EXACT TRAVELING WAVE SOLUTIONS OF MODIFIED ZAKHAROV EQUATIONS FOR PLASMAS WITH A QUANTUM CORREC

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一个无穷维动力系统的行波解

NONLINEARITY doi:10.1088/0951-7715/24/10/012
Travelling wave solutions for an infection-age structured epidemic model with external supplies
Arnaud Ducrot and Pierre Magal
2 ∂t S(t, x) = dS ∂x S(t, x) +
− γ S(t, x),
for t > 0,
x ∈ R,
(1.1)
where the parameter > 0 represents the entering ﬂux of susceptible individuals, while γ > 0 is the rate at which individuals die or leave the population. If we assume that the process of contamination occurs by mass action law, then the model we will consider in this work reads as
ቤተ መጻሕፍቲ ባይዱ
where the function β(a) denotes the infection age-speciﬁc contamination rate and the function µ(a) represents an additional mortality (or exit) rate due to the disease. Here the diffusion coefﬁcients dS > 0 and dI > 0 describe the spatial motility of each class. Note that the diffusion coefﬁcients may be different from susceptible to infected depending on the disease. Throughout this work, we will make the following assumption. Assumption 1.1. We assume that dS , dI , and γ are positive constants, and the maps a → β(a) and a → µ(a) are almost everywhere bounded, and, respectively, belong to L∞ (0, ∞) and L∞ loc ([0, ∞)). The corresponding spatially homogeneous system reads as

BIFURCATIONS OF THE TRAVELLING WAVE SOLUTIONS TO A GENERALIZED CAMASSA-HOLM EQUATION

An n. of D iﬀ. Eqs . 24:3(2008) , 317-325
BIFURCATIONS OF THE TRAVELLING WAVE SOLUTIONS TO A GENERALIZED CAMASSA-HOLM EQUATION ∗ †
Rui Weiguo, Long Yao, He Bin
1
Introduction
In th is pap e r, we c onside r a ge ne raliz ed Camassa-H olm e quation (see [1,2,6,7]) u t − u x xt + ωu x + 3 uu x = γ (2 u x u xx + uu xx x ) , (1 . 1)
( D ep t. o f Ma th ., Ho ngh e U niversit y, Me ngzi 6 61 1 00 , Yun nan )
A bs tra ct
In t his pa per, w e stud y som e g eneralized C a ma ssa -H olm equa tion . T hroug h the a nalysis of t he ph ase-po rtrait s, t he exist ence o f solit a ry w ave , cusp wave, pe rio dic w ave , period ic cusp wave and co m pact o ns w ere discu ssed. In so m e c ert ain pa ra me tric co ndition s, m a ny exact so lut ions t o t he a bo ve travelling wa ves w ere g iven. Furth erm ore, the 3D a nd 2 D pictures of t he a bo ve t rave lling wa ve so lut ions a re d ra wn using M aple softw are. Key wo rds t ra velling wa ve so lut io n; Ca m assa-H o lm equa tio n; ela st ic ro d m o del 20 00 M athem atic s Sub jec t C lassiﬁc ation 3 7 K40 ; 37 K5 0

