in these papers[19,21,23,24,27].Except global Lipschitz continuity assumption,these activation functions are assumed to be bounded,and monotonic increasing and bounded in[19,21,24,27],respectively.One of the assumptions of activation functions is stronger than global Lipschitz continuity in[23].However,our model only assume the activation functions to be globally Lipschitz continuous.consequently,our results are new compared with ones in these papers.Remark 3 The papers[16,17]have discussed the stability of the equilibrium point.However,the cNN models in[16,17]are dif f erent from the model(1)in this paper because the coefficients in the former are constants and the ones in the latter are pared with them,our model and results are new.Remark 4 If the assumption of ai(t)>0 is abandoned and the inequalities(16)are accordingly changed intoit is easily proved that the condition(22)guarantees the global exponential stability of the model(1)without the assumption of ai(t)>0 from the proof of Theorem 2.In additional,the paper[20]has dealt with the global exponential stability of generalized delayed and periodic cellular neural networks without the assumption of ai(t)>0.On the one hand,our model additionally consider the ef f ect of impulsive perturbations compared with the paper[20].In the sense of model,our model is the generalization of onein[20].On the other hand,the condition(16)is weaker than the one(3.7)of Theorem 3.1 and 3.2 in[20]even if we do not consider impulsive ef f ect,which means that our result is improvement of Theorem 3.1 and 3.2 in[20].The following example is presented to illustrate that our result is generalization and improvement of Theorem 3.1 and 3.2 in the paper[20]. Example 1 consider the following delayed and impulsive cNNs with periodiccoefficientswhere i=1,2,fj(x)=tanhx,andIn the model(23),the coefficients ai(t),bij(t)and cij(t)areperiodic for i,j=1,2 and τ=2.L(fj)=1 and L(gj)=for j=1,2.It is obvious that the assumptionsof(H1)–(H3)are satis fi ed.Furthermore,taking d1=2 and d2=5,we enjoyeven if there exists t0such thatThis means that the inequalities(16)hold.Henceperiodic solution of the model(23)is globally exponentially stable.Figure 1 is the simulation of the model(23).Figure 1:The simulation for the solutions to delayed impulsive cNNs(23) If we do not consider the impulsive ef f ect of the model(23),we takeThis implies that A(t)<B(t)for any t≥0.It is obvious that the conditionm[A(t)]>m[B(t)]impossibly holds.consequently,Theorem 3.1 and 3.2 in the paper[20]are not valid to this example.5 conclusionThis paper has discussed global exponential stability of non-autonomous impulsive and delayed cNNs with periodic coefficients.The new integral average criteria for global exponential stability of periodic solutions of this cNNs model and a general nonautonomous periodic impulsive functional dif f erential equation have been obtained by means of nonlinear measure method and periodic Halanay dif f erential inequality.Our method does not require the assumptions on boundedness and monotonicity of activation functions,which demonstrates that our derived criteria are less restrictive than some existing ones and can be applied to more general practical problems.Moreover,the stability criterion of this cNNs is the generalization and improvements of some existing ones.The example and its simulation have illustrated that the ef f ectiveness of the proposed method and the correctness of our results.References:[1]chua L O,Yang L.cellular neural networks:theory[J].IEEE Transactions on circuits and Systems,1988,35(10):1257-1272[2]chua L O,Yang L.cellular neural networks:applications[J].IEEE Transactions on circuits and Systems,1988,35(10):1273-1290[3]Gilli M,et al.Equilibrium analysis of cellular neural networks[J].IEEE Transactions on circuits and Systems I:Fundamental Theory and Applications,2004,51(5):903-912[4]Chen W H,Zheng W X.A new method for complete stability analysis of cellular neural networks with time delay[J].IEEE Transactions on Neural Networks,2010,21(7):1126-1138[5]Ahmada S,Stamovab I M.Global exponential stability for impulsive cellular neural networks with timevarying delays[J].Nonlinear Analysis:Real World Applications,2008,69(3):786-795[6]Gilli M.Stability of cellular neural networks and delayed cellular neural networks with nonpositive templates and nonmonotonic output functions[J].IEEE Transactions on circuits and Systems I:Fundamental Theory and Applications,1994,41(8):518-528[7]He Y,et al.An improved global asymptotic stability criterion for delayed cellular neural networks[J].IEEE Transactions on NeuralNetworks,2006,17(1):250-252[8]Hu L,et al.Novel stability of cellular 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of nonautonomous cellular neural networks with impulses and delays[J].Nonlinear Analysis:Real World Applications,2009,10(3):1891-1903[15]Feng G,Lam J.Stability and dissipativity analysis of distributed delay cellular neural networks[J].IEEE Transactions on NeuralNetworks,2011,22(6):976-981[16]Song X L,Peng J G.Global asymptotic stability of impulsive cNNs with proportional delays and partially Lipschitz activation functions[J].Abstract and Applied Analysis,2014,doi:10.1155/2014/832892[17]Zhou L Q.Delay-dependent exponential stability of cellular neural networks with multi-proportional delays[J].Neural ProcessingLetters,2013,38(3):347-359[18]Gopalsamy K.Stability and Oscillations in Delay Dif f erential Equations of Population Dynamics[M].Kluwer,Dordrecht:The Netherlands,1992 [19]Wang H,et al.Existence and global exponential stability of periodic solution of cellular neural networks with delay and impulses[J].Results in Mathematics,2010,58(1-2):191-204[20]Lisena B.Average criteria for 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北师大考博辅导班：2019北师大基础数学考博难度解析及经验分享

北师大考博辅导班：2019北师大基础数学考博难度解析及经验分享根据教育部学位与研究生教育发展中心最新公布的第四轮学科评估结果可知，在2018-2019年基础数学专业学校排名中，排名第一的是北京大学，排名第二的是复旦大学，排名第三的是中山大学。

作为北京师范大学实施国家“211工程”和“985工程”的重点学科，数学科学学院的基础数学一级学科在历次全国学科评估中均名列第四。

下面是启道考博辅导班整理的关于北京师范大学基础数学考博相关内容。

一、专业介绍基础数学专业是一级学科数学下设的二级学科。

它包含了诸多的研究方向和新的、有活力的交叉学科研究方向。

基础数学最新的研究方向主要有：应用动力系统、小波分析、非线性泛函分析与代数表示论。

本专业培养掌握数学科学的基本理论与基本方法具备运用数学知识使用计算机解决实际问题的能力受到科学研究的初步训练能在科技教育和经济部门从事研究教学工作或在制造业生产经营及管理部门从事实际应用开发研究和管理工作。

IT业职员、商务人员、教师都是不错的选择。

北京师范大学数学科学学院的基础数学专业在博士招生方面，划分为11个研究方向：070101基础数学研究方向：01调和分析及其应用考试科目：①1101英语②2027现代分析基础或2212泛函分析或2213实分析③3337调和分析02常微分方程与动力系统考试科目：①1101英语②2212泛函分析或2213实分析③3342微分方程定性理论03代数组合论考试科目：①1101英语或1102俄语②2201抽象代数③3068组合数学04偏微分方程及其应用考试科目：①1101英语②2212泛函分析③3348偏微分方程05拓扑学和微分几何考试科目：①1101英语②2230代数拓扑③3117同伦论06函数空间及其应用考试科目：①1101英语②2212泛函分析或2213实分析③3014Fourier分析07数理逻辑考试科目：①1101英语②2201抽象代数或2213实分析③3213数理逻辑08函数逼近论考试科目：①1101英语②2212泛函分析或2213实分析③3307函数逼近论基础09同调代数与代数表示论考试科目：①1101英语②2201抽象代数③3343代数表示论基础或3346同调代数10复分析考试科目：①1101英语②2228复分析③3338解析函数论11辛几何拓扑与非线性分析考试科目：①1101英语②2206微分几何或2212泛函分析③3317非线性泛函分析或3318拓扑学二、考试内容北京师范大学基础数学专业博士研究生招生包括初试和复试。

广义Boussinseq波动方程的周期波解

广义Boussinseq波动方程的周期波解郑珊【摘要】运用动力系统分支方法研究非线性发展方程的精确行波解，获得了一些孤立波解和椭圆函数形式的周期波解的显示表达式。

并且证明了在某种意义下，孤立波解是周期波解的极限，表明在某些情形下可以通过周期波解得到孤立波解。

%We employ bifurcation method of dynamical systems to investigate exact traveling wave solutions of a nonlinear evolution equation. We obtain some exact explicit expressions of solitary wave solutions and some new exact periodic wave solutions in parameter forms of Jacobian elliptic function. We point out that the solitary waves are limits of the periodic waves in some sense,the results infer that the periodic waves degenerate solitary waves in some conditions.【期刊名称】《广州航海高等专科学校学报》【年(卷),期】2015(000)004【总页数】4页(P43-46)【关键词】分支方法;周期波;孤立波【作者】郑珊【作者单位】广州航海学院基础部，广东广州510000【正文语种】中文【中图分类】O130 引言随着海洋知识不断发展，流体力学和水动力学的研究与波浪数学模型与波动方程的研究紧密联系．当波浪受地形或建筑影响，产生浅水变形等现象，浅水波的非线性波动方程就成为一种重要的研究模型．关于非线性浅水波的Boussinseq波动方程［1］及其解的情况成为本文研究的重点对象．考虑如下Boussinseq波动方程［1］其中α，β，γ是常数．对方程(1)作行波变换得:其中c表示波速常数．将方程(2)带入(1)，得:方程(2)的导数是关于变量ξ的导数，对(3)关于ξ积分两次，取第一次积分常数为0，第二次积分常数为 g，得:将方程(4)转变成平面系统系统(5)的首次积分为:其中h表示积分常数．本文的内容安排如下，在第二部分描绘系统(5)的分支相图，画出同宿轨和周期轨线，第三部分根据轨线求出孤立波解和周期波解，第四部分给出孤立波解和周期波解之间的关系，最后对本文小结．1 平面系统(5)的分支相图记Δ=(c2+α)2+4βg，当Δ ＞0时，系统(5)有两个奇点(Φ1，0)和(Φ2，0)，其中Φ1=，它们的哈密顿量分别记为h1=H(Φ1，0)和h2=H(Φ2，0)．根据微分方程定性理论［2－4］，系统(5)有同宿轨和周期轨，如图1 所示．图1 系统在一定参数条件下的分支相图2 方程的周期波解在本节，模为m的Jacob ran椭圆函数sn(l，m)记作snl．设f(Φ)=，则f(Φ)满足如下引理．引理1 若相轨上点的哈密顿量h=H(Φ，0)满足h1＜h＜h2或h2＜h＜h1，则函数f(Φ)必有3个不同的实的零点．证明仅证的情况，当时与此类似．在上述条件下，经计算可得:因此，f(Φ1)·f(Φ2)=4(h－h1)(h－h2)＜0．又f(Φ)满足f(－∞)＞0，f(Φ1)＜0，f(Φ2)＞0，f(+∞)＜0，而且f'(Φ)=Φ2)在区间(－∞，Φ1)，(Φ1，Φ2)，(Φ2，+∞)是单调的．由连续函数的零点定理，f(Φ)必存在3个不同的零点位于上述3个不同的区间，引理得证．设c1＜c2＜c3是f(Φ)的3个不同的实的零点，引理1说明(6)与Φ 轴有 3个交点(c1，0)，(c2，0)和(c3，0)．因而(6)可改写成如下形式:其中c1＜Φ1＜c2＜Φ2＜c3．当将(8)代入并沿着周期轨积分，得:时，由(7)可得周期轨的表达式为:由文献［5］中公式(236)得:其中由(9)可解得周期波解为Φ=c3－(c3－，即 u1(x，t)=c3－(c3－c2)sn2，其中sn的模是m1=类似的，当将(10)代入时，周期轨的表达式为:并沿着周期轨积分，得:相应的周期波解的表达式为:其中sn的模是m2=3 方程的精确孤立波解本节利用椭圆函数的性质，当模m→1时，sn→tanh［1］，可由周期波解得孤立波解．定理1 设ui(i=1，2)是方程(1)的周期波解，mi(i=1，2)是对应解中椭圆函数sn的模．则如下结论成立．(1)在g=0且的条件下，当mi(i=1，2)→1时，由ui(i=1，2)可以得到孤立波解u3(x，t)=(2)在g=0且的条件下，当mi(i=1，2)→1时，由ui(i=1，2)可以得到孤立波解u4(x，t)=(3)在g≠0且的条件下，当m1→1时，由u1可以得到孤立波解 u5(x，t)=(4)在g≠0且的条件下，当m2→1时，由u2可以得到孤立波解 u6(x，t)=证明 (1)当 m1= 时，c1=c2，sn→tanh，又 g=0，经计算得，c1=c2=0，c3=将 c1，c2，c3 代入 u1，得:当m2→1且g=0时，c2=c3=0，c1= 将 c1，c2，c3 代入 u2(x，t)，得 u2=u1．类似地，可以证明(2)成立．对于(3)，在g≠0的她条件下，当m1=→1时，c1=c2，sn→tanh，经计算得:c1=c2=将c1，c2，c3代入 u1(x，t)得:类似地，可以证明(4)成立．4 小结本文运用动力系统分支方法，首先得到了方程(1)的椭圆函数形式的周期波解，然后由椭圆函数的性质，由周期波解推导出孤立波解，从这两种的关系可以看出，孤立波解可以看作是周期波解的极限形式．参考文献:［1］ ABDOU M A．Exact periodic wave solutions to some nonlinear evolution equations［J］．International Journal of Nonlinear Science，2008(6):145－153．［2］ CHOW S N，HALE J K．Method of Bifurcation Theory ［M］．Berlin:Springer，1981．［3］ LI J B，LIU Z R．Smooth and non-smooth traveling waves in a nonlinearly dispersive equation［J］．Applied Mathematical Modeling，2000(25):41－56．［4］ LI J B，LIU Z R．Travelling wave solutions for a class of nonlinear dispersive equations［J］．Chinese Annals of Mathematics，2002(23):397－418．［5］ BYRD P F，FRIEDMAN M D．Handbook of elliptic integrals for engineers and scientists［M］．Berlin:Springer，1971．。

(1+1)维混合KdV方程的精确解

第39卷第1期注為科修 Vol. 39 No. 12021 年 2 月JIANGXI SCIENCE Fl. 2021doi ：10.13990/j. it y l001 -3679.2021.01.005(1+1)维混合KdV 方程的精确解翟子璇，李琪8(东华理工大学理学院,330013,南昌)摘要：讨论一类混合KdV 方程，通过F-展开法及辅助常微分方程,成功得到该方程的精确解。

关键词:F-展开法；混合KdV 方程;精确解中图分类号：0175.29文献标识码:A 文章编号：1001 -3679(2021)01 -022-03Exact Solutions of Mixed (1+1) - Dimensional KdV EquationZHAI Zixuan, LI Qi **收稿日期:2020 -12 -10；修订日期:2021 -01 -12作者简介:翟子璇(1996—),男，硕士研究生，研究方向为非线性可积系统及应用。

基金项目：国家自然科学基金(11561002、11861006)；江西省教育厅科技项目(GJJ191419)。

*通信作者:李琪(1973—)，女，博士，教授，研究方向为非线性可积系统及应用。

E - mail :qli@ ecut. edu. cn 。

( School of Science , East China University of Technology, 330013 , Nanchang )Abstract : The exact solution' of mixed ( 1 + 1) - dimensional KdV equation are obtained by using F- expansion method and auxiliaie ordinary dVTerentiaO equation .Key words : F 一 expansion method ； mixed ( 1 + 1) 一 dimensional KdV equation ； exact solutions0引言随着人们对自然界更加深入的研究,许多非线性现象逐渐进入研究者的视野,而非线性发展方程则是对这些现象进行客观描述的有力工具。

Dielectric-fibre surface waveguides for optical frequencies中文翻译

光频率介质纤维表面波导Dielectric-fibre surface waveguides for optical frequencies高锟(G.A. Hockham)关键词：光学纤维，波导摘要：折射率高于周围区域的介质纤维是作为在光频段引导传输的可能的介质的一种介电波导形式。

文章中讨论的这种特殊的结构形式是圆的横截面。

用作通信目的的光波导传播模式的选择通常主要考虑损耗特性和信息容量。

文章中讨论了介电损耗，弯曲损耗和辐射损耗并且讨论了与信息容量相关的模式稳定，色散和功率控制，同时也讨论了物理实现方面，也包含了对对光学和微波波长的实验研究。

主要符号列表：n J = n 阶的第一类贝塞尔函数n K = 2π修正的第二类n 阶的变型贝塞尔函数β = g2λπ，波导的相位系数 n J ' = n J 的一阶导数n K ' = n K 的一阶导数i h = 衰减系数或辐射波数i ε = 相对介电常数0k = 自由空间传播系数a = 光纤半径γ = 纵向传播系数k = 波耳兹曼常数T = 绝对温度，Kc β = 等温可压缩性λ = 波长n = 折射率)(H i υ = 第υ阶Hankel 函数的第i 阶导数υH ' = υH 的导数 υ = 方位角传播系数=21υυj -L = 调制周期下标n 是整数，下标m 是n J = 0的第m 个根。

1. 简介折射率高于周围区域的介质纤维是一种介电波导，它代表了光频段中能量有向传输的一种媒介。

这种结构形式引导电磁波沿着不同折射率区域的特定边界传播，相关电磁场部分在光纤内部分在光纤外。

外部电磁场在垂直于传播方向上是逐渐消失的，以且在无穷远处以近似指数的形式衰减到零。

这种结构经常被称为开放波导，以表面波模式传播。

下面要讨论的是具有圆形截面的特种介质纤维波导。

2．介质纤维波导具有圆形截面的介质纤维能够传输所有的H 0m 模、E 0m 模和HE nm 混合模。

流体力学常用名词中英文对照

流体力学常用名词流体动力学fluid dynamics连续介质力学mechanics of continuous介质medium流体质点fluid particle无粘性流体nonviscous fluid, inviscid连续介质假设continuous medium hypothesis流体运动学fluid kinematics水静力学hydrostatics液体静力学hydrostatics支配方程governing equation伯努利方程Bernoulli equation伯努利定理Bernonlli theorem毕奥-萨伐尔定律Biot-Savart law欧拉方程Euler equation亥姆霍兹定理Helmholtz theorem开尔文定理Kelvin theorem涡片vortex sheet库塔-茹可夫斯基条件Kutta-Zhoukowski condition 布拉休斯解Blasius solution达朗贝尔佯廖d'Alembert paradox雷诺数Reynolds number施特鲁哈尔数Strouhal number随体导数material derivative不可压缩流体incompressible fluid质量守恒conservation of mass动量守恒conservation of momentum能量守恒conservation of energy动量方程momentum equation能量方程energy equation控制体积control volume液体静压hydrostatic pressure涡量拟能enstrophy压差differential pressure流[动] flow流线stream line流面stream surface流管stream tube迹线path, path line流场flow field流态flow regime流动参量flow parameter流量flow rate, flow discharge 涡旋vortex涡量vorticity涡丝vortex filament涡线vortex line涡面vortex surface涡层vortex layer涡环vortex ring涡对vortex pair涡管vortex tube涡街vortex street卡门涡街Karman vortex street 马蹄涡horseshoe vortex对流涡胞convective cell卷筒涡胞roll cell涡eddy涡粘性eddy viscosity环流circulation环量circulation速度环量velocity circulation 偶极子doublet, dipole驻点stagnation point总压[力] total pressure总压头total head静压头static head总焓total enthalpy能量输运energy transport速度剖面velocity profile库埃特流Couette flow单相流single phase flow单组份流single-component flow均匀流uniform flow非均匀流nonuniform flow二维流two-dimensional flow三维流three-dimensional flow准定常流quasi-steady flow非定常流unsteady flow, non-steady flow 暂态流transient flow 周期流periodic flow振荡流oscillatory flow分层流stratified flow无旋流irrotational flow有旋流rotational flow轴对称流axisymmetric flow不可压缩性incompressibility不可压缩流[动] incompressible flow 浮体floating body定倾中心metacenter阻力drag, resistance减阻drag reduction表面力surface force表面张力surface tension毛细[管]作用capillarity来流incoming flow自由流free stream自由流线free stream line外流external flow进口entrance, inlet出口exit, outlet扰动disturbance, perturbation分布distribution传播propagation色散dispersion弥散dispersion附加质量added mass ,associated mass 收缩contraction镜象法image method无量纲参数dimensionless parameter 几何相似geometric similarity运动相似kinematic similarity动力相似[性] dynamic similarity平面流plane flow势potential势流potential flow速度势velocity potential复势complex potential复速度complex velocity流函数stream function源source汇sink速度[水]头velocity head拐角流corner flow空泡流cavity flow超空泡supercavity超空泡流supercavity flow空气动力学aerodynamics低速空气动力学low-speed aerodynamics 高速空气动力学high-speed aerodynamics 气动热力学aerothermodynamics亚声速流[动] subsonic flow跨声速流[动] transonic flow超声速流[动] supersonic flow锥形流conical flow楔流wedge flow叶栅流cascade flow非平衡流[动] non-equilibrium flow细长体slender body细长度slenderness钝头体bluff body钝体blunt body翼型airfoil翼弦chord薄翼理论thin-airfoil theory构型configuration后缘trailing edge迎角angle of attack失速stall脱体激波detached shock wave波阻wave drag诱导阻力induced drag诱导速度induced velocity临界雷诺数critical Reynolds number 前缘涡leading edge vortex 附着涡bound vortex约束涡confined vortex气动中心aerodynamic center气动力aerodynamic force气动噪声aerodynamic noise气动加热aerodynamic heating离解dissociation地面效应ground effect气体动力学gas dynamics稀疏波rarefaction wave热状态方程thermal equation of state 喷管Nozzle 普朗特-迈耶流Prandtl-Meyer flow瑞利流Rayleigh flow可压缩流[动] compressible flow可压缩流体compressible fluid绝热流adiabatic flow非绝热流diabatic flow未扰动流undisturbed flow等熵流isentropic flow匀熵流homoentropic flow兰金-于戈尼奥条件Rankine-Hugoniot condition 状态方程equation of state量热状态方程caloric equation of state完全气体perfect gas拉瓦尔喷管Laval nozzle马赫角Mach angle马赫锥Mach cone马赫线Mach line马赫数Mach number马赫波Mach wave当地马赫数local Mach number冲击波shock wave激波shock wave正激波normal shock wave斜激波oblique shock wave头波bow wave附体激波attached shock wave 激波阵面shock front激波层shock layer压缩波compression wave反射reflection折射refraction散射scattering衍射diffraction绕射diffraction出口压力exit pressure超压[强] over pressure反压back pressure爆炸explosion爆轰detonation缓燃deflagration水动力学hydrodynamics液体动力学hydrodynamics泰勒不稳定性Taylor instability 盖斯特纳波Gerstner wave斯托克斯波Stokes wave瑞利数Rayleigh number自由面free surface波速wave speed, wave velocity 波高wave height波列wave train波群wave group波能wave energy表面波surface wave表面张力波capillary wave规则波regular wave不规则波irregular wave浅水波shallow water wave深水波deep water wave重力波gravity wave椭圆余弦波cnoidal wave潮波tidal wave涌波surge wave破碎波breaking wave船波ship wave非线性波nonlinear wave孤立子soliton水动[力]噪声hydrodynamic noise 水击water hammer空化cavitation空化数cavitation number空蚀cavitation damage超空化流supercavitating flow 水翼hydrofoil水力学hydraulics洪水波flood wave涟漪ripple消能energy dissipation海洋水动力学marine hydrodynamics 谢齐公式Chezy formula 欧拉数Euler number弗劳德数Froude number水力半径hydraulic radius水力坡度hvdraulic slope高度水头elevating head水头损失head loss水位water level水跃hydraulic jump含水层aquifer排水drainage排放量discharge壅水曲线back water curve压[强水]头pressure head过水断面flow cross-section明槽流open channel flow孔流orifice flow无压流free surface flow有压流pressure flow缓流subcritical flow急流supercritical flow渐变流gradually varied flow急变流rapidly varied flow临界流critical flow异重流density current, gravity flow 堰流weir flow掺气流aerated flow含沙流sediment-laden stream降水曲线dropdown curve沉积物sediment, deposit沉[降堆]积sedimentation, deposition 沉降速度settling velocity流动稳定性flow stability不稳定性instability奥尔-索末菲方程Orr-Sommerfeld equation 涡量方程vorticity equation 泊肃叶流Poiseuille flow奥辛流Oseen flow剪切流shear flow粘性流[动] viscous flow层流laminar flow分离流separated flow二次流secondary flow近场流near field flow远场流far field flow滞止流stagnation flow尾流wake [flow]回流back flow反流reverse flow射流jet自由射流free jet管流pipe flow, tube flow内流internal flow拟序结构coherent structure 猝发过程bursting process表观粘度apparent viscosity 运动粘性kinematic viscosity 动力粘性dynamic viscosity泊poise厘泊centipoise厘沱centistoke剪切层shear layer次层sublayer流动分离flow separation层流分离laminar separation 湍流分离turbulent separation 分离点separation point附着点attachment point再附reattachment再层流化relaminarization起动涡starting vortex驻涡standing vortex涡旋破碎vortex breakdown涡旋脱落vortex shedding压[力]降pressure drop压差阻力pressure drag压力能pressure energy型阻profile drag滑移速度slip velocity无滑移条件non-slip condition壁剪应力skin friction, frictional drag 壁剪切速度friction velocity磨擦损失friction loss磨擦因子friction factor耗散dissipation滞后lag相似性解similar solution局域相似local similarity气体润滑gas lubrication液体动力润滑hydrodynamic lubrication浆体slurry泰勒数Taylor number纳维-斯托克斯方程Navier-Stokes equation 牛顿流体Newtonian fluid 边界层理论boundary later theory边界层方程boundary layer equation边界层boundary layer附面层boundary layer层流边界层laminar boundary layer湍流边界层turbulent boundary layer温度边界层thermal boundary layer边界层转捩boundary layer transition边界层分离boundary layer separation 边界层厚度boundary layer thickness 位移厚度displacement thickness动量厚度momentum thickness能量厚度energy thickness焓厚度enthalpy thickness注入injection吸出suction泰勒涡Taylor vortex速度亏损律velocity defect law形状因子shape factor测速法anemometry粘度测定法visco[si] metry流动显示flow visualization油烟显示oil smoke visualization孔板流量计orifice meter频率响应frequency response油膜显示oil film visualization阴影法shadow method纹影法schlieren method烟丝法smoke wire method丝线法tuft method 说明氢泡法nydrogen bubble method相似理论similarity theory相似律similarity law部分相似partial similarity定理pi theorem, Buckingham theorem 静[态]校准static calibration动态校准dynamic calibration风洞wind tunnel激波管shock tube激波管风洞shock tube wind tunnel 水洞water tunnel拖曳水池towing tank旋臂水池rotating arm basin扩散段diffuser测压孔pressure tap皮托管pitot tube普雷斯顿管preston tube斯坦顿管Stanton tube文丘里管Venturi tubeU形管U-tube压强计manometer微压计micromanometer多管压强计multiple manometer静压管static [pressure]tube流速计anemometer风速管Pitot- static tube激光多普勒测速计laser Doppler anemometer,laser Doppler velocimeter 热线流速计hot-wire anemometer热膜流速计hot- film anemometer流量计flow meter粘度计visco[si] meter涡量计vorticity meter传感器transducer, sensor压强传感器pressure transducer 热敏电阻thermistor示踪物tracer时间线time line脉线streak line尺度效应scale effect壁效应wall effect堵塞blockage堵寒效应blockage effect动态响应dynamic response响应频率response frequency底压base pressure菲克定律Fick law巴塞特力Basset force埃克特数Eckert number格拉斯霍夫数Grashof number努塞特数Nusselt number普朗特数prandtl number雷诺比拟Reynolds analogy施密特数schmidt number斯坦顿数Stanton number对流convection自由对流natural convection, free convec-tion 强迫对流forced convection热对流heat convection质量传递mass transfer传质系数mass transfer coefficient热量传递heat transfer传热系数heat transfer coefficient对流传热convective heat transfer辐射传热radiative heat transfer 动量交换momentum transfer能量传递energy transfer传导conduction热传导conductive heat transfer 热交换heat exchange临界热通量critical heat flux浓度concentration扩散diffusion扩散性diffusivity扩散率diffusivity扩散速度diffusion velocity分子扩散molecular diffusion沸腾boiling蒸发evaporation气化gasification凝结condensation成核nucleation计算流体力学computational fluid mechanics多重尺度问题multiple scale problem伯格斯方程Burgers equation对流扩散方程convection diffusion equationKDU方程KDV equation修正微分方程modified differential equation拉克斯等价定理Lax equivalence theorem数值模拟numerical simulation大涡模拟large eddy simulation数值粘性numerical viscosity非线性不稳定性nonlinear instability希尔特稳定性分析Hirt stability analysis相容条件consistency conditionCFL条件Courant- Friedrichs- Lewy condition ,CFL condition 狄里克雷边界条件Dirichlet boundary condition熵条件entropy condition远场边界条件far field boundary condition流入边界条件inflow boundary condition无反射边界条件nonreflecting boundary condition 数值边界条件numerical boundary condition流出边界条件outflow boundary condition冯.诺伊曼条件von Neumann condition近似因子分解法approximate factorization method 人工压缩artificial compression人工粘性artificial viscosity边界元法boundary element method配置方法collocation method能量法energy method有限体积法finite volume method流体网格法fluid in cell method,FLIC method通量校正传输法flux-corrected transport method 通量矢量分解法flux vector splitting method伽辽金法Galerkin method积分方法integral method标记网格法marker and cell method, MAC method 特征线法method of characteristics直线法method of lines矩量法moment method多重网格法multi- grid method板块法panel method质点网格法particle in cell method, PIC method 质点法particle method预估校正法predictor-corrector method投影法projection method准谱法pseudo-spectral method随机选取法random choice method激波捕捉法shock-capturing method激波拟合法shock-fitting method谱方法spectral method稀疏矩阵分解法split coefficient matrix method不定常法time-dependent method时间分步法time splitting method变分法variational method涡方法vortex method隐格式implicit scheme显格式explicit scheme交替方向隐格式alternating direction implicit scheme, ADI scheme 反扩散差分格式anti-diffusion difference scheme紧差分格式compact difference scheme守恒差分格式conservation difference scheme克兰克-尼科尔森格式Crank-Nicolson scheme杜福特-弗兰克尔格式Dufort-Frankel scheme指数格式exponential scheme戈本诺夫格式Godunov scheme高分辨率格式high resolution scheme拉克斯-温德罗夫格式Lax-Wendroff scheme蛙跳格式leap-frog scheme单调差分格式monotone difference scheme保单调差分格式monotonicity preserving diffe-rence scheme 穆曼-科尔格式Murman-Cole scheme半隐格式semi-implicit scheme斜迎风格式skew-upstream scheme全变差下降格式total variation decreasing scheme TVD scheme 迎风格式upstream scheme , upwind scheme计算区域computational domain物理区域physical domain影响域domain of influence依赖域domain of dependence区域分解domain decomposition维数分解dimensional split物理解physical solution弱解weak solution黎曼解算子Riemann solver守恒型conservation form弱守恒型weak conservation form强守恒型strong conservation form散度型divergence form贴体曲线坐标body- fitted curvilinear coordi-nates [自]适应网格[self-] adaptive mesh适应网格生成adaptive grid generation自动网格生成automatic grid generation数值网格生成numerical grid generation交错网格staggered mesh网格雷诺数cell Reynolds number数植扩散numerical diffusion数值耗散numerical dissipation数值色散numerical dispersion数值通量numerical flux放大因子amplification factor放大矩阵amplification matrix阻尼误差damping error离散涡discrete vortex熵通量entropy flux熵函数entropy function分步法fractional step method。