InternationalJournalofNonlinearScience：国际非线性科学杂志

257ISSN: 1749-3889 ( print ) 1749-3897 (online)BimonthlyVol.7 (2009) No.3JuneEngland, UK ***************************.uk International Journal ofN o n l i n e a r S c i e n c e Edited by International Committee for Nonlinear Science, WAUPublished by World Academic Union (World Academic Press)CONTENTS259.New Exact Travelling Wave Solutions for Some Nonlinear Evolution EquationsA. Hendi268.A New Hierarchy of Generalized Fisher Equations and Its Bi- Hamiltonian StructuresLu Sun274.New Exact Solutions of Nonlinear Variants of the RLW, the PHI-four and Boussinesq Equations Based on Modified Extended Direct Algebraic MethodA. A. Soliman, H. A. Abdo283. Monotone Methods in Nonlinear Elliptic Boundary Value ProblemG.A.Afrouzi, Z.Naghizadeh, S.Mahdavi290. Influence of Solvents Polarity on NLO Properties of Fluorone Dye Ahmad Y. Nooraldeen1, M. Palanichant, P. K. Palanisamy301.Projective Synchronization of Chaotic Systems with Different Dimensions via Backstepping DesignXuerong Shi, Zhongxian Wang307.Adaptive Control and Synchronization of a Four-Dimensional Energy Resources System of JiangSu ProvinceLin Jia , Huanchao Tang312.Optimal Control of the Viscous KdV-Burgers Equation Using an Equivalent Index MethodAnna Gao, Chunyu Shen，Xinghua Fan319.Adaptive Control of Generalized Viscous Burgers’ Equation Xiaoyan Deng, Wenxia Chen, Jianmei Zhang327. Wavelet Density Degree of a Class of Wiener Processes Xuewen Xia, Ting Dai332.Niches’ Similarity Degree Based on Type-2 Fuzzy Niches’ Model Jing Hua, Yimin Li340.Full Process Nonlinear Analysis the Fatigue Behavior of the Crane Beam Strengthened with CFRPHuaming Zhu, Peigang Gu, Jinlong Wang, Qiyin Shi345.The Infinite Propagation Speed and the Limit Behavior for the B-family Equation with Dispersive TermXiuming Li353.The Classification of all Single Traveling Wave Solutions to Fornberg-Whitham EquationChunxiang Feng, Changxing Wu360.New Jacobi Elliptic Functions Solutions for the Higher-orderNonlinear Schrodinger EquationBaojian Hong, Dianchen Lu368.A Counterexample on Gap Property of Bi-Lipschitz Constants Ying Xiong, Lifeng Xi371.A Method for Recovering the Shape for Inverse Scattering Problem of Acoustic WavesLihua Cheng, Tieyan Lian, Ping Li379.An Approach of Image Hiding and Encryption Based on a New Hyper-chaotic SystemHongxing Yao, Meng Li258International Journal of Nonlinear Science (IJNS)BibliographicISSN: 1749-3889 (print), 1749-3897 (online), BimonthlyEdited by International Editorial Committee of Nonlinear Science, WAUPublished by World Academic Union (World Academic Press)Publisher Contact:Academic House, 113 Mill LaneWavertree Technology Park, Liverpool L13 4AH, England, UKEmail:******************.uk,*********************************URL: www. World Academic Union .comContribution enquiries and submittingThe paper(s) could be submitted to the managing editor ***************************.uk. Author also can contact our editorial offices by mail or email at addresses below directly.For more detail to submit papers please visit Editorial BoardEditor in Chief: Boling Guo, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China;************.cnCo-Editor in Chief: Lixin Tian, Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu,212013;China;**************.cn,************.cnStanding Members of Editorial Board:Ghasem Alizahdeh Afrouzi, Department ofMathematics, Faculty of Basic Sciences, Mazandaran University,Babolsar,Iran;**************.ir Stephen Anco, Department of Mathematics, Brock University, 500 Glenridge Avenue St. Catharines, ON L2S3A1,Canada;***************Adrian Constantin ,Department of Mathematics, Lund University,22100Lund,****************.seSweden;*************************.se,Ying Fan, Department of Management Science, Institute of Policy and Management, ChineseAcademy of Sciences, Beijing 100080,China,**************.cn.Juergen Garloff, University of Applied Sciences/ HTWG Konstanz, Faculty of Computer Science, Postfach100543, D-78405 Konstanz, Germany;************************Tasawar Hayat, Department of mathematics,Quaid-I-AzamUniversity,Pakistan,*****************Y Jiang, William Lee Innovation Center, University of Manchester, Manchester, M60 1QD UK;*******************Zhujun Jing,Institute of Mathematics, Academy of Mathematics and Systems Sciences, ChineseAcademy of Science, Beijing, 100080,China;******************Yue Liu,Department of Mathematics, University of Texas, Arlington,TX76019,USA;************Zengrong Liu,Department of Mathematics, Shanghai University, Shanghai, 201800,China;******************.cnNorio Okada, Disaster Prevention Research Institute, KyotoUniversity,****************.kyoto-u.ac.jp Jacques Peyriere,Université Paris-Sud, Mathématique, bˆa t. 42591405 ORSAY Cedex , France;************************,****************************.frWeiyi Su, Department of Mathematics, NanjingUniversity, Nanjing，Jiangsu, 210093,China;*************.cnKonstantina Trivisa ,Department of Mathematics, University of Maryland College Park，MD20472-4015，USA;****************.eduYaguang Wang,Department of Mathematics, Shanghai Jiao Tong University, Shanghai， 200240，China;***************.cnAbdul-Majid Wazwaz, 3700 W. 103rd Street Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655 ,USA;**************Yiming Wei, Institute of Policy and management, Chinese Academy of Science, Beijing, 100080,China;*****************Zhiying Wen,Department of Mathematics, Tsinghua University, Beijing, 100084, China;*******************Zhenyuan Xu, Faculty of Science, Southern Yangtze University, Wuxi , Jiangsu 214063 ,China;*********************Huicheng Yi n, Department of Mathematics, Nanjing University, Nanjing， Jiangsu, 210093, China;****************.cnPingwen Zhang, School of Mathematic Sciences, Peking University, Beijing, 100871, China;**************.cnSecretary: Xuedi Wang, Xinghua FanEditorial office:Academic House, 113 Mill Lane Wavertree Technology Park Liverpool L13 4AH, England, UK Email:***************************.uk **************************.uk ************.cn。

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．。

非线性离散薛定谔方程的显式精确解

cosh ( k) ) , r = 1, a0 = a1 = 0,
b1 = ±
( (
co sh ( k) 1 + cosh
- 1) ( k) )
,
c = 4 sin ( p) sinh ( k) / ( 1 + cosh ( k) ) ;
( 5) q = 2 ( 1 + cosh ( k) - 2cos( p) ) /
第 4期朱加民 :非线性离散薛定谔方程的显式精确解
·403·
到了方程的精确解。
1 双曲函数方法简述
为方便对改进的双曲函数方法的论述 ,以 ( 2
+ 1)维非线性差分微分方程为例。设非线性差
分微分方程 (DDE)为
Δ( un + p1 ( x, t) , un + p2 ( x, t) , …,
+ co sh ( k) ) , a0 = 0, a1 = a1 ,
b1 = ±
( co sh ( k) - 1) (1 + cosh ( k) )
r= ±
( a21
-
1
+ cosh ( k) + a21 cosh ( k) ( cosh ( k) - 1
) ,
c = 4 sin ( p) sinh ( k) / ( 1 + cosh ( k) ) 。
( cosh2 ( k) - 1) cosh ( k)
tanh
(
kn
+ 2 sin
( p) sinh co sh ( k)
( k)
t
+ζ)
ei (
pn
+
2
( cosh2 ( k) - co s( p) cosh2 ( k)