黄冈市人民政府关于颁授黄冈市第十届自然科学优秀学术论文的通报-黄政发〔2019〕13号

黄冈市人民政府关于颁授黄冈市第十届自然科学优秀学术论文的通报正文：----------------------------------------------------------------------------------------------------------------------------------------------------黄冈市人民政府关于颁授黄冈市第十届自然科学优秀学术论文的通报各县、市、区人民政府，龙感湖管理区、黄冈高新区管委会、黄冈白潭湖片区筹委会、白莲河示范区管委会，市直各单位：近年来，全市广大科技工作者潜心钻研，大胆创新，取得了一批自然科学成果及优秀学术论文。

为进一步营造崇尚科学、尊重知识、尊重人才、鼓励创造的科学文化氛围，鼓励全市科技工作者不断加强学术创新，更好地服务于黄冈市高质量发展，经各县（市、区）科协、市直各有关单位推荐、初评，经黄冈市第十届自然科学优秀学术论文评审委员会评审确定，并经公示无异议，市政府同意颁授万柳撰写的《Nitrogen, sulfur co-doped hierarchically porous carbon from rape pollen as high-performance supercapacitor electrode》、万美南撰写的《Observation of reduced phase transition temperature in N-doped thermochromic film of monoclinic VO2 》、丁秀娟撰写的《超声引导下置入PICC导管异位的原理分析及护理体会》等250 篇论文为黄冈市第十届自然科学优秀学术论文。

希望获奖的同志珍惜荣誉，再接再厉，不断探索奋进，在各自工作领域作出新的更大成绩。

市政府号召全市广大科技工作者要以优秀论文撰写者为榜样，进一步解放思想，紧紧围绕我市经济社会发展中的重大课题，深入研究，克难攻关，锐意进取，勇于创新，为推动黄冈在湖北高质量发展中力争上游作出新的更大贡献。

辅助函数法求解非线性偏微分方程精确解

辅助函数法求解非线性偏微分方程精确解杨健;赖晓霞【摘要】在数学和物理学领域,将含有非线性项的偏微分方程称为非线性偏微分方程.非线性偏微分方程用于描述物理学中许多不同的物理模型,范围涉及从引力到流体动力学的众多领域,还在数学中用于验证庞加莱猜想和卡拉比猜想.在求解非线性偏微分方程的过程中,几乎没有通用的求解方法能够应用于所有的方程.通常,可依据模型方程的数学物理背景来先验地假设非线性偏微分方程解的形式,并根据解的特点给出辅助方程.非线性偏微分方程可通过行波变换转化为常微分方程,再借助辅助方程来求解常微分方程.为此,借助行波变换及辅助方程的求解思路对BBM方程和Burgers方程进行了研究,并获得了其双曲正切函数及三角函数形式的精确解.研究结果表明,所采用的方法可广泛应用于若干在数学物理中有典型应用背景的非线性偏微分方程的精确解求解中.%In mathematics and physics,a nonlinear partial differential equation is a partial differential equation with nonlinear terms,which can describe many different physical models ranging from gravitation to fluid dynamics,and have been adopted in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. There are almost no general solutions that can be applied for all equa-tions. Nonlinear partial differential equation usually originates from mathematical and physical fields,such that the ansatz of the solutions has been given and an auxiliary function has been provided according to its mathematical and physical features. They can be transmitted to an ordinary differential equations via a traveling wave transformation. Through introduction of the auxiliary function into the ordinary dif-ferential equation a set of nonlinear algebra equations is acquired,which can supply solutions original partial differential equation in sol-ving process. Therefore,BBM equation and Burgers equation can be solved with the auxiliary function. The exact solutions include tan-gent function and trigonometric functions. The research shows that the proposed auxiliary function method can be applied to solve some other nonlinear partial differential equations with mathematical and physical background.【期刊名称】《计算机技术与发展》【年(卷),期】2017(027)011【总页数】5页(P196-200)【关键词】非线性偏微分方程;辅助函数法;BBM方程;Burgers方程;精确解【作者】杨健;赖晓霞【作者单位】陕西师范大学计算机科学学院,陕西西安 710119;陕西师范大学计算机科学学院,陕西西安 710119【正文语种】中文【中图分类】TP39非线性方程广泛应用于物理学和应用数学的许多分支，尤其在流体力学、固态物理学、等离子物理和非线性光学等。

(2+1)维非线性演化方程的显示解

Vol. 33,No. 1Mar. 2021第33卷第1期2021年3月河南工程学院学报（自然科学版）JOURNAL OF HENAN UNIVERSITY OF ENGINEERING （2+1 ）维非线性演化方程的显示解刘小平（电子科技大学中山学院计算机学院，广东中山528406）摘要：首先借助规范变换并利用Himta 双线性算子的特性,推导出（2 + 1）维非线性方程的双线性形式，然后利用Himta 直接法求出该方程的孤予解和奇异解，包括单孤予解、二孤予解、单奇异解和二奇异解，最后给出新的变换，求出该方程的有理周期解。

关键词：双线性形式；有理周期解;孤子解；奇异解中图分类号：029 文献标志码:A 文章编号：1674 -330X （2021）01 -0074 -03Explicit solutions of the （2+1） dimensional nonlinear evolution equationLIU Xiaoping（^Department of Computer Science , Zhongshan Institute , University of Electronic Science and Technology of China ,Zhongshart 528406, China ）Abstract : In tlie paper, the bilinear form of the （2 + 1） dimensional nonlinear evolution equation is established by utilizing tlie characteristic of Hirota bilinear operators with aid of gauge transfonnation. Tlie soliton solutions and singular solutions are derived by u- sing Hirota direct method , including one soliton solution » two soliton solution » one singular and two singular solution. In the end » a new rational periodic solution is obtained according to a new transformation.Keywords : Hirota bilinear form ； rational periodic solution ； soliton solution ； singular solution寻求物理、数学上有重耍意义方程的显示解一宜是热门话题，现已形成许多成熟的方法，比如Hirota 宜接法「匕穿衣方法［6-12\Riemam-theta 函数与直接法相结合的方法”叭其中,Hirota 直接法提供了一个强有力的获得非线性演化方程的方法。

FMCW雷达周期非线性相位估计与矫正

FMCW雷达周期非线性相位估计与矫正李公全;赖涛;靳科;赵拥军【摘要】In order to obtain a high-quality range image,frequency modulated continuous wave(FMCW) radar usually requires modulation nonlinearity correction under the mechanism of dechirp reception.T raditional methods based on the polynomial phase model can correct most nonlinear distortion in the system,but it is hard to eliminate the paired echoes invited by period nonlinearity,w hich reveals great improvement still exists in range compression.The residual nonlinearity error is modeled as a multi-component sinusoidal FM signal,and after that,a parameter estimation method combining initial value matching and nonlinear least squares optimiza-tion is proposed.The method can achieve high efficiency in the estimation of sinusoidal frequency modulation pa-rameters via Fourier transform and one-dimensional phase searching.Then,the nonlinear least squares is used to further improve the estimation accuracy.Finally the range-dependent nonlinearity is corrected by match Fou-rier transform.Simulation and radar echo processing results demonstrate that the proposed method can effec-tively eliminate periodic phase nonlinearity,suppress paired echoes,and significantly improve range compression quality.%为获取高质量距离像,解线频调接收体制的调频连续波雷达通常需要进行非线性矫正.传统基于多项式模型的方法虽然能够矫正系统中绝大部分的非线性失真,但难以抑制由周期非线性引起的成对回波,距离压缩质量仍有较大提升空间.本文将剩余的周期非线性误差建模为多分量正弦调频信号,提出一种匹配初值与非线性最小二乘相结合的参数估计方法.所提算法仅采用傅里叶变换以及一维相位搜索即可实现正弦调频参数的快速估计,然后利用非线性最小二乘进一步提高估计精度,最后利用匹配傅里叶变换进行空变非线性矫正.仿真以及实测数据均表明,本文所提算法能够有效消除周期相位误差,抑制成对回波,显著改善距离压缩质量.【期刊名称】《系统工程与电子技术》【年(卷),期】2018(040)003【总页数】8页(P538-545)【关键词】调频连续波雷达;非线性矫正;多分量正弦调频;非线性最小二乘;匹配傅里叶变换【作者】李公全;赖涛;靳科;赵拥军【作者单位】信息工程大学导航与空天目标工程学院,河南郑州450001;信息工程大学导航与空天目标工程学院,河南郑州450001;信息工程大学导航与空天目标工程学院,河南郑州450001;信息工程大学导航与空天目标工程学院,河南郑州450001【正文语种】中文【中图分类】TP7510 引言调频连续波(frequency modulated continuous wave,FMCW)雷达相对脉冲体制雷达具有体积小重量轻,结构简单,成本低,低截获概率等优点,因此FMCW雷达在近距离安全检测、机场异物识别、形变测量等领域具有广泛的应用[1-4]。

非线性Schrodinger-MKdV方程的Hamilton结构及代数几何解

非线性Schrodinger-MKdV方程的Hamilton结构及代数几何解岳超【摘要】由3×3等谱Lax矩阵导出了非线性Schr?dinger-MKdV(NLS-MKdV)方程族,应用迹恒等式得到了其Hamilton结构.为方便构造代数几何解,我们将3×3矩阵等谱问题转化为等价的2×2问题,借助Riemann theta函数,求出了耦合的NLS方程及耦合的MKdV方程的代数几何解.【期刊名称】《聊城大学学报（自然科学版）》【年(卷),期】2019(032)001【总页数】8页(P30-37)【关键词】迹恒等式;Hamilton结构;代数几何解;Riemanntheta函数【作者】岳超【作者单位】泰山医学院医学信息工程学院,山东泰安271016【正文语种】中文【中图分类】O175.20 IntroductionSearching for the exact solutions of nonlinear equations has been important and interesting in the areas of the mathematics and physics，and several systematic methods have been developed to obtain explicit solutions of soliton equations， for instance， the inverse scattering method[1]，Darboux and Bäcklund transformations[2]，Hirota’s bilinear method[2-4]， Lie symmetry analysis etc [5-12]. The algebraic-geometric method was first developed by Matveev et al.as an analog of the inverse scattering theory.As a degenerated case of the algebro-geometric solutions， the multi-soliton solution and periodic solution in elliptic function type may be worked out.A systematic approach， proposed by Gesztesy and Holden to construct algebro-geometric solutions for integrable equations， has been extended to the whole (1+1) dimensional integrable hierarchy， such as the AKNS hierarchy， the Camassa-Holm hierarchy etc[13].Recently， Fan etc.investigated algebro-geometric solutions for the Gerdjikov-Ivanov hierarchy， the Hunter-Saxton hierarchy and so on[14-17].In this paper，we first use a 3×3 isospectral Lax matrix to obtain a NLS-MKdV hierarchy by use of the Tu scheme[18-23]， which can reduce to the coupled NLS equation and coupled MKdV equation and whose Hamiltonian structure can be generated by applying the trace identity.As we know， constructing algebro-geometric solutions associated with the 3×3 matrix isosp ectral problem is more complicated than that related to the 2×2 case，hence we transform the above 3×3 matrix isospectral problem into an equivalent 2×2 one， by using Riemann theta functions，the algebro-geometric solutions of the coupled NLS equation and coupled MKdV equation are obtained easily.1 The NLS-MKdV hierarchy and its Hamiltonian structureConsider the 3×3 isospectral Lax matrix(1)solving the equation Vx=[U，V]，leads to(2)(3)Notea direct calculation may show that the compatibility conditions of the Lax pairs engenders the integrable hierarchy，(4)where J is a Hamiltonian operator， from (3)， we obtain a recurrence operatorTherefore， expression (4) can be written as(5)Reduction case 1 When n=2， the system (5) reduces to the following coupled NLS equation(6)Taking we obtain the NLS equationiRt2+2Rxx+R=0.Reduction case 2 When taking n=3 in (5)， we have the coupled MKdV equation(7)Taking β=1，q=0， we get the MKdV equation rt3-3r2rx-2rxxx=0.Hence we call the system (5) NLS-MKdV hierarchy.A direct computation yields=2b+2c，substituting the above equations into the trace identity [18]， we get(8)Comparing the coefficients of λ-n-1 on both sides in (8) leads toit is easy to find that γ=0， then we haveHence， we obtain the Hamiltonian structure of (5)(9)It is easy to verify that JL=L*J， so the NLS-MKdV hierarchy is integrable in Liouville sense.In the following section， we are interested in constructing algebro-geometric solutions of the coupled NLS equation (6) and the coupled MKdV equation (7).2 Algebro-geometric solutions of the coupled NLS equation (6) and the coupled MKdV equation (7)For calculation convenience，we transform 3×3 matrix isospectral problem (1) into an equivalent 2×2 one，(10)which can also generate equations (6) and (7).We consider the Lenard gradient sequence by the recursion relationKSj-1=JSj，Sj|(q，r)=0，S0=(2β，0，0)T ，(11)whereIt is easy to find that Sj is uniquely determined by the recursion relation (10).Here the condition Sj |(q，r)=0 is used to select the integration constant to be zero.A direct computation shows from (11) thatWe suppose (10) has two basic solutions X=(X1，X2)Tand Y=(Y1，Y2)T，thensatisfies the Lax equationWx=[U，W]， Wtm=[V(m)，W]，(12)which implies that the function det(W) is a constant independent of x andtm.From (12)， we get2gx=(q+r)h+(r-q)f，fx=λf-(q+r)g，hx=-λh+(q-r)g，(13)andgtm=B(m)h-C(m)f， ftm=2A(m)f-2B(m)g， htm=2C(m)g-2A(m)h，(14)where(15)and N is an arbitrary positive integer value.Substituting (15) into (13) gives KQj-1=JQj， JQ0=0 ， KQN=0， Qj=(aj，bj，cj)T.(16)It is clear to find JQ0=0 that has the general solutionQ0=α0S0=α0(2β，0，0)T，(17)from (11) and (16)， we have(18)where a0，…，ak+1 are integral constants.Substituting Eq.(18) into Eq.(16) yields the following certain stationary evolution equation(19)without loss of generality we set a0=1， from Eqs.(15)， (16)and (18)， we then have(20)By applying (15) we can rewrite f and h as the following finite products(21)By comparing the coefficients of λN-1，λN-2 and combining Eqs.(15) and (21)， we get(22)(23)since det (W) is a (2N + 2) th-order polynomial in λ with constants， we have(24)Substituting Eq.(15) into Eq.(24) and comparing the coefficient of λ2N+2，λ2N,gives(25)hence we obtain(26)From (24)we have(27)Again utilizing (13) and (21)， we find(28)which together with (27) leads to(29)Similarly， by use of (14)， (21) and (27)， we obtain(30)thus(31)(32)hence let μk(x，tm)，vk(x，tm) be distinct solutions of the ordinary differential Eqs.(29) and(30)，then (q，r) determined by (22) is a solution of Eq.(6)with n=m=2 or a solution of Eq.(7) with n=m=3.Based on the form of the function det (W)in Eq.(24)， we introduce the hyperelliptic Riemann surfacewith genus g=N.For the same ， there are two points and on differentsheets of Γ.Since R(λ) is a polynomial of order 2N+2 in terms of λ， there are two infinite points ∞1 and ∞2 which are not branch points of Γ.On Γ we fix a set of regular cycle paths：a1，a2，...，aN；b1，b2，...，bN，which are independent and have the intersection numbers as followsak∘aj=bk∘bj=0，ak∘bj=δkj，k，j=1，…，N.The holomorphic differentials on Γ are chosen to beLet N×N matrices A=(Akj) and B=(Bkj) are invertible.Define matrices C and τ by C=A-1，τ=A-1B.The matrix τ can be shown to be symmetric and has a positive definite imaginary part.We normalize into the following new basisThen we findʃbkωj=τjk.For a fixed point p0, the Abel-Jacobi coordinate are given as follows(33)(34)By using (33) and the first expression of (29), we havewhich givesby use of the following equalityIn a similar way, we get from(29)-(34)Based on the above results we have the followingρ1=Ω1x+Ωmtm+γ1,ρ2=-Ω1x-Ωmtm+γ2,whereWe define an Abel map on Γ as followsA(p)=ω,ω=(ω1，…,ωN)T,A(∑nkpk)=∑nkA(pk).Consider two special divisors m (m=1,2); then we getWe define the Riemann theta function of Γ as(πiτz,z+2πiζ,z),ζ∈CN,in which ζ=(ζ,…,ζN)T,ζ,z terms of the Riemann theorem in algebraic geometry, there exist two constant vectors M1,M2∈CNsuch thatF1=θ(A(p)-ρ1-M1)has exactly N zeros at λ=μ1,...,μN ; andF2=θ(A(p)-ρ2-M2)has exactly N zeros at λ=ν1,...,νN.In order to make these functions single valued, the surface Γ is cut along all ak,bk to form a simply connected region, whose boundary is denoted by γ.Notice the fact that the integralsare constants independent of ρ1 and ρ2withApplying the residue theorem, we have(35)(36)In order to compute the residues in (35) and (36), we first introduce local coordinates z=λ-1at infinity.Then the hyperelliptic curve ξ2=R(λ) in the neighborhood of infinity can be expressed as with is easy to see thatSince the Riemann theta function is an even function，Fm(λ) can be written as(37)where Dj signifies its derivative with respect to the j th argument of is easy to compute that(38)Substituting Eq.(38) into Eq.(37), we arrive atwhich leads toHence we have(39)whereand πs and ηs are constants.From Eqs.(35), (36) and (39), we get(40)Substituting Eq.(40) into Eqs.(22), we finally obtain the following algebraic-geometric solutions of Eq.(6) with n=m=2 or of Eq.(7) with n=m=3,rwhere q0(tm)and r0(tm)are two arbitrary complex functions about variable tm .3 ConclusionsWe obtained a nonlinear NLS-MKdV hierarchy and its Hamiltonian structure by use of the Tu scheme, furthermore, for the convenience of obtaining algebro-geometric solutions, we transform the3×3matrix isospectral problem into an equivalent2×2one, then the algebro-geometric solutions of the coupled NLS equation and coupled MKdV equation are constructed in terms of Riemann theta functions easily.References【相关文献】[1] Gardner C S, Greene J M, Kruskal M D, et al.Method for solving the Korteweg-de Vries equation[J].Phys Rev Lett, 1967,19: 1095-1097.[2] Li Y S.Soliton and Integrable Systems,Advanced Series in NonlinearScience[M].Shanghai: Shanghai Scientific and Technological Education Publishing House, 1999.[3] Hirota R, Satsuma J.A variety of nonlinear network equations generated from theBäcklund transformation f or the Tota lattice[J].Suppl Prog Theor Phys,1976, 59: 64-100.[4] Liu H Z.Exact periodic wave solutions for the hKdV equation[J].Nonlinear Anal, 2009, 70: 2379-2381.[5] Olver P J.Applications of Lie Groups to Differential Equations, in: Grauate Texts in Mathematics[M].New York: Springer, 1993.[6] Bluman G W, Kumei S.Symmetries and Differential Equations[M].Berlin:Springer-Verlag World Publishing Gorp, 1989.[7] Cantwell B J.Introduction to Symmetry Analysis[M].Cambridge:Cambridge University Press, 2002.[8] Liu H Z.Lie symmetry analysis and exact explicit solutions for general Burgers’ equation[J].J Comput Appl Math,2008,228: 1-9.[9] Liu Y, Liu X Q.Symmetry reduction, exact solutions and conservation laws of the(1+1)dimension Caudrey-Dodd-Gibbon equation[J].Journal of Liaocheng University (Natural Science Edition)，2014，27(01)：8-12+24.[10] Li Z Q，Li u H Z.The application of (G′/G)-expansion method for fifth-order dispersive equation [J].Journal of Liaocheng University(Natural Science Edition)，2016，29(04)：1-3+7.[11] Li Y, Li L Q, Li H H，et al.Exact solutions and conservation laws of the(3+1)dimensional Zakharov-Kuznetsov-Burgers equation [J].Journal of Liaocheng University(Natural Science Edition)，2017，30(01)：10-17.[12] Xin X P.Non-local symmetries and exact solutions of nonlinear development equations[J].Journal of Liaocheng University(Natural Science Edition),2018,31(1):15-20.[13] Gesztesy F,Holden H.Soliton Equations and their Algebro-GeometricSolutions[M].Cambridge: Cambridge University Press, 2003.[14] Hou Y,Fan E G.Algebro-geometric solutions for the Gerdjikov-Ivanov hierarchy[J].J Math Phys, 2013, 54: 073505-073530.[15] Hou Y, Fan E G, Zhao P.The algebro-geometric solutions for Hunter-Saxton hierarchy[J].Z Angew Math Phys,2013，65(3)：487-520.[16] Hou Y, Zhao P, Fan E G,et al.Algebro-geometric solutions for the Degasperis-Procesi hierarchy[J].SIAM J Math Anal, 2013, 45: 1216-1266.[17] Zhao P, Fan E G,Hou Y.Algebro-Geometric solutions for the Ruijsenaars-Toda hierarchy[J].Chaos Solitons Fractals, 2013, 54: 8-25.[18] Tu G Z.The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems[J].J Math Phys, 1989, 30: 330-338.[19] Zhang Y F,Hon Y C.Some evolution hierarchies derived from self-dual Yang-Mills equations[J].Commun Theor Phys, 2011, 56: 856-872.[20] Zhang Y F,Feng B L.A few Lie algebras and their applications for generating integrable hierarchies of evolution types[J].Commun Nonlinear Sci Numer Simulat,2011, 16(8): 3045-3061.[21] Zhang Y F, Tam H H.An integrable hierarchy and Darboux transformations, bilinear Bäcklund transformations of a reduced equation[J].Applied Mathematics and Computation, 2013, 219:5837-5848.[22] Wang X Z, Dong H H, Li Y X.Some reductions from a lax integrable system and their hamiltonian structures[J]. Applied Mathematics and Computation,2012, 218:10032-10039.[23] Yang H X, Du J, Xu X X,et al.Hamiltonian and Super-hamiltonian systems of a hierarchy of soliton equations[J]. Applied Mathematics and Computation, 2010, 217: 1497-1508.。

浅水波方程的暗孤子解

浅水波方程的暗孤子解欧阳正勇;郑珊【摘要】This paper studied the soliton solutions of a nonlinear shallow water wave equation. By using the qualitative theorem of differential equations we prove the existence of dark soliton solutions and discussed some of their qualitative characteristics. The dark soliton solutions are symmetric on both sides of the crest and the amplitude increases with the increase of wave speed. Dark soliton solutions of different speeds intersect each other in two symmetrical spots and decay exponentially to zero in infinity.%研究了一个非线性浅水波动方程的孤立子解，运用微分方程定性理论，证明了向左迁移的暗孤子解的存在性，并分析了暗孤子解的一些定性特征：该解具有对称性，其振幅随着波速的增大而增加，不同波速的暗孤子解必相交于对称的两点，在无穷远处呈指数衰减到零。