Equilibrium and traveling-wave solutions of plane Couette flow

To be submitted to J.Fluid Mech.1Equilibrium and traveling-wave solutions ofplane Couette ﬂowBy J.H A L C R O W,J. F.G I B S O N A N D P.C V I T A N O V I ´C School of Physics,Georgia Institute of Technology,Atlanta,GA 30332,USA (Printed 25August 2008)We survey equilibria and traveling waves of plane Couette ﬂow in small periodic cells at moderate Reynolds number Re ,adding in the process eight new equilibrium and two new traveling-wave solutions to the four solutions previously known.Bifurcations un-der changes of Re and spanwise period are examined.These non-trivial ﬂow-invariant unstable solutions can only be computed numerically,but they are ‘exact’in the sense that they converge to solutions of the Navier-Stokes equations as the numerical resolution increases.We ﬁnd two complementary visualizations particularly insightful.Suitably cho-sen sections of their 3D -physical space velocity ﬁelds are helpful in developing physical intuition about coherent structures observed in moderate Re turbulence.Projections of these solutions and their unstable manifolds from ∞-dimensional state space onto suit-ably chosen 2-or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear ﬂows.1.Introduction In Gibson et al.(2008b )(henceforth referred to as GHC )we have utilized exact equilib-rium solutions of the Navier-Stokes equations for plane Couette ﬂow in order to illustrate a visualization of moderate Re turbulent ﬂows in an inﬁnite-dimensional state space,in terms of dynamically invariant,intrinsic and representation independent coordinate frames.The state space portraiture (ﬁgure 1)oﬀers a visualization of numerical (or ex-perimental)data of transitional turbulence in boundary shear ﬂows,complementary to 3D visualizations of such ﬂows (ﬁgure 2).Side-by-side animations of the two visualiza-tions illustrate their complementary strengths (see Gibson (2008b )online simulations).In these animations,3D spatial visualization of instantaneous velocity ﬁelds helps elu-cidate the physical processes underlying the formation of unstable coherent structures,such as the Self-Sustained Process (SSP)theory of Waleﬀe (1990,1995,1997).Running concurrently,the ∞-dimensional state space representation enables us to track unstable manifolds of equilibria of the ﬂow,the heteroclinic connections between them (Halcrowet al.2008),and gain new insights into the nonlinear state space geometry and dynamics of moderate Re wall-bounded shear ﬂows such as plane Couette ﬂow.Here we continue our investigation of exact equilibrium and traveling-wave solutions of Navier-Stokes equations,this time tracking them adiabatically as functions of Re and periodic cell size [L x ,2,L z ]and,in the process,uncovering new invariant solutions,and determining new relationships between them.The history of experimental and theoretical advances is reviewed in GHC ,Sect.2;here we cite only the work on equilibria and traveling waves directly related to this investigation.Nagata (1990)found the ﬁrst pair of nontrivial equilibria,as well as the ﬁrst traveling wave in plane Couette ﬂow (Nagata 1997).Waleﬀe (1998,2003)computed a r X i v :0808.3375v 1 [p h y s i c s .f l u -d y n ] 25 A u g 20082J.Halcrow,J.F.Gibson,and P.Cvitanovi´cFigure1.A3-dimensional projection of the∞-dimensional state space of plane Couetteﬂow in the periodic cellΩW03at Re=400,showing all equilibria and traveling waves discussed in§4.Equilibria are marked: EQ0,◦EQ1,•EQ2, EQ3, EQ4,♦EQ5, EQ7, EQ9,EQ10, EQ11.Traveling waves trace out closed orbits:the spanwise-traveling TW1(blue loops),streamwise TW2(green lines),and TW3(red lines).In this projection the latter twostreamwise traveling waves appear as line segments.The EQ1→EQheteroclinic connectionsand the S-invariant portion of EQ1and EQ2unstable manifolds are shown with black lines.The cloud of dots are temporally equispaced points on a long transiently turbulent trajectory, indicating the natural measure.The projection is onto the translational basis(3.13)constructed from equilibrium EQ2.Nagata and other equilibria guided by the SSP theory.Other traveling waves were com-puted by Viswanath(2008)and Jim´e nez et al.(2005).Schmiegel(1999)computed and investigated a large number of equilibria.His1999Ph.D.provides a wealth of ideas and information on solutions to plane Couetteﬂow,and in many regards the published lit-erature is still catching up this work.GHC added the dynamically important‘newbie’u NB equilibrium(labeled EQ4in this paper)to the stable.We review plane Couetteﬂow in§2and its symmetries in§3.The main advance reported in this paper is the determination of a number of new moderate-Re plane Couetteﬂow equilibria and traveling waves(§4),as well as explorations of the Re(§5) and spanwise cell aspect dependence(§6)of these solutions.Outstanding challenges are discussed in§7.Detailed numerical results such as stability eigenvalues and symmetries of corresponding eigenfunctions are given in Halcrow(2008),while the complete data sets for the invariant solutions can be downloaded from .2.Plane Couetteﬂow–a reviewPlane Couetteﬂow is comprised of an incompressible viscousﬂuid conﬁned between two inﬁnite parallel plates moving in opposite directions at constant and equal velocities, with no-slip boundary conditions imposed at the walls.The plates move along in the streamwise or x direction,the wall-normal direction is y,and the spanwise direction is z.Theﬂuid velocityﬁeld is u(x)=[u,v,w](x,y,z).We deﬁne the Reynolds number as Re=Uh/ν,where U is half the relative velocity of the plates,h is half the distance between the plates,andνis the kinematic viscosity.After non-dimensionalization,the plates are positioned at y=±1and move with velocities u=±1ˆx,and the Navier-StokesEquilibria and traveling waves of plane Couetteﬂow3Figure2.A snapshot of a typical turbulent state in a large aspect cell[L x,2,L z]=[15,2,15], Re=400.The walls at y=±1move away/towards the viewer at equal and opposite velocities U=±1.The color indicates the streamwise(u,or x direction)velocity of theﬂuid:red shows ﬂuid moving at u=+1,blue,at u=−1.The colormap as a function of u is indicated by the laminar equilibrium inﬁgure4.Arrows indicate in-plane velocity in the respective planes:[v,w] in(y,z)planes,etc.The top half of theﬂuid is cut away to show the[u,w]velocity in the y=0 midplane.See Gibson(2008b)for movies of the time evolution of such states.equations are∂u ∂t +u·∇u=−∇p+1Re∇2u,∇·u=0.