【期刊名称】《厦门理工学院学报》【年(卷),期】2015(000)005【总页数】5页(P89-93)【关键词】浅水波方程;微分定性理论;暗孤子【作者】欧阳正勇;郑珊【作者单位】佛山科学技术学院理学院，广东佛山528000;广州航海学院基础部，广东广州520725【正文语种】中文【中图分类】O193考虑浅水波动方程［1－2］其中：u表示波动函数；ε和δ表示与振幅、水深和波长相关的参数,当参数δ≪1时,对应的水波称为浅水波,当参数ε满足ε＝O(δ)时,对应的水波称为小振幅水波,例如Camassa－Holm(CH)方程［3］和Degraspris－Process(DP)方程［4］都是描述小振幅的浅水波.这类小振幅的浅水波方程具有很好的结构,具有孤立子和尖孤立子解［5－12］,特别是具有爆破解［13］,此外还存在整体弱解［14］.解的稳定性问题如光滑孤立波、孤立尖波和多孤子解的轨道稳定性也得到证明［15－20］.如上述介绍,小振幅浅水波的许多结果已经通过CH方程和DP方程及其广义形式等获得.然而对振幅略大的浅水波研究较少,振幅略大的浅水波也是客观存在的一类波动现象,因此研究振幅稍大的浅水波是有意义和有必要的.本文中所说的振幅略大的浅水波动方程(1)是指参数ε满足条件ε＝O(δ),即其振幅略有增大.文献［2］研究了当波速c＞1时,方程(1)存在孤立子解,本文对方程(1)(见图1)进行了进一步的研究,证明了当波速c＜－1时,方程(1)存在暗孤立子解,并给出了暗孤子解的一些定性特征.作变换［2］方程(1)可转换成如下常微分方程注波速c大于0时表示右行波,波速c小于0时表示左行波,本文研究的孤立子解满足条件φ, φ(n)→0(ξ→∞),其中n∈N.对方程(3)两边关于ξ积分,得其中C为积分常数.因为本文中考虑的暗孤子解满足φ,φ′→0(ξ→∞),故方程(4)中的积分常数C为0.将方程(4)转变成对应的平面系统如下：系统(5)的首次积分为其中h为积分常数.定理1当波速c＜－1时,方程(1)存在整体暗孤子解.证明当波速c＜－1时,系统(5)有两个平衡点P1(0,0)和P2(φc,0),其中φc是多项式唯一的负实根.事实上P(φ)关于φ是单调递减的,且P(φ)→＋∞(φ→－∞),P(φ)→－∞(φ→＋∞),故P(φ)有唯一的零点.又因为当c＜－1时,所以φc＜0.下面计算系统(5)的线性化系统在平衡点P1(0,0)和P2(φc,0)处的特征值并判断奇点的类型,为了方便计算,对系统(5)作变换,设dξ＝(1＋c＋7φ)dτ,系统(5)可转换为除了奇直线φ＝－(1＋c)/7,系统(5)和(9)的分支相图是一样的,因而可用系统(9)代替(5)进行计算.设奇点的特征值为λ,则λ满足其中当c＜－1时,平衡点P1(0,0)有一正一负两个不同的实特征值,因而是鞍点.在平衡点P2(φc,0)处的特征值为在平衡点P1(0,0)处的特征值为其中f′(φ)＝12(c－1)－36φc＋18φc2－18φc3,因为φc＜0,所以f′(φc)＞0,又c＜－1,从而λ3,4为一对纯虚根,故P2(φc,0)为中心.系统(5)的同宿轨对应着方程(1)的孤立子解,为了证明方程(1)的暗孤子解的存在性,需要在φ－y平面上找出一条从鞍点P1(0,0)向φ轴负方向出发,环绕中心P2(φc,0)之后再回到P1(0,0)的同宿轨.在φ－y平面的下半平面,因为φ′＝y＜0,φ随着ξ的增大而减小,所以存在一条轨线从P1(0,0)向左出发,且必定穿过直线φ＝φc.事实上, 其中MP是φP(φ)的最大值,由式(13)可知y′有下界,故轨线必穿过直线φ＝φc.当轨线穿过φ＝φc之后 (即φ＜φc),有φ′＝y＜0,y′＞0,轨线接着向左上方走,且穿过φ轴,不然,假设y单增趋于某常数(y′→0),那么φ→－∞,由式(5)可推出y′→∞,矛盾.再利用系统(5)关于φ轴的对称性,即系统(5)在变换ξ→－ξ下是不变的,在上半平面,轨线按相同的方式回到P1(0,0).这样就证明了存在一条同宿轨,而该轨线就对应着方程(1)的一个暗孤子解.虽然不能直接求出暗孤子解的表达式,但仍可以应用微分方程定性理论分析暗孤子解的一些特征,然后画出暗孤子解的平面图.本文讨论的暗孤子解具有如下定性特征：定理2方程(1)的暗孤子解具有对称性,其波峰随着波速的增大而增大,且在无穷远处呈指数衰减到0.证明由于本文所求的暗孤子解满足φ′,φ→0(ξ→∞),因而式(6)中的积分常数h＝0,上节中的同宿轨满足如下等式即其中易判断Q(φ)关于φ是单减的三次多项式,因而Q(φ)有唯一的实零点φQ,φ′在点φQ处等于零,从而该孤立子解存在一个波峰,且φQ满足如下方程其中当波速越大时,波峰将会越高.事实上,由于φQ是Q(φ)唯一的实根,当时,Q(φ,c2)－Q(φ,c1)＝12(c2－c1)＜0,Q(φ)的图像向左平移,当增大时,也增大.暗孤子解关于纵轴两边是对称的,即φ(ξ)关于ξ是偶函数.事实上,由式(15)可将φ′看成关于φ的函数,那么对波上的每一个高度,都对应着两个绝度值相同的斜率,但符号相反,因此在纵轴两侧,波的陡峭程度保持一致,从而说明φ(ξ)是关于ξ的偶函数.当时,φ→0,由式(15)进一步可知,由式(17)可近似解得这表明暗孤子解在无穷远处按指数衰减到0,也能反映出对应的同宿轨从鞍点P1(0,0)发出时与纵轴的夹角.定理3设φ(ξ,c1)和φ(ξ,c2)表示在不同波速c1,c2下的两个暗孤子解,那么这两个解必相交于两点.证明本小节考虑不同波速下波形的比较,设φ(ξ)是方程(1)的暗孤子解,且在ξ＝0处达到波峰.由定理2知,波峰的高度是波速的函数,定义如下函数因φ(ξ)是暗孤子解,且波高随着增大而增大,所以在ξ＝0处有f(0)＝∂cφ(0,c)＞0.从而当时,即式(20)中c*∈(c2,c1).当时,所以足够大时满足f(ξ)＜0,且当时,即式(22)中c**∈(c2,c1).结合暗孤子解得对称性,由上所述可知,不同波速的暗孤子解的平面图必交于两点 (图3).本文从方程本身的结构出发,结合微分方程定性理论,分析并证明了在波速c＜－1时暗孤子解的存在性,给出了暗孤子解的定性特征,拓展了文献中的相关结论.当波速－1≤c≤1的时候,孤立子解的存在性有待进一步研究.【相关文献】［1］CONSTANTIN A，LANNES D.The hydrodynamical relevance of the Camassa－Holm and Degasperis-Processi equations［J］.Arch Ration Mech Anal，2009，192：165-186. ［2］GEYER A.Solitary traveling water waves of moderate amplitude［J］.J Nonlinear Math Phys，2012，19：104-115.［3］CAMASSA R，HOLM D.An integrable shallow water equation with peaked solitons ［J］.Phys Rev Lett，1993，71：1 661-1 664.［4］DEGASPERIS A，PROCESI M.Asymptotic integrability［C］//Symmetry and Perturbation Theory.London：World Scientific Publishing Company，1999：23-37. ［5］CONSTANTIN A.On the scattering problem for the Camassa-Holm equation ［J］.Math Phys Eng Sci，2001，457：953-970.［6］CONSTANTIN A，GERDJIKOV V S，IVANOV R I.Inverse scattering transform for the Camassa-Holm equation［J］. Inverse Problems，2006，22：2 197-2 207.［7］BOUTET DE MONVEL A，KOSTENKO A，SHEPELSKY D，et al.Long-time asymptotics for the Camassa-Holm equation［J］.SIAM J Math Anal，2009，41：1 559-1 588.［8］EL DIKA K，MOLINET L.Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation［J］.Math Phys Eng Sci，2007，365：2 313-2 331.［9］LIU Z R，QIAN T F.Peakons of the Camassa-Holm equation［J］.Appl Math Model，2002，26（3）：473-480.［10］ZHANG W L.General expressions of peaked traveling wave solutions of CH-gamma and CH equations［J］.Sci China Ser A Math，2004，47（6）：862-873.［11］BRESSAN A，CONSTANTIN A.Global conservative solutions of the Camassa Holm equation［J］.Arch Ration Mech Anal，2007，183：215-239.［12］CONSTANTIN A，ESCHER J.Analyticity of periodic traveling free surface water waves with vorticity［J］.Ann of Math，2011，173：559-568.［13］CONSTANTIN A，ESCHER J.Wave breaking for nonlinear nonlocal shallow water equations［J］.Acta Math，1998，181：229-243.［14］CONSTANTIN A，MOLINET L.Global weak solutions for a shallow water equation ［J］.Commun Math Phys，2000，211：45-61.［15］CONSTANTIN A，STRAUSS W.Stability of peakons［J］.Comm Pure Appl Math，2000，53：603-610.［16］CONSTANTIN A，STRAUSS W.Stability of the Camassa-Holm solitons［J］.J Nonlinear Sci，2002，12：415-422.［17］OUYANG Z Y，ZHEN S，LIU Z R.Orbital stability of peakons with nonvanishing boundary for CH and CH-gamma equations［J］.Phys Lett A，2008，372：7 046-7 050. ［18］BENJAMIN T B.The stability of solitary waves［J］.Proc R Soc Lond，1972，328：153-183.［19］EL DIKA K，MOLINET L.Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation［J］.Math Phys Eng Sci，2007，365：2 313-2 331.［20］EL DIKA K，MOLINET L.Stability of multipeakons［J］.Ann I H Poincar AN，2009，26：1 517-1 532.。

Nonlinear Systems and Dynamics

Nonlinear Systems and Dynamics Nonlinear systems and dynamics are an essential aspect of modern science and engineering. These systems are characterized by their complex behavior, which cannot be described by simple linear equations. Nonlinear systems are ubiquitous in nature, from the behavior of living organisms to the dynamics of the universe. Understanding these systems is crucial for developing new technologies, predicting and controlling complex phenomena, and advancing scientific knowledge.One of the most important features of nonlinear systems is their sensitivity to initial conditions. Small changes in the initial conditions of a nonlinear system can lead to significant differences in the system's behavior over time. This phenomenon, known as the butterfly effect, is a fundamental aspect of chaos theory. Chaos theory has broad applications in various fields, including meteorology, economics, and biology. The butterfly effect can be seen in many natural phenomena, such as weather patterns, the growth of populations, and the behavior of ecosystems.Nonlinear systems are also characterized by the presence of feedback loops. Feedback loops are essential for maintaining stability in complex systems. They allow the system to adjust its behavior in response to changes in its environment. Feedback loops can be positive or negative, depending on whether they amplify or dampen the system's response to external stimuli. Positive feedback loops can lead to runaway behavior, while negative feedback loops can stabilize the system.Another important aspect of nonlinear systems is their ability to exhibit oscillatory behavior. Oscillations are a fundamental feature of many natural phenomena, from the beating of a heart to the oscillation of a pendulum. Nonlinear systems can exhibit a wide range of oscillatory behavior, from simple periodic oscillations to chaotic oscillations. Understanding the dynamics of oscillatory systems is crucial for developing new technologies, such as electronic circuits, and for predicting and controlling complex phenomena, such as the spread of diseases.Nonlinear systems also play a crucial role in the study of complex networks. Complex networks are ubiquitous in nature and society, from the networks of neurons in the brain to the networks of social interactions between individuals.Understanding the dynamics of complex networks is crucial for developing new technologies, such as the internet, and for predicting and controlling complex phenomena, such as the spread of information or the emergence of new social behaviors. Nonlinear dynamics plays a central role in the study of complex networks, as it allows us to understand how the behavior of individual nodes inthe network affects the behavior of the network as a whole.In conclusion, nonlinear systems and dynamics are a crucial aspect of modern science and engineering. These systems are characterized by their complex behavior, sensitivity to initial conditions, presence of feedback loops, ability to exhibit oscillatory behavior, and importance in the study of complex networks. Understanding the dynamics of nonlinear systems is crucial for developing new technologies, predicting and controlling complex phenomena, and advancingscientific knowledge. Nonlinear dynamics is a rapidly growing field with broad applications in various fields, including physics, biology, engineering, andsocial science. As such, it is an essential area of study for anyone interested in understanding the world around us.。

非线性波方程的奇异曲线和奇异行波

II
Abstract
The results show that the equation exist a new double compacton solutions. The new compacton solutions are different from the famous Rosenau-Hyman compacton solutions, because they are derived by a singular elliptic curve tangenting to the homoclinic orbits rather than the singular straight line in the phase space. In Chapter 6, we study the relasionship of the parabola singular curves and the singular traveling wave. The deformed Hunter-Zheng equation is specifically studied and corresponding travelling wave system possing the parabola singular curves is well derived. This is the first time that this type of equations has been found. Futher studying the periodic curves tangent to the parabola singular curves, we obtain the new singular periodic wave and analyze its dynamical behaviors. In Chapter 7, we summarize the work of the paper and put forward the research work in the future. Keywords: peakon solutions; compacton solutions; cuspon solutions; pseudo-cuspon solutions；periodic wave solutions

PKP-方程的精确周期孤子解和双周期解

PKP-方程的精确周期孤子解和双周期解李自田【摘要】应用同宿测试方法研究并获得了PKP-方程的新的精确周期孤子解和双周期解,同时得出了该方程在点p2=4处具有衰减性.从平衡点的左侧到右侧,方程的解从周期孤子解衰变为双周期解.【期刊名称】《山西大学学报（自然科学版）》【年(卷),期】2010(033)002【总页数】3页(P166-168)【关键词】周期孤子解;双周期;同宿测试法;衰减【作者】李自田【作者单位】曲靖师范学院,数学与信息科学学院,云南,曲靖,655011【正文语种】中文【中图分类】O175.23在过去的二十年里,在非线性发展方程广泛出现的应用领域引起了数学和物理工作者的普遍关注,许多学者在这一领域进行了卓有成效的研究.特别是在精确解的寻求和获得方面开辟和发展了许多方法.诸如, F-扩展法[1];齐次平衡法[2]以及逆散射法[3]等.在本文中,我们将研究如下形式的PKP-方程:其中u:Rx×Ry×Rt→R.并且:取自“+”和“-”被分别称为PKP-I方程和PKP-II方程.众所周知,该模型属于潘勒卫不可积类型.但通过应用潘勒卫扩展变换,我们可将该方程转换为双线性方程,进而通过对双线性方程的研究,可找出并获得该方程的解.最近以来,该方程在诸多方面获得了较广泛的研究[3-6].许多学者在这些方面取得了很大的进展.文献[5]通过应用F-扩展函数的方法研究并获得了用椭圆函数表示的一系列周期波解;在文献[6]中,文章的作者给出了该系统的N-孤子解,并得出了该系统可简化为Melnikov-方程和KP-方程的特殊类型的结论.本文通过对双线形方程的研究,应用文献[7]发展起来的方法,即同宿测试法,获得了该方程的新的周期解和双孤子解,其中的一些方法的应用和结论在解决其他同类型的问题中将具有十分深远的意义.首先,我们考虑PKP-I方程:引入变换:将(3)代入方程(2),则方程(2)可化为:随后,我们采用下面的变换:将变换(5)代入方程(4),则我们得到如下形式的双线性方程:这里,算子“D”定义为:引入测试函数:其中b1,b2,Ω,τ,p是实数.将(7)式代入方程(6),通过计算,我们得到如下的关系式:从而,将(8)代入(7)并代入(5),我们得到方程的周期孤子解:显然,我们要求条件:以便使式(8)中的Ω2>0,从而确保Ω能取到实数.把ζ=x+t代入(9)中,并令b2=1.从而,我们得到下列形式的周期孤子解:考察如下形式的PKP-II方程:应用和上面使用的相同的变换以及处理PKP-I方程所用的类似的方法,我们有双线性方程:设:将(13)代入方程(12)并应用符号计算系统,我们获得了方程(11)的精确解:其中系数满足:同理,要求条件:p2>4,从而使得(15)中Ω的满足Ω2>0.类似地,我们取b2=1.则PKP-II方程的精确解具有如下表达式:考虑变换:(ζ,y)→(ζ,iy)将它代入(9)并令τ=0.我们得到了一个新解,它是一个双周期解:比较方程(2)和方程(11),我们不难发现,只要我们应用时间和空间的变换(ζ,y)→(ζ,iy),方程(2)可以转换为方程(11),反之亦然.这样,我们获得了PKP-I方程的双周期解: 其中:p2-4>0.注意到,(18)是PKP-I方程的奇性周期解.为了避免奇性,我们令cos(Ωy)>0和cosp(x+t)>0.此外,同理可得PKP-II方程的双周期解:其中,要求条件:p2-4<0.依据讨论,我们得出结论:p2=4是PKP-I方程和PKP-II方程的唯一周期分歧点.在p2=4的两侧, PKP-I方程和PKP-II方程的解的性质发生了改变.当平衡点p2从4的一侧变到另一侧,周期孤子解衰变为双周期解.【相关文献】[1] ZHANG Hui-qun.New Exact Travelling Wave Solutions for Some Nonlinear Evolution Equations,Part II[J].Chaos, Solitons and Fractals,2008,37:1328-1334.[2] ZHOU Yu-bin,WNAG Ming-liang,MIAO Tian-de.The Periodic Wave Solutions and Solitary Wave Solutions for a Class of Nonlinear Partial Differential Equations[J].Phys Lett A,2004,323(1-2):77-88.[3] ABLOWITZ M J,CLARKSON P A.Solitons,Nonlinear Evolution Equations and Inverse Scattering Transform[M]. Cambridge University Press,1990:8-17[4] CARIELLO F,TABOR M.Painleve Expansions for Non-integrable Evolution Equations[J].Physica D:Nonlinear Phenomena,1989,39(1):77-94.[5] AKHMEDIEV N,ANKIEWICZ A.Solitons,Nonlinear Pulses and Beams[M].Chapman and Hall,London,1997:124-127.[6] ZHOU Yu-bin,WANG Ming-liang.Periodic Wave Solutions to a Coupled Kdv Equations with Variable Coefficients[J]. Phys Lett A,2003,308(1):31-36.[7] DAI Zheng-de,LI Shao-ling,ZHU Ai-jun.Singular Periodic Soliton Solutions and Resonance for the Kadomtsev-Petviashvili Equation[J].Chaos,Solitons andFractals,2007,34(4):1148.。

附子的强心作用及其机理研究进展

★于武华钟凌云（江西中医药大学［摘要］附子是回阳救逆的第一要药。

现代临床运用中，学参数和减轻心衰发生时心肌细胞的损伤，其作用途径较广，调节神经内分泌因子、增强心肌细胞抗氧化能力、对附子的强心作用及其机理研究进行了系统综述，［关键词］附子；配伍运用；强心；作用机理中图分类号：R282.71 文献标识码：A通信作者：钟凌云，博士，教授，博士生导师。

E-mail ：*****************。

附子为毛茛科植物乌头子根的加工品。

该药始载于《神农本草经》，味辛、甘，性大热，有毒，归心、脾、肾经，被誉为“乱世之良将，回阳救逆之第一品，补命门真火第一要药”。

现代学者认为，附子的强心作用是其中医临床疗效（回阳救逆、补火助阳）的主要药理学作用。

附子含有多种强心成分，如去甲乌药碱、撑棍碱、去甲猪毛菜碱等［1-3］。

关于附子强心作用的机理研究可进一步揭示附子的作用基础，提高附子的临床运用。

笔者检索了国内外有关附子强心作用、炮制研究及其机理研究的文献，对附子的强心作用做出系统的综述，希望为日后附子的研究提供一定帮助。

1 附子的强心作用附子的强心作用主要表现在增强心率、升高心室内压变化速率、增强心室收缩压和舒张压、改善血流动力学，从而达到治疗心衰的目的。

研究发现，生附子和附子炮制品或附子配伍使用均具有强心作用。

1.1 生附子的强心作用实验证实，附子对蛙、兔、蟾蜍等动物具有一定的强心作用，尤其在心功能不全时效果更为显著［4］。

附子能降低戊巴比妥钠致大鼠心衰细胞及缺氧/复氧乳鼠心肌细胞死亡率［5-6］，对大鼠离体心脏的心率、左心室收缩压、舒张末压、左心室内压变化速率均有显著性提高［7］。

附子能降对急性心衰大鼠心率、左心室内压变化速率等血流动力学参数均有显著性增强，且增强程度与用药后时间有关［11-12］。

附子经不同生姜炮制后毒性降低，对急性心衰大鼠血流动力学参数及心脏功能的改善作用并未减弱［13］。

1.3 附子配伍的强心作用附子与人参不同比例配伍均能改善阿霉素致心衰大鼠的血流动力学参数，改善心肌收缩和心肌舒张功能［14］。

强非线性论文阅读笔记

文献阅读笔记1、杨世平，吴晓。

强非线性振动系统的渐近解。

佳木斯工学院学报，1998，16（3）：303~307 摘要：对强非线性振动系统进行参数变换，把强非线性振动来统转化为弱非线性振动系统，利用参数待定法即可方便求出强非线性振动来统的渐近解。

在现行的机械振动专著中，基本上都讨论弱非线性振动系统，而对强非线性振动系统很少讨论研究.众所周知，摄动展开法是研究弱非线性振动系统的一个强有力教学工具.而对工程实际中存在的大量非线性振动系统，摄动展开法却难以见效.本文对强非线性振动系统进行参数变换，把强非线性振动系统转化为弱非线性振动系统，利用傅立叶级数展开成一个小参数的幕级数，采用系数待定法即可方便求出强非线性振动系统的渐近解.本文方法不但求解过程简单，精度高（比KBM 法的计算精度高），而且回避了解微分方程和依靠消除永年项建立补充方程的复杂过程。

parameter undetermined method ：参数待定法parameter transformation ：参数变换 strongly nonlinear vibration system ：强非线性振动系统2、李银山，郝黎明，树学锋。

强非线性Duffing 方程的摄动解。

太原理工大学学报，2000，31（5）：516~520摘要：用参数展开摄动法和改进的L-P 方法求解强非线性Duffing 方程。

与寻常的摄动法相比,具有较高的精度。

G ．Duffing 方程是具有非线性恢复力的强迫振动系统，在工程各领域中具有广泛的代表性，从它被提出（1918年）到现在，已经90多年了，但对它的解的性质并未完全清楚。

对于形如()()2,01x x f x x ωεε+=＜的弱非线性自治系统，以及形如()()2,,01x x f x x t ωεε+=Ω＜的弱非线性非自治系统，目前已经有多种有效的近似解法，如Lindstedt —Poincare(L--P)法、平均法、时间变换法、KBM 法和多尺度法等。

On the asymptotic expansion of the solutions of the separated nonlinear Schroedinger equati

a r X i v :n l i n /0012025v 3 [n l i n .S I ] 10 M a y 2001On the Asymptotic Expansion of the Solutions of the Separated Nonlinear Schr¨o dinger EquationA.A.Kapaev,St Petersburg Department of Steklov Mathematical Institute,Fontanka 27,St Petersburg 191011,Russia,V.E.Korepin,C.N.Yang Institute for Theoretical Physics,State University of New York at Stony Brook,Stony Brook,NY 11794-3840,USAAbstractNonlinear Schr¨o dinger equation with the Schwarzian initial data is important in nonlinear optics,Bose condensation and in the theory of strongly correlated electrons.The asymptotic solutions in the region x/t =O (1),t →∞,can be represented as a double series in t −1and ln t .Our current purpose is the description of the asymptotics of the coeﬃcients of the series.MSC 35A20,35C20,35G20Keywords:integrable PDE,long time asymptotics,asymptotic expansion1IntroductionA coupled nonlinear dispersive partial diﬀerential equation in (1+1)dimension for the functions g +and g −,−i∂t g +=12∂2x g −+4g 2−g +,(1)called the separated Nonlinear Schr¨o dinger equation (sNLS),contains the con-ventional NLS equation in both the focusing and defocusing forms as g +=¯g −or g +=−¯g −,respectively.For certain physical applications,e.g.in nonlin-ear optics,Bose condensation,theory of strongly correlated electrons,see [1]–[9],the detailed information on the long time asymptotics of solutions with initial conditions rapidly decaying as x →±∞is quite useful for qualitative explanation of the experimental phenomena.Our interest to the long time asymptotics for the sNLS equation is inspired by its application to the Hubbard model for one-dimensional gas of strongly correlated electrons.The model explains a remarkable eﬀect of charge and spin separation,discovered experimentally by C.Kim,Z.-X.M.Shen,N.Motoyama,H.Eisaki,hida,T.Tohyama and S.Maekawa [19].Theoretical justiﬁcation1of the charge and spin separation include the study of temperature dependent correlation functions in the Hubbard model.In the papers[1]–[3],it was proven that time and temperature dependent correlations in Hubbard model can be described by the sNLS equation(1).For the systems completely integrable in the sense of the Lax representa-tion[10,11],the necessary asymptotic information can be extracted from the Riemann-Hilbert problem analysis[12].Often,the fact of integrability implies the existence of a long time expansion of the generic solution in a formal series, the successive terms of which satisfy some recurrence relation,and the leading order coeﬃcients can be expressed in terms of the spectral data for the associ-ated linear system.For equation(1),the Lax pair was discovered in[13],while the formulation of the Riemann-Hilbert problem can be found in[8].As t→∞for x/t bounded,system(1)admits the formal solution given byg+=e i x22+iν)ln4t u0+∞ n=12n k=0(ln4t)k2t −(1t nv nk ,(2)where the quantitiesν,u0,v0,u nk and v nk are some functions ofλ0=−x/2t.For the NLS equation(g+=±¯g−),the asymptotic expansion was suggested by M.Ablowitz and H.Segur[6].For the defocusing NLS(g+=−¯g−),the existence of the asymptotic series(2)is proven by P.Deift and X.Zhou[9] using the Riemann-Hilbert problem analysis,and there is no principal obstacle to extend their approach for the case of the separated NLS equation.Thus we refer to(2)as the Ablowitz-Segur-Deift-Zhou expansion.Expressions for the leading coeﬃcients for the asymptotic expansion of the conventional NLS equation in terms of the spectral data were found by S.Manakov,V.Zakharov, H.Segur and M.Ablowitz,see[14]–[16].The general sNLS case was studied by A.Its,A.Izergin,V.Korepin and G.Varzugin[17],who have expressed the leading order coeﬃcients u0,v0andν=−u0v0in(2)in terms of the spectral data.The generic solution of the focusing NLS equation contains solitons and radiation.The interaction of the single soliton with the radiation was described by Segur[18].It can be shown that,for the generic Schwarzian initial data and generic bounded ratio x/t,|c−xthese coeﬃcients as well as for u n,2n−1,v n,2n−1,weﬁnd simple exact formulaeu n,2n=u0i n(ν′)2n8n n!,(3)and(20)below.We describe coeﬃcients at other powers of ln t using the gener-ating functions which can be reduced to a system of polynomials satisfying the recursion relations,see(24),(23).As a by-product,we modify the Ablowitz-Segur-Deift-Zhou expansion(2),g+=exp i x22+iν)ln4t+i(ν′)2ln24t2] k=0(ln4t)k2t −(18t∞n=02n−[n+1t n˜v n,k.(4)2Recurrence relations and generating functions Substituting(2)into(1),and equating coeﬃcients of t−1,weﬁndν=−u0v0.(5) In the order t−n,n≥2,equating coeﬃcients of ln j4t,0≤j≤2n,we obtain the recursion−i(j+1)u n,j+1+inu n,j=νu n,j−iν′′8u n−1,j−2−−iν′8u′′n−1,j+nl,k,m=0l+k+m=nα=0, (2)β=0, (2)γ=0, (2)α+β+γ=ju l,αu k,βv m,γ,(6) i(j+1)v n,j+1−inv n,j=νv n,j+iν′′8v n−1,j−2++iν′8v′′n−1,j+nl,k,m=0l+k+m=nα=0, (2)β=0, (2)γ=0, (2)α+β+γ=ju l,αv k,βv m,γ,(7)where the prime means diﬀerentiation with respect toλ0=−x/(2t).Master generating functions F(z,ζ),G(z,ζ)for the coeﬃcients u n,k,v n,k are deﬁned by the formal seriesF(z,ζ)= n,k u n,k z nζk,G(z,ζ)= n,k v n,k z nζk,(8)3where the coeﬃcients u n,k,v n,k vanish for n<0,k<0and k>2n.It is straightforward to check that the master generating functions satisfy the nonstationary separated Nonlinear Schr¨o dinger equation in(1+2)dimensions,−iFζ+izF z= ν−iν′′8zζ2 F−iν′8zF′′+F2G,iGζ−izG z= ν+iν′′8zζ2 G+iν′8zG′′+F G2.(9) We also consider the sectional generating functions f j(z),g j(z),j≥0,f j(z)=∞n=0u n,2n−j z n,g j(z)=∞n=0v n,2n−j z n.(10)Note,f j(z)≡g j(z)≡0for j<0because u n,k=v n,k=0for k>2n.The master generating functions F,G and the sectional generating functions f j,g j are related by the equationsF(zζ−2,ζ)=∞j=0ζ−j f j(z),G(zζ−2,ζ)=∞j=0ζ−j g j(z).(11)Using(11)in(9)and equating coeﬃcients ofζ−j,we obtain the diﬀerential system for the sectional generating functions f j(z),g j(z),−2iz∂z f j−1+i(j−1)f j−1+iz∂z f j==νf j−z iν′′8f j−ziν′8f′′j−2+jk,l,m=0k+l+m=jf k f lg m,2iz∂z g j−1−i(j−1)g j−1−iz∂z g j=(12)=νg j+z iν′′8g j+ziν′8g′′j−2+jk,l,m=0k+l+m=jf kg l g m.Thus,the generating functions f0(z),g0(z)for u n,2n,v n,2n solve the systemiz∂z f0=νf0−z (ν′)28g0+f0g20.(13)The system implies that the product f0(z)g0(z)≡const.Since f0(0)=u0and g0(0)=v0,we obtain the identityf0g0(z)=−ν.(14) Using(14)in(13),we easilyﬁndf0(z)=u0e i(ν′)28n n!z n,4g0(z)=v0e−i(ν′)28n n!z n,(15)which yield the explicit expressions(3)for the coeﬃcients u n,2n,v n,2n.Generating functions f1(z),g1(z)for u n,2n−1,v n,2n−1,satisfy the diﬀerential system−2iz∂z f0+iz∂z f1=νf1−z iν′′8f1−ziν′8g0−z(ν′)24g′0+f1g20+2f0g0g1.(16)We will show that the diﬀerential system(16)for f1(z)and g1(z)is solvable in terms of elementary functions.First,let us introduce the auxiliary functionsp1(z)=f1(z)g0(z).These functions satisfy the non-homogeneous system of linear ODEs∂z p1=iν4−ν′′4f′0z(p1+q1)−i(ν′)28−ν′g0,(17)so that∂z(q1+p1)=−(ν2)′′8z,p1(z)= −iνν′′8−ν′u′032z2,g1(z)=q1(z)g0(z),g0(z)=v0e−i(ν′)24−ν′′4v0 z+i(ν′)2ν′′4−ν′′4u0 ,v1,1=v0 iνν′′8−ν′v′0u n,2n −1=−2u 0i n −1(ν′)2(n −1)n −1ν′′u 0,n ≥2,v n,2n −1=−2v 0(−i )n −1(ν′)2(n −1)n −1ν′′v 0,n ≥2.Generating functions f j (z ),g j (z )for u n,2n −j ,v n,2n −j ,j ≥2,satisfy the diﬀerential system (12).Similarly to the case j =1above,let us introduce the auxiliary functions p j and q j ,p j =f jg 0.(21)In the terms of these functions,the system (12)reads,∂z p j =iνz(p j +q j )+b j ,(22)wherea j =2∂z p j −1+i (ν′)28−j −14(p j −1f 0)′8f 0+iν4−ν′′zq j −1−−ν′g 0+i(q j −2g 0)′′zj −1 k,l,m =0k +l +m =jp k q l q m .(23)With the initial condition p j (0)=q j (0)=0,the system is easily integrated and uniquely determines the functions p j (z ),q j (z ),p j (z )= z 0a j (ζ)dζ+iνzdζζζdξ(a j (ξ)+b j (ξ)).(24)These equations with expressions (23)together establish the recursion relationfor the functions p j (z ),q j (z ).In terms of p j (z )and q j (z ),expansion (2)readsg +=ei x22+iν)ln 4t +i(ν′)2ln 24tt2t−(18tv 0∞ j =0q j ln 24tln j 4t.(25)6Let a j (z )and b j (z )be polynomials of degree M with the zero z =0of multiplicity m ,a j (z )=M k =ma jk z k,b j (z )=Mk =mb jk z k .Then the functions p j (z )and q j (z )(24)arepolynomials of degree M +1witha zero at z =0of multiplicity m +1,p j (z )=M +1k =m +11k(a j,k −1+b j,k −1)z k ,q j (z )=M +1k =m +11k(a j,k −1+b j,k −1) z k.(26)On the other hand,a j (z )and b j (z )are described in (23)as the actions of the diﬀerential operators applied to the functions p j ′,q j ′with j ′<j .Because p 0(z )=q 0(z )≡1and p 1(z ),q 1(z )are polynomials of the second degree and a single zero at z =0,cf.(19),it easy to check that a 2(z )and b 2(z )are non-homogeneous polynomials of the third degree such thata 2,3=−(ν′)4(ν′′)2210(2+iν),(27)a 2,0=−iνν′′8−ν′u ′08u 0,b 2,0=iνν′′8−ν′v ′08v 0.Thus p 2(z )and q 2(z )are polynomials of the fourth degree with a single zero at z =0.Some of their coeﬃcients arep 2,4=q 2,4=−(ν′)4(ν′′)24−(1+2iν)ν′′8u 0−ν(u ′0)24−(1−2iν)ν′′8v 0−ν(v ′0)22.Proof .The assertion holds true for j =0,1,2.Let it be correct for ∀j <j ′.Then a j ′(z )and b j ′(z )are deﬁned as the sum of polynomials.The maximal de-grees of such polynomials are deg (p j ′−1f 0)′/f 0 =2j ′−1,deg (q j ′−1g 0)′/g 0 =72j′−1,anddeg 1z j′−1 α,β,γ=0α+β+γ=j′pαqβqγ =2j′−1. Thus deg a j′(z)=deg b j′(z)≤2j′−1,and deg p j′(z)=deg q j′(z)≤2j′.Multiplicity of the zero at z=0of a j′(z)and b j′(z)is no less than the min-imal multiplicity of the summed polynomials in(23),but the minor coeﬃcients of the polynomials2∂z p j′−1and−(j−1)p j′−1/z,as well as of2∂z q j′−1and −(j−1)q j′−1/z may cancel each other.Let j′=2k be even.Thenm j′=min m j′−1;m j′−2+1;minα,β,γ=0,...,j′−1α+β+γ=j′mα+mβ+mγ =j′2 . Let j′=2k−1be odd.Then2m j′−1−(j′−1)=0,andm j′=min m j′−1+1;m j′−2+1;minα,β,γ=0,...,j′−1α+β+γ=j′mα+mβ+mγ =j′+12]p j,k z k,q j(z)=2jk=[j+12]z nn−[j+18k k!,g j(z)=v0∞n=[j+12]k=max{0;n−2j}q j,n−k(−i)k(ν′)2k2]k=max{0;n−2j}p j,n−ki k(ν′)2k2]k=max{0;n−2j}q j,n−k(−i)k(ν′)2kIn particular,the leading asymptotic term of these coeﬃcients as n→∞and j ﬁxed is given byu n,2n−j=u0p j,2j i n−2j(ν′)2(n−2j)n) ,v n,2n−j=v0q j,2j (−i)n−2j(ν′)2(n−2j)n) .(32)Thus we have reduced the problem of the evaluation of the asymptotics of the coeﬃcients u n,2n−j v n,2n−j for large n to the computation of the leading coeﬃcients of the polynomials p j(z),q j(z).In fact,using(24)or(26)and(23), it can be shown that the coeﬃcients p j,2j,q j,2j satisfy the recurrence relationsp j,2j=−i (ν′)2ν′′2jj−1k,l,m=0k+l+m=jp k,2k p l,2l q m,2m++ν(ν′)2ν′′4j2j−1k,l,m=0k+l+m=jp k,2k(p l,2l−q l,2l)q m,2m,q j,2j=i (ν′)2ν′′2jj−1k,l,m=0k+l+m=jp k,2k q l,2l q m,2m−(33)−ν(ν′)2ν′′4j2j−1k,l,m=0k+l+m=jp k,2k(p l,2l−q l,2l)q m,2m.Similarly,the coeﬃcients u n,0,v n,0for the non-logarithmic terms appears from(31)for j=2n,and are given simply byu n,0=u0p2n,n,v n,0=v0q2n,n.(34) Thus the problem of evaluation of the asymptotics of the coeﬃcients u n,0,v n,0 for n large is equivalent to computation of the asymptotics of the minor coeﬃ-cients in the polynomials p j(z),q j(z).However,the last problem does not allow a straightforward solution because,according to(8),the sectional generating functions for the coeﬃcients u n,0,v n,0are given byF(z,0)=∞n=0u n,0z n,G(z,0)=∞n=0v n,0z n,and solve the separated Nonlinear Schr¨o dinger equation−iFζ+izF z=νF+18zG′′+F G2.(35)93DiscussionOur consideration based on the use of generating functions of diﬀerent types reveals the asymptotic behavior of the coeﬃcients u n,2n−j,v n,2n−j as n→∞and jﬁxed for the long time asymptotic expansion(2)of the generic solution of the sNLS equation(1).The leading order dependence of these coeﬃcients on n is described by the ratio a n2+d).The investigation of theRiemann-Hilbert problem for the sNLS equation yielding this estimate will be published elsewhere.Acknowledgments.We are grateful to the support of NSF Grant PHY-9988566.We also express our gratitude to P.Deift,A.Its and X.Zhou for discussions.A.K.was partially supported by the Russian Foundation for Basic Research under grant99-01-00687.He is also grateful to the staﬀof C.N.Yang Institute for Theoretical Physics of the State University of New York at Stony Brook for hospitality during his visit when this work was done. 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分数阶非线性迭代方程的周期解