(2.1)We seek spatially periodic equilibrium and traveling-wave solutions to(2.1)for the do-mainΩ=[0,L x]×[−1,1]×[0,L z](orΩ=[L x,2,L z]),with periodic boundary conditions in x and z.Equivalently,the periodicity of solutions can be speciﬁed in terms of their fundamental wavenumbersαandγ.A given solution is compatible with a given domain ifα=mL x/2πandγ=nL z/2πfor integer m,n.In this study the spatial mean of the pressure gradient is heldﬁxed at zero.Most of this study is conducted at Re=400in one of the two small aspect ratio cells:ΩW03=[2π/1.14,2,2π/2.5]≈[5.51,2,2.51]≈[190,68,86]wall unitsΩHKW=[2π/1.14,2,2π/1.67]≈[5.51,2,3.76]≈[190,68,128]wall units(2.2) where the wall units are in relation to a mean shear rate of ∂u/∂y =2.9in non-dimensionalized units computed for a large aspect ratio simulation at Re=400.Em-pirically,at this Reynolds number theΩHKWcell sustains turbulence for arbitrarily longtimes(Hamilton et al.1995),whereas theΩW03cell(Waleﬀe2003)exhibits only short-lived transient turbulence(GHC).Unless stated otherwise,all calculations are carriedout for Re=400and theΩW03cell.In the notation of this paper,the Nagata(1990) solutions have wavenumbers(α,γ)=(0.8,1.5)andﬁt in the cell[2π/0.8,2,2π/1.5]≈[7.85,2,4.18].†Schmiegel(1999)’s study of plane Couette solutions and their bifurca-tions was conducted in the cell of sizeΩ=[4π,2,2π]≈[12.57,2,6.28].We were not able to to obtain data for Schmiegel’s solutions,or to continue more than a few of our solutions to his much larger cell size.Although the cell aspect ratios studied in this paper are small,the3D states explored by equilibria and their unstable manifolds explored here are strikingly similar to typical states in larger aspect cells,such asﬁgure2.Kim et al.(1971)observed that stream-†Note also that Reynolds number in Nagata(1990)is based on the full wall separation and the relative wall velocity,making it a factor of four larger than the Reynolds number used in this paper.4J.Halcrow,J.F.Gibson,and P.Cvitanovi´cwise instabilities give rise to pairwise counter-rotating rolls whose spanwise separation is approximately100wall units.These rolls,in turn,generate streamwise streaks of high and low speedﬂuid,by convectingﬂuid alternately away from and towards the walls. The streaks have streamwise instabilities whose length scale is roughly twice the roll separation.These‘coherent structures’are prominent in numerical and experimental observations(seeﬁgure2and Gibson(2008b)animations),and they motivate our inves-tigation of how equilibrium and traveling-wave solutions of Navier-Stokes change with Re and cell size.Fluid states are characterized by their energy E=12 u 2and energy dissipation rateD= ∇×u 2,deﬁned in terms of the inner product and norm(u,v)=1VΩd x u·v, u 2=(u,u).(2.3)The rate of energy input is I=1/(L x L z)dxdz∂u/∂y,where the integral is taken overthe upper and lower walls at y=±1.Normalization of these quantities is set so thatI=D=1for laminarﬂow and˙E=I−D.In some cases it is convenient to considerﬁelds as diﬀerences from the laminarﬂow.We indicate such diﬀerences with hats:ˆu=u−yˆx.3.Symmetries and isotropy subgroupsIn an inﬁnite domain and in the absence of boundary conditions,the Navier-Stokes equations are invariant under any3D translation,3D rotation,and x→−x,u→−u inversion through the origin,which we shall callσxz(Frisch1996).In plane Couette ﬂow,the counter-moving walls restrict the rotation symmetry to rotation byπabout the z-axis,which we shall callσz.Theσxz andσz symmetries generate a discrete dihedral group D1×D1={e,σx,σz,σxz}of order4,whereσz[u,v,w](x,y,z)=[u,v,−w](x,y,−z)σx[u,v,w](x,y,z)=[−u,−v,w](−x,−y,z)(3.1)σxz[u,v,w](x,y,z)=[−u,−v,−w](−x,−y,−z).The walls also restrict the translation symmetry to2D in-plane translations.With pe-riodic boundary conditions,these translations become the SO(2)x×SO(2)z continuous two-parameter group of streamwise-spanwise translationsτ( x, z)[u,v,w](x,y,z)=[u,v,w](x+ x,y,z+ z).(3.2) The equations of plane Couetteﬂow are thus invariant under the group GPCF=O(2)x×O(2)z=D1,x×SO(2)x×D1,z×SO(2)z,where x subscripts indicate streamwise trans-lations and(x,y)→(−x,−y)inversion,and z subscripts indicate spanwise translations and z→−z inversion.The invariance of the equations does not imply invariance of their solutions.In general, a solution u of an invariant equation is only equivariant;that is,the action of the symme-try group G maps u into a set of distinct but equivalent solutions G u.However,solutions of an equation can be invariant under a subgroup of the equations’full symmetry group. Such subgroups are called‘isotropy’or‘stabilizer’subgroups(Marsden&Ratiu1999; Golubitsky&Stewart2002;Gilmore&Letellier2007).For example,a typical turbulent trajectory u(x,t)has no symmetry beyond the identity,so its isotropy group is{e}. At the other extreme is the laminar equilibrium,whose isotropy group is the full planeCouette symmetry group GPCF.In between,the isotropy subgroup of the Nagata equilibria and most of the equilibriaEquilibria and traveling waves of plane Couetteﬂow5 reported is S={e,s1,s2,s3},wheres1[u,v,w](x,y,z)=[u,v,−w](x+L x/2,y,−z)s2[u,v,w](x,y,z)=[−u,−v,w](−x+L x/2,−y,z+L z/2)(3.3)s3[u,v,w](x,y,z)=[−u,−v,−w](−x,−y,−z+L z/2).These particular combinations ofﬂips and shifts match the symmetries of instabilities of streamwise-constant streakyﬂow(Waleﬀe1997,2003)and are well suited to the wavy streamwise streaks observable inﬁgure2.But S is one choice of amongst intermediate isotropy groups,and other subgroups might also play important role in the dynamics.In this section we provide a partial