第59卷第3期吉林大学学报(理学版)V o l .59 N o .32021年5月J o u r n a l o f J i l i nU n i v e r s i t y (S c i e n c eE d i t i o n )M a y2021研究简报d o i :10.13413/j .c n k i .jd x b l x b .2020423分数阶非线性迭代方程的周期解李翠英1,吴睿2,程毅1(1.渤海大学数学科学学院,辽宁锦州121013;2.长春财经学院数学教研部,长春130122)摘要:考虑一类C a p u t o 型分数阶导数意义下非线性迭代微分方程的周期问题,在非线性项满足单边L i p s c h t i z 条件下,应用L e r a y -S c h a u d e r 不动点定理和拓扑度理论,证明该类非线性分数阶迭代微分方程解的存在性和唯一性.关键词:存在性;唯一性;分数阶;迭代方程中图分类号:O 175.14 文献标志码:A 文章编号:1671-5489(2021)03-0555-04P e r i o d i c S o l u t i o n s f o r aF r a c t i o n a lN o n l i n e a r I t e r a t i v eE qu a t i o n s L IC u i y i n g 1,WU Ru i 2,C H E N G Y i 1(1.C o l l e g e o f M a t h e m a t i c a lS c i e n c e s ,B o h a i U n i v e r s i t y ,J i n z h o u 121013,L i a o n i n g Pr o v i n c e ,C h i n a ;2.D e p a r t m e n t o f M a t h e m a t i c s ,C h a n g c h u nU n i v e r s i t y o f F i n a n c e a n dE c o n o m i c s ,C h a n gc h u n 130122,C h i n a )A b s t r a c t :W e c o n s ide r e d t h e p e r i o d i c p r o b l e m of ac l a s so fn o n l i n e a r i t e r a t i v ed i f f e r e n t i a l e qu a t i o n i n t h es e n s eo fC a p u t ot y p ef r a c t i o n a ld e r i v a t i v e .U n d e ro n es i d e d -L i p s c h t i zc o n d i t i o n so n n o n l i n e a r t e r m ,t h ee x i s t e n c ea n d u n i qu e n e s s o fs o l u t i o nf o rt h e n o n l i n e a rf r a c t i o n a li t e r a t i v e d i f f e r e n t i a l e q u a t i o n s i s p r o v e d b y a p p l y i n g t h eL e r a y -S c h a u d e r f i x e d p o i n t t h e o r e ma n d t o p o l o g i c a l d e g r e e t h e o r y .K e y w o r d s :e x i s t e n c e ;u n i q u e n e s s ;f r a c t i o n a l o r d e r ;i t e r a t i v e e q u a t i o n 收稿日期:2020-12-18. 网络首发日期:2021-04-06.第一作者简介:李翠英(1983 ),女,汉族,硕士,讲师,从事分数阶微分动力系统的研究,E -m a i l :l n c y999@126.c o m.通信作者简介:吴睿(1978 ),女,汉族,博士,教授,从事微分方程理论的研究,E -m a i l :w u r u i 0221@s i n a .c o m.基金项目:吉林省自然科学基金(批准号:20200201274J C ).网络首发地址:h t t ps ://k n s .c n k i .n e t /k c m s /d e t a i l /22.1340.O.20210406.0958.001.h t m l .0 引言目前,关于迭代微分方程的边值㊁周期问题研究已得到广泛关注.P e t u k h o v [1]研究了如下二阶迭代微分方程周期边值问题:x ᵡ=λx (x (t)),x (0)=x (T )=α,得到了参数λ,α在不同范围内方程解的存在唯一性;K a u f m a n n [2]考虑一类二阶迭代微分方程的边值问题,用S c h a u d e r 不动点定理证明了该问题解的存在性;文献[3-6]用不同的不动点理论(K r a s n o s e l s k i i 不动点定理㊁S c h a u d e r 不动点定理等)证明了若干类迭代微分方程周期解或拟周期解的存在唯一性.分数阶微分方程在数学㊁化学㊁物理和工程等许多领域应用广泛,但关于分数阶迭代微分方程边值㊁周期问题的研究成果目前报道较少.当非线性函数满足L i ps c h i t z 条件时,I b r a h i m 等[7]将整数阶的一些结果推广到分数阶迭代微分方程中.但目前已有结果均为处理一维的迭代微分系统,对于向量迭代方程的研究尚未见文献报道.本文在非线性函数满足单边L i p s c h i t z 条件时,证明一类非线性C a pu t o 型分数阶迭代向量微分系统周期解的存在唯一性.本文令Tʒ=[0,b],ℝn为n维E u c l i d空间,<㊃,㊃>表示ℝn中的内积, ㊃表示ℝn空间的范数.设C(T,ℝn)表示从T到ℝn全体连续函数组成的空间,其范数定义为 x C=m a x tɪT x(t) .关于分数阶微积分的基础知识可参见文献[8-9].1主要结果考虑如下分数阶迭代向量微分方程:C Dαx(t)+A x(t)=f(t,x(t),x[2]+(t)), x(0)=x(b),(1)其中:x[2]+(t)=(x1( x ),x2( x ), ,x n( x ));C Dαx(t)=1Γ(1-α)ʏt0(t-s)-αxᶄ(s)d s, ∀αɪ(0,1);A:ℝnңℝn是一个线性算子;f:Tˑℝnˑℝnңℝn是一个连续函数.下面给出假设条件:(H1)设A:ℝnңℝn是一个有界㊁线性的正定算子,即对任意的zɪℝn,存在常数cɪℝ+,使得<A z,z>ȡc z 2;(H2)设f:Tˑℝnˑℝnңℝn是一个连续函数,且:(i)对任意的u,vɪℝn,存在一个非负函数λɪLɕ+[0,b],使得∀tɪ[0,b], f(t,u,v) ɤλ(t);(i i)对任意的tɪ[0,b],u1,u2,v1,v2ɪℝn存在函数μɪLɕ+[0,b],使得<f(t,u1,v1)-f(t,u2,v2),u1-u2>ɤμ(t) u1-u2 2,其中 μ ɕ<c,c是H1中的常数.定理1假设条件(H1),(H2)成立,且b大于某常数M1/(1-α),则分数阶迭代微分系统(1)存在唯一解.证明:由文献[10]中推论7.1知,问题(1)等价于如下积分迭代方程:x(t)=Eα(A tα)x(0)+ʏt0(t-τ)α-1Eα,α(A(t-τ)α)f(τ,x(τ),x[2]+(τ))dτ.(2)定义算子T1:C(T,ℝn)ңC(T,ℝn),且T1(x(t))ʒ=Eα(A tα)x(0)+ʏt0(t-τ)α-1Eα,α(A(t-τ)α)f(τ,x,x[2]+)dτ.首先,证明解的先验有界性.根据算子T1的定义和假设条件(H2)中(i),可推出T1(x(t)) ɤ x(0) Eα(A Tα) +m a x tɪT Eα,α(A tα)Γ(α)ʏt0(t-s)α-1f(s,x(s),x[2](s))d sɤx(0) Mα+ λ ɕ^MαΓ(α)ʏt0(t-s)α-1d sɤx(0) Mα+ λ ɕ^MααΓ(α)bα,(3)其中Mα= Eα(A Tα) ,^Mα=m a x tɪT Eα,α(A tα) .下面估计初值 x(0) .在式(2)中令t=b,可得x(b)=Eα(A bα)x(0)+ʏb0(b-τ)α-1Eα,α(A(b-τ)α)f(τ,x(τ),x[2]+(τ))dτ.由x(0)=x(b)和假设条件(H1)易知,行列式E-Eα(A bα)ʂ0,其中E表示单位矩阵.故x(0)=(E-Eα(A bα))-1b0(b-τ)α-1Eα,α(A(b-τ)α)f(τ,x(τ),x[2]+(τ))dτ.655吉林大学学报(理学版)第59卷根据假设条件(H 2)中(i ),类似式(3),可得 x (0) ɤM E ^M α λ ɕb ααΓ(α),(4)其中M E = (E -E α(A b α))-1 .将式(4)代入式(3),可得 T 1(x (t )) ɤ(M E M α+1)^M α λ ɕαΓ(α)b α, ∀t ɪT .记M =(M E M α+1)^M α λ ɕαΓ(α),由于b >M 1/(1-α),因此 T 1(x (t )) ɤM b α<b .其次,证明非线性算子T 1是全连续算子,从而得到解的存在性.先证明∀x ɪC (T ,ℝn ),T 1(x (t ))ɪC (T ,ℝn ).对任意的t ,t +δɪ[0,b ],且δ>0,满足T 1(x (t +δ))-T 1(x (t ))=1Γ(α)ʏt +δ0(t +δ-s )α-1E α,α(A (t +δ-τ)α)f (s ,x (s ),x [2](s ))d s -1Γ(α)ʏt(t -s )α-1E α,α(A (t -τ)α)f (s ,x (s ),x [2](s ))d s +[E α(A (t +δ)α)-E α(A t α)]x (0)ɤ λ ɕ^M αΓ(α)ʏt +δ0(t +δ-s )α-1d s +ʏt 0(t +δ-s )α-1-(t -s )α-1d s +[E α(A (t +δ)α)-E α(A t α)]x (0)ɤ2 λ ɕ^M ααΓ(α)δα+2 λ ɕ^M ααΓ(α)(t +δ-a )α-(t -a )α+[E α(A (t +δ)α)-E α(A t α)]x (0).当δң0时,有T 1(x (t +δ))-T 1(x (t ))ң0,故T 1(x (t ))ɪC (T ,ℝn ).取x n ңx ɪC (T ,ℝn ),则易推出T 1(x n )-T 1(x )ң0,从而T 1:C (T ,ℝn )ңC (T ,ℝn )是连续的.根据1)中先验估计,应用A r z e l a -A s c o l i 定理易知,算子T 1:ΩңΩ是全连续的,其中Ωʒ={u ɪC (T ,ℝn ): u C ɤb +1}.从而可将微分迭代系统(1)解的存在性转化为T 1的不动点问题.定义映射h ε(x )=x -εT 1(x ),其中εɪ[0,1].取p ∉h (∂Ω),则对任意的εɪ[0,1],可得d e g (h ε,Ω,p )=d e g (h 1,Ω,p )=d e g (I -T 1,Ω,p )=d e g (h 0,Ω,p )=d e g (I ,Ω,p )=1ʂ0,其中I 是恒等映射.因此T 1在Ω上存在不动点,即x =T 1(x ).从而微分迭代系统(1)至少存在一个解.最后,证明微分迭代系统(1)解的唯一性.假设x 1,x 2ɪC (T ,ℝn )是问题(1)的两个解,对这两个解做差再与x 1-x 2做内积,可得<x 1(t )-x 2(t ),D α(x 1(t )-x 2(t ))>+<x 1(t )-x 2(t ),A (x 1(t )-x 2(t ))>=<x 1(t )-x 2(t ),f (t ,x 1(t ),x [2]1+(t ))-f (t ,x 2(t ),x [2]2+(t))>.根据假设条件(H 1)和(H 2)中(i i),利用分数阶微分不等式[10],可推出D α x 1(t )-x 2(t ) 2ɤ2<x 1(t )-x 2(t ),D α(x 1(t )-x 2(t ))>ɤ2μ(t ) x 1-x 2 2-2c x 1-x 2 2.(5)为方便,令S (t )= x 1(t )-x 2(t ) 2,式(5)可简化为D αS (t )ɤ2(μ(t )-c )S (t ),755 第3期李翠英,等:分数阶非线性迭代方程的周期解855吉林大学学报(理学版)第59卷从而S(t)ɤS(0)Eα(2( μ ɕ-c)tα), ∀tɪT.再令t=b,得S(b)ɤS(0)Eα(2( μ ɕ-c)bα).(6)由S(t)= x1-x2 2及边界条件x(b)=x(0)可知,S(b)=S(0),整理可得S(0){1-Eα[2( μ ɕ-c)bα]}ɤ0.由M i t t a g-L e f f l e r函数Eα(t)(tȡ0)的单调性和 μ ɕ<c知,Eα[2( μ ɕ-c)bα]<1.又由S(0)= x1(0)-x2(0) 2ȡ0,可推出S(0)=0.由式(6)知,S(t)ɤ0,从而S(t)恒为0,即x1=x2,故迭代微分方程(1)有唯一解.参考文献[1] P E T U K HO V V R.O n a B o u n d a r y V a l u e P r o b l e m[J].T r u d y S e m T e o r D i f f e r e n c i a l U r a v n e n iǐO t k l o nA r g u m e n t o m U n i vD r užb y N a r o d o vP a t r i s aL u m u m b y,1965,3:252-255.[2] K A U F MA N N ER.E x i s t e n c e a n dU n i q u e n e s s o f S o l u t i o n s f o r aS e c o n d-O r d e r I t e r a t i v eB o u n d a r y-V a l u eP r o b l e m[J].E l e c t r o n i c JD i f f e r e n t i a l E q u a t i o n s,2018,2018(150):1-6.[3] Z HA O H Y,L I UJ.P e r i o d i cS o l u t i o n s o f a n I t e r a t i v eF u n c t i o n a lD i f f e r e n t i a l E q u a t i o nw i t hV a r i a b l eC o e f f i c i e n t s[J].M a t h M e t hA p p l S c i,2016,40(1):286-292.[4] B O U A K K A Z A,A R D J O U N I A,D J O U D I A.P e r i o d i c S o l u t i o n sf o ra S e c o n d O r d e r N o n l i n e a r F u n c t i o n a lD i f f e r e n t i a lE q u a t i o nw i t h I t e r a t i v eT e r m sb y S c h a u d e r sF i x e dP o i n tT h e o r e m[J].A c t a M a t hU n i vC o m e n i a n,2018,87(2):223-235.[5] L I UB W,T U NÇC.P s e u d oA l m o s tP e r i o d i cS o l u t i o n s f o r aC l a s s o fF i r s tO r d e rD i f f e r e n t i a l I t e r a t i v eE q u a t i o n s[J].A p p lM a t hL e t t,2015,40:29-34.[6] F E㊅C K A N M,WA N GJR,Z HA O H Y.M a x i m a l a n d M i n i m a lN o n d e c r e a s i n g B o u n d e dS o l u t i o n so f I t e r a t i v eF u n c t i o n a lD i f f e r e n t i a l E q u a t i o n s[J].A p p lM a t hL e t t,2021,113:106886-1-106886-7.[7]I B R A H I M R W,K I L IÇMA N A,D AMA G F H.E x i s t e n c ea n d U n i q u e n e s sf o raC l a s so f I t e r a t i v eF r a c t i o n a lD i f f e r e n t i a lE q u a t i o n s[J/O L].A d vD i f f e rE q u,2015-03-05[2017-11-10].h t t p s://d o i.o r g/10.1186/s13662-015-0421-y.[8] M I L L E R KS,R O S SB.A nI n t r o d u c t i o nt ot h eF r a c t i o n a lC a l c u l u sa n dF r a c t i o n a lD i f f e r e n t i a lE q u a t i o n s[M].N e w Y o r k:W i l e y,1993:80-121.[9] P O D L U B N Y I.F r a c t i o n a l D i f f e r e n t i a l E q u a t i o n s:A n I n t r o d u c t i o n t o F r a c t i o n a l D e r i v a t i v e s,F r a c t i o n a lD i f f e r e n t i a lE q u a t i o n s,t o M e t h o d s o fT h e i r S o l u t i o na n dS o m e o fT h e i rA p p l i c a t i o n s[M].S a nD i e g o:A c a d e m i cP r e s s,1998:41-119.[10] C H E NJJ,Z E N G Z G,J I A N G P.G l o b a l M i t t a g-L e f f l e rS t a b i l i t y a n d S y n c h r o n i z a t i o n o f M e m r i s t o r-B a s e dF r a c t i o n a l-O r d e rN e u r a lN e t w o r k s[J].N e u r a lN e t w,2014,51:1-8.(责任编辑:李琦)。