classiﬁcation of the isotropy groups of GPCF,suﬃcient to classify all invariant solutions encountered so far and to guide the search for new solutions with other symmetries.We focus on isotropy groups involving at most half-cell shifts. The main result is that,among these,there are onlyﬁve inequivalent isotropy groups in which we should expect toﬁnd equilibria.3.1.Flips and half-shiftsA few observations will be useful in what follows.First,we note the key role played by the inversion and rotation symmetriesσx andσz(3.1)in the classiﬁcation of solutions and their isotropy groups.The invariance of plane Couetteﬂow under continuous translations allows for traveling-wave solutions,i.e.,solutions that are steady in a frame moving with a constant velocity in[x,z].In state space,such solutions either trace out circles or wind around tori,and are both continuous-translation and time invariant.The sign changes underσx,σz,andσxz,however,imply particular centers of symmetry in x,z, and both x and z,respectively,and thusﬁx the translational phases of aﬁeld having these symmetries.Thus the presence ofσx orσz in an isotropy group prohibits traveling waves in x or z,and the presence ofσxz prohibits any form of traveling wave.Guided by this observation,we will seek equilibria whose isotropy subgroups contain theσxz inversion symmetry.Second,the periodic boundary conditions on the domain impose discrete translation symmetries ofτ(L x,0)andτ(0,L z)on the velocityﬁelds.In addition to this full-period translation symmetry,a solution can also be invariant under a rational translation,such asτ(mL x/n,0)or a continuous translationτ( x,0).Invariance under continuous trans-lation implies theﬁeld is constant along the given spatial variable.Invariance under rational translationτ(mL x/n,0)implies symmetry underτ(mL x/n,0)for m∈[1,n−1] as well,provided that m and n are relatively prime.For this reason the subgroups of the continuous translation SO(2)x consist of the discrete cyclic groups C n,x for n=2,3,4,... together with the trivial subgroup{e}and the full group SO(2)x itself,and similarly for z.For rational shifts x/L x=m/n we simplify the notation a bit by rewriting(3.2)asτm/n x =τ(mL x/n,0),τm/nz=τ(0,mL z/n).(3.4)Since m/n=1/2will loom large in what follows,we omit exponents of1/2:τx=τ1/2x ,τz=τ1/2z,τxz=τxτz.(3.5)Invariance of aﬁeld u under a rational shiftτ(L x/n)implies periodicity on the smaller spatial domain x∈[0,L x/n].For this reason we can exclude from our searches all equi-librium whose isotropy subgroups contain rational translations in favor of equilibria computed on smaller domains.However,as we need to study bifurcations into states with wavelengths longer than the initial state,the linear stability computations need to be carried out in the full[L x,2,L z]cell.For example,if EQ is an equilibrium so-lution in theΩ1/3=[L x/3,2,L z]cell,we refer to the same solution repeated thrice in6J.Halcrow,J.F.Gibson,and P.Cvitanovi´cΩ=[L x,2,L z]as“spanwise-tripled”or3×EQ.Such solution is by construction invariantunder C3,x={e,τ1/3x ,τ2/3x}subgroup.Third,some isotropy groups are equivalent under coordinate transformations.Note that O(2)is not an abelian group,since reﬂectionsσand translationsτalong the same axis do not commute(Harter1993).Insteadστ=τ−1σ.Rewriting this relation as στ2=τ−1στ,we note thatσxτx( x,0)=τ−1( x/2,0)σxτ( x/2,0).(3.6) The right-hand side of(3.6)is a similarity transformation that translates the origin of coordinate system.For x=L x/2we haveτ−1/4 x σxτ1/4x=σxτx,(3.7)and similarly for the spanwise shifts/reﬂections.Thus in classifying subgroups we can trade shift-reﬂectσxτx in favor of reﬂectionσx in a shifted coordinate system.However,if bothσx andσxτx are elements of a subgroup,neither can be eliminated,as the similarity transformation simply interchanges them.Fourth,for x=L x,we haveτ−1x σxτx=σx,so that,in the special case of half-cellshifts,σx andτx commute.The same considerations apply to reﬂections and translations in z,soσz andτz commute as well,and the order-16isotropy subgroupG=D1,x×C2,x×D1,z×C2,z⊂GPCF(3.8) is abelian.3.2.The67-fold pathWe now undertake a partial classiﬁcation of the lattice of isotropy subgroups of plane Couetteﬂow.We focus on isotropy groups involving at most half-cell shifts,with SO(2)x×SO(2)z translations restricted to order4subgroup of spanwise-streamwise translations (3.5)of half the cell length,T=C2,x×C2,z={e,τx,τz,τxz}.(3.9)All such isotropy subgroups of GPCFare contained in the subgroup G(3.8).Let usﬁrst enumerate all subgroups H⊂G.The subgroups can be of order|H|= {1,2,4,8,16}.A subgroup is generated by multiplication of a set of generator elements, with the choice of generator elements unique up to a permutation of subgroup elements.A subgroup of order|H|=2has only one generator,since every group element is its own inverse.There are15non-identity elements to choose from,so there are15subgroups of order2.Subgroups of order4are generated by multiplication of two group elements. There are15choices for theﬁrst and14choices for the second.Equivalent subgroup generators can be picked in3·2diﬀerent ways,so there are(15·14)/(3·2)=35subgroups of order4.Subgroups of order8have three generators.There are15choices for theﬁrst, 14for the second,and12for the third(If the element were the product of theﬁrst two,we would get a subgroup of order4).There are7·6·4ways of picking equivalent generators,so there are(15·14·12)/(7·6·4)=15subgroups of order8.In summary: there is the group itself,of order16,15subgroups of order8,35of order4,15of order 2,and1(the identity)of order1,or67subgroups in all(Halcrow2008).This is whole lot of subgroups to juggle;fortunately,the observations of§3.1reduce this number to5 inequivalent isotropy subgroups that suﬃce to describe all equilibria studied here.The equivalence ofσxτx toσx under coordinate transformation(3.7)(and similarly for z)allows the15order-2groups to be reduced to the8groups generated individually byEquilibria and traveling waves of plane Couetteﬂow7 the8generatorsσx,σz,σxz,σxτz,σzτx,τx,τz,andτxz.Of these,the latter three imply periodicity on smaller domains.Of the remainingﬁve,σx andσxτz support traveling waves in z,σz andσzτx support traveling waves in x.Only a single isotropy subgroup of order2,{e,σxz},(3.10) generated by inversionσxz,supports equilibria.EQ9,EQ10,and EQ11described below are examples of equilibria with isotropy group{e,σxz}.Of the35subgroups of order4we need to identify here only those that containσxz and thus support equilibria.Four isotropy subgroups of order4are generated by picking σxz as theﬁrst generator,andσz,σzτx,σzτz orσzτxz as the second generator(R for reﬂect-rotate):R={e,σx,σz,σxz}={e,σxz}×{e,σz}R x={e,σxτx,σzτx,σxz}={e,σxz}×{e,σxτx}(3.11)R z={e,σxτz,σzτz,σxz}={e,σxz}×{e,σzτz}R xz={e,σxτxz,σzτxz,σxz}={e,σxz}×{e,σzτxz} S.These are the only inequivalent subgroups of order4containingσxz and no isolated trans-lation elements.Together with{e,σxz}these yield the total of5inequivalent isotropy subgroups in which we can expect toﬁnd equilibria.The R xz isotropy subgroup is particularly important,as the Nagata(1990)equilibria are invariant in R xz or an equivalent subgroup(Waleﬀe1997;Clever&Busse1997;Wal-eﬀe2003),as are most of the solutions reported here.The‘NBC’isotropy subgroup of Schmiegel(1999)and S of Gibson et al.(2008b)are equivalent to R xz under quarter-cell coordinate transformations.In keeping with previous literature,most of our results are presented in terms of the isotropy subgroup S={e,s1,s2,s3}={e,σzτx,σxτxz,σxzτz} rather than the equivalent R xz.Schmiegel’s‘I’isotropy group is equivalent to our R z; Schmiegel(1999)contains many examples of R z-invariant equilibria.R-invariant equi-libria were found by Tuckerman&Barkley(2002)for plane Couetteﬂow in which the translation symmetries were broken by a streamwise ribbon.We have not searched for R x-invariant solutions,and are not aware of any published in the literature.The remaining subgroups of orders4and8all involve{e,τi}factors and by(3.4) are equivalent by translation to a state repeated twice spanwise,streamwise,or both. For example,the isotropy subgroup of EQ7and EQ8studied below is S×{e,τxz} R×{e,τxz}and thus these are repeats of solutions on half-domains.For the detailed count of all67subgroups,see Halcrow(2008).3.3.State-space visualizationGHC presents a method for visualizing low-dimensional projections of trajectories in the inﬁnite-dimensional state space of the Navier-Stokes equations.Brieﬂy,we construct an orthonormal basis{e1,e2,···,e n}that spans a set of physically importantﬂuid states ˆu A,ˆu B,...,such as equilibrium states and their eigenvectors,and we project the evolv-ingﬂuid stateˆu(t)=u−yˆx onto this basis using the L2inner product(2.3).It is convenient to use diﬀerences from laminarﬂow,sinceˆu forms a vector space with the laminar equilibrium at the origin,closed under addition.This produces a low-dimensional projectiona(t)=(a1,a2,···,a n,···)(t),a n(t)=(ˆu(t),e n),(3.12) which can be viewed in2d planes{e m,e n}or in3d perspective views{e ,e m,e n}.The state-space portraits are dynamically intrinsic,since the projections are deﬁned in terms8J.Halcrow,J.F.Gibson,and P.Cvitanovi´cof intrinsic solutions of the equations of motion,and representation independent,since the inner product(2.3)projection is independent of the numerical or experimental repre-sentation of theﬂuid state data.Such bases are eﬀective because moderate-Re turbulence explores a small repertoire of unstable coherent structures(rolls,streaks,their mergers), so that the trajectory a(t)does not stray far from the subspace spanned by the key structures.There is no a priori prescription for picking a‘good’set of basisﬂuid states,and construction of{e n}set requires some experimentation.The plane Couette system at hand has a total of29known equilibria within the S-invariant subspace:four translated copies each of EQ1-EQ6,two translated copies of EQ7(which have an additionalτxz symmetry),plus the laminar equilibrium EQ0at the origin.As shown in GHC,the dynamics of diﬀerent regions of state space can be elucidated by projections onto basis sets constructed from combinations of equilibria and their eigenvectors.In this paper we present global views of all invariant solutions in terms of the or-thonormal‘translational basis’constructed in GHC from the four translated copies of EQ2:τxτzτxze1=c1(1+τx+τz+τxz)ˆuEQ2+++e2=c2(1+τx−τz−τxz)ˆuEQ2+−−(3.13)e3=c3(1−τx+τz−τxz)ˆuEQ2−+−e4=c4(1−τx−τz+τxz)ˆuEQ2−−+,where c n is a normalization constant determined by e n =1.The last3columns indicate the symmetry of the basis vector under half-cell translations;e.g.±1in theτx column impliesτx e j=±e j.4.Equilibria and traveling waves of plane CouetteﬂowWe seek equilibrium solutions to(2.1)of the form u(x,t)=uEQ(x)and traveling-waveor relative equilibrium solutions of the form u(x,t)=uTW(x−c t)with c=(c x,0,c z). Let F NS(u)represent the Navier-Stokes equations(2.1)for the given geometry,boundaryconditions,and Reynolds number,and f tNSits time-t forward map∂u ∂t =F NS(u),f t NS(u)=u+tdτF NS(u).(4.1)Then for anyﬁxed T>0,equilibria satisfy f T(u)−u=0and traveling waves satisfy f T(u)−τu=0,whereτ=τ(c x T,c z T).When u is approximated with aﬁnite spectral expansion and f t with CFD algorithm,these equations become set of nonlinear equations in the expansion coeﬃcients for u and,in the case of traveling waves,the wave velocities (c x,0,c z).Viswanath(2007)presents an algorithm for computing solutions to these equations based on Newton search,Krylov subspace methods,and an adaptive‘hookstep’trust-region limitation to the Newton steps.This algorithm can provide highly accurate so-lutions from even poor initial guesses.The high accuracy stems from the use of Krylov subspace methods,which can be eﬃcient with105or more spectral expansion coeﬃ-cients.The robustness with respect to initial guess stems from the hookstep algorithm. The hookstep limitation restricts steps to a radius r of estimated validity for the local linear approximation to the Newton equations.As r increases