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into Eq.(2.1) and integrating once again, we then get the following bilinear form
2 4 G(Dx , Dt )f · f = (Dx Dt + 6u0 Dx + Dx + c)f · f = 0
(2.3)
where c = c(t) is a integration constant and u0 is a constant solution of KdV equation. The Hirota bilinear diﬀerential operator is deﬁned by
We consider Riemann theta function solution of KdV equation eπi<τ n,n>+2πi<ξ,n>
n∈Z N
f=
(2.5)
where n = (n1 , · · · , nN ), ξ = (ξ1 , · · · , ξN ), τ is a symmetric matrix and Imτ > 0, ξj = kj x + ωj t, j = 1, · · · , N. Consider the case when N = 1, then (2.5) becomes
∞
f=
n=−∞
e2πinξ+πin
2τ
(2.6)
4
Substitutiபைடு நூலகம்g (2.6) into (2.3) gives
∞ ∞
Gf · f = G(Dx , Dt )
n=−∞ ∞ ∞
e
2πinξ +πin2 τ
·
m=−∞
e2πimξ+πim
2τ
2τ
=
n=−∞ m=−∞ ∞ ∞
G(Dx , Dt )e2πinξ+πin τ · e2πimξ+πim
Constructing periodic wave solutions of nonlinear equations by Hirota bilinear method
Huihui Daia , Engui Fanb , Xianguo Gengc
arXiv:nlin/0602015v1 [nlin.SI] 8 Feb 2006
m n ξ1 Dx Dt e · eξ2 = (k1 − k2 )m (ω1 − ω2 )n eξ1 +ξ2
where ξj = kj x + ωj t, j = 1, 2. More general, we have G(Dx , Dt )eξ1 · eξ2 = G(k1 − k2 , ω1 − ω2 )eξ1 +ξ2 (2.4)
a. Department of Mathematics, City University of Hong Kong, Hong Kong SAR, China b. Institute of Mathematics, Fudan University, Shanghai 200433, P. R. China. c. Department of Mathematics, Zhengzhou University, Zhengzhou, P. R. China
1
2
applied in the integrable nonlinear evolution equations admitting Lax pairs representation and involves complicated algebraic geometry theory. These have been used far less than their soliton counterparts. The main reason for this is that they are far more complicated. Soliton solutions are typically expressed in terms of rational or hyperbolic functions, whereas qusi-periodic solutions require the use of Riemann theta functions and calculus on Riemann surfaces. Recently, Deconinck, Hoeij et al proposed an algorithm to compute the Riemann theta function and Riemann constants [25,26]. It is well known that the bilinear derivative method developed by Hirota is a powerful and direct approach to construct exact solution of nonlinear equations [79]. Once a nonlinear equation is written in bilinear forms by a dependent variable transformation, then multi-soliton solutions and rational solutions can be obtained. In recent years, Hirota method also has been developed for obtaining Wronskian and Pfaﬃan forms of N-soliton solution [10,11]. In this paper, we propose Hirota method for directly constructing periodic wave solutions in Riemann theta functions. It is shown that the periodic solutions can be reduced to classical soliton solutions under a certain limit. The appeal and success of this method lies in the fact we circumvent complicated algebro-geometric theory to directly get explicit periodic wave solutions. Moreover, all parameters appearing in the solutions are free variables, whereas usual quasi-periodic solutions involve Riemann constants which are diﬃcult to be determined and need to make complicated Abel transformation on Riemann surface [13-18]. As illustrative examples, we then consider KdV equation and KP equation in this paper. Other many equations also can be dealt with this way. 2. Periodic wave solutions of KdV equation and their reduction Consider KdV equation ut + 6uux + uxxx = 0. Substituting transformation u = u0 + 2(ln f )xx (2.2) (2.1)
m n Dx Dt f (x, t) · g (x, t) = (∂x − ∂x′ )m (∂t − ∂t′ )n f (x, t)g (x, t)|x=x′,t=t′
The D-operator have good property when acting on exponential functions
Abstract. Hirota bilinear method is proposed to directly construct periodic wave solutions in terms of Riemann theta functions for nonlinear equations. The asymptotic property of periodic waves are analyzed in detail. It is shown that well-known soliton solutions can be reduced from the periodic wave solutions. 1. Introduction The investigation of the exact solutions of nonlinear equations plays an important role in the study of nonlinear physical phenomena. For example, the wave phenomena observed in ﬂuid dynamics, plasma and elastic media are often modelled by the bell shaped sech solutions and the kink shaped tanh traveling wave solutions. The exact solution, if available, of those nonlinear equations facilitates the veriﬁcation of numerical solvers and aids in the stability analysis of solutions. In the past decades, there has been signiﬁcant progression in the development of these methods such as inverse scattering method [1,2], Darboux transformation [3-6], Hirota bilinear method [7-12], algebro-geometric method [13-18] and others. Among them, the algebro-geometric method is an analogue of inverse scattering transformation, which was ﬁrst developed by Matveev, Its, Novikov, and Dubrovin et al. The method can derive an important class of exact solutions, which is called quasi-periodic or algebro-geometric solution, to many soliton equations such as KdV equation, sin-Gordon equation, and Schrodinger equation. In recent years, such solutions of nonlinear equations have been aroused much interest in the mathematical physics [19-24]. However, this method usually is