from zero,the hookstep varies smoothly from the Krylov-subspace gradient direction to the Newton step,so thatEquilibria and traveling waves of plane Couette ﬂow9Figure 3.Four projections of equilibria,traveling waves and their half-cell shifts onto trans-lational basis (3.13)constructed from equilibrium EQ 4.Equilibria are marked EQ 0,◦EQ 1,•EQ 2, EQ 3, EQ 4,♦EQ 5, EQ 7, EQ 9, EQ 10,and EQ 11.Traveling waves trace out closed loops.In some projections the loops appear as line segments or points.TW 1(blue)is a spanwise-traveling,symmetry-breaking bifurcation oﬀEQ 1,so it passes close to diﬀerent translational phases of ◦EQ 1.Similarly,TW 3(red)bifurcates oﬀ EQ 3and so passes near its translations.TW 2(green)was not discovered through bifurcation (see §4);it appears as the shorter,isolated line segment in (a 1,a 4)and (a 2,a 4).The EQ 1→EQ 0relaminarizing hetero-clinic connections are marked by dashed lines.A long-lived transiently turbulent trajectory is plotted with a dotted line.The EQ 4-translational basis was chosen here since it displays the shape of traveling waves more clearly than the projection on the EQ 2-translational basis of ﬁgure 1.the hookstep algorithm behaves as a gradient descent when far away from a solution and as the Newton method when near,thus greatly increasing the algorithm’s region of convergence around solutions,compared to the Newton method (J.E.Dennis,Jr.,&Schnabel 1996).The choice of initial guesses for the search algorithm is one of the main diﬀerences between this study and previous calculations of equilibria and traveling waves of shear ﬂows.Prior studies have used homotopy,that is,starting from a solution to a closely related problem and following it through small steps in parameter space to the problem of interest.Equilibria for plane Couette ﬂow have been continued from Taylor-Couette ﬂow (Nagata 1990),Rayleigh-B´e nard ﬂow (Clever &Busse 1997),and from plane Couette with imposed body forces (Waleﬀe 1997).Equilibria and traveling waves have also been found using “edge-tracking”algorithms,that is,by adjusting the magnitude of a perturbation of the laminar ﬂow until it neither decays to laminar nor grows to turbulence,but instead converges toward a nearby weakly unstable solution (Skufca et al.(2006);Viswanath (2008);Schneider et al.(2008)).In this study,we take as initial guesses samples of velocity ﬁelds generated by long-time simulations of turbulent dynamics.The intent is to10J.Halcrow,J.F.Gibson,and P.Cvitanovi´cﬁnd the dynamically most important solutions,by sampling the turbulentﬂow’s natural measure.We discretize u with a spectral expansion of the formu(x)=Jj=−JKk=−KL=03m=1u jkl T (y)e2πi(jx/L x+kz/L z),(4.2)where the T are Chebyshev polynomials.Time integration of f t is performed with aprimitive-variables Chebyshev-tau algorithm with tau correction,inﬂuence-matrix en-forcement of boundary conditions,and third-order backwards diﬀerentiation time step-ping,and dealiasing in x and z(Kleiser&Schuman(1980);Canuto et al.(1988);Peyret (2002)).We eliminate from the search space the linearly dependent spectral coeﬃcients of u that arise from incompressibility,boundary conditions,and complex conjugacies that arise from the real-valuedness of velocityﬁelds.Our Navier-Stokes integrator,im-plementation of the Newton-hookstep search algorithm,and all solutions described in this paper are available for download from website(Gibson2008a). For further details on the numerical methods see GHC and Halcrow(2008).Solutions presented in this paper are use spatial discretization(4.2)with(J,K,L)= (15,15,32)(or32×33×32gridpoints)and roughly60k expansion coeﬃcients.The esti-mated accuracy of each solution is listed in table1.As is clear from Schmiegel(1999) Ph.D.thesis,ours is almost certainly an incomplete inventory;while for anyﬁnite Re,ﬁnite-aspect ratio cell the number of distinct equilibrium and traveling wave solutions isﬁnite,we know of no way of determining or bounding this number.It is diﬃcult to compare our solutions directly to those of Schmiegel since those solutions were computed in a[4π,2,2π]cell(roughly twice our cell size in both span and streamwise directions) and with lower spatial resolution(2212independent expansion functions versus our60k for a cell of one-fourth the volume).We expect that many of Schmiegel’s equilibria could be continued to higher resolution and smaller cells.4.1.Equilibrium solutionsOur primary focus is on the S-invariant subspace(3.3)of theΩW03cell at Re=400.We initiated28equilibrium searches at evenly spaced intervals∆t=25along a trajectory in the unstable manifold of EQ4that exhibited turbulent dynamics for800nondimension-alized time units after leaving the neighborhood of EQ4and before decaying to laminar ﬂow.The solutions are numbered in order of discovery,adjusted so that lower,upper branch solutions are labeled with consecutive numbers.EQ0is the laminar equilibrium,EQ1and EQ2are the Nagata lower and upper branch,and EQ4is the uNBsolution reported in GHC.The rest are new.Only one of the28searches failed to converge onto an equilibrium;the successful searches converged to equilibria with frequencies listed in table1.The higher frequency of occurrence of EQ1and EQ4suggests that these arethe dynamically most important equilibria in the S-invariant subspace for theΩW03cell at Re=400.Stability eigenvalues of known equilibria are plotted inﬁgure7.Tables of stability eigenvalues and other properties of these solutions are given in Halcrow(2008), while the images,movies and full data sets are available online at .All equilibrium solutions have zero spatial-mean pressure gradient,which was imposed in theﬂow conditions,and,due to their symmetry,zero mean velocity.EQ1,EQ2equilibria.This pair of solutions was discovered by Nagata(1990),recom-puted by diﬀerent methods by Clever&Busse(1997)and Waleﬀe(1998,2003),and found multiple times in randomly initiated searches as described above.The lower branch EQ1 and the upper branch EQ2are born together in a saddle-node bifurcation at Re≈218.5.。

An exactly soluble noisy traveling wave equation appearing in the problem of directed polym

´ Eric Brunet∗ and Bernard Derrida†
ቤተ መጻሕፍቲ ባይዱ
arXiv:cond-mat/0409261v1 [cond-mat.dis-nn] 10 Sep 2004
We calculate exactly the velocity and diﬀusion constant of a microscopic stochastic model of N evolving particles which can be described by a noisy traveling wave equation with a noise of order N −1/2 . Our model can be viewed as the inﬁnite range limit of a directed polymer in random medium with N sites in the transverse direction. Despite some peculiarities of the traveling wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoﬀ approximation and to show that, in the weak noise limit, the position of the front can be completely described by the eﬀect of the noise on the ﬁrst particle.
2 For a given front conﬁguration ut (x), the random variables {xi (t + 1)} are uncorrelated and the probability to be on the right of x is given by Prob xi (t + 1) > x = 1 − µ(x − xj (t)), (5) From now on, we will limit our discussion to the case where ρ(s) is a Gumbel distribution: ρ(s) = exp −s − e−s . (12)

推广的Tanh函数法与_2_1_维Burgers方程组新的精确行波解

［13 ］
2 c2 2（ 1 － m ） c0 = ， c2 ＞ 0 ， c4 （ 2 － m 2 ） 2
槡
－
槡
c2 ξ）， 2 － m2 （ 13 ）
c0 槡
+ c1
+ c2
2
+ c3 ω 3 + c4 ω 4 ，（ 5）
当 m→1 时，周期解（ 11 ）退化为钟状孤子解（ 5 ），周期解（ 12 ）退化为扭状孤子解（ 8 ）．（ iv）当 c4 = c0 = c1 = 0 时，方程（ 1 ）具有如下钟状孤子解，三角函数周期解和有理解 =－ =－ c2 c2 sech2 （槡 ξ）， c2 ＞ 0 ， c3 2 （ 14 ）
sech［ c2 （ x + 槡－ α y + λ t）］，槡 c2 ＞ 0 ， c4 ＜ 0 ， u12 = a0 + 槡－ α b1
（ iii）当 c3 = c1 = 0 时，有解为 u31 = a0 + 槡－ α b1 cn［ v31 = c2
2
槡
c2 × c4 c2 － × c4
（ School of Mathematics and Informations，Henan Polytechnic University，Jiaozuo 454000 ，Henan，China）
Abstract： Based on an extension of the Hyperbolic Tangent Function Method， which is a direct and unified algebraic method for constructing more general form travelling wave solutions of nonlinear partial differential equations and implemented in a computer algebraic system． More general form solutions are obtained， including kindshaped solitons， bellshaped solitons，singular solitons and periokdic solitions． The properties of some new formal solitary wave solutions are shown by some figures． Key words： an extension of the hyperbolic tangent function method；（ 2 + 1 ） dimensional burgers equation； exact solution； soliton solution； clock shape solution； periodic solution

2020新译林版高中英语选修一Unit2课文原文及翻译

Unit 2Working the landReadingPrecision farming hits its target 精准农业正中靶心When we think of farming, the first image that springs to mind might be of a farmer working in a field under the baking sun. Face covered in sweat, he might be walking through the field, carefully checking his crops before deciding what needs to be done. In modern times, however, this deep-rooted image of a traditional farmer is being changed. The collaboration between farming and technology has given rise to precision farming, an approach that equips farmers with the tools and data they need to make reliable decisions with remarkable accuracy. This evolution is having a positive impact on farming, while also providing better solutions to the world's pressing food problems.当我们想到农业时，脑海中浮现的第一个画面可能就是烈日下一个农民在田地里干活。

他满脸大汗，可能正在田间走动，仔细检查他的作物，然后决定需要做什么。

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。

Kdv方程

Original Research Article
Received: 22 June 2013 Accepted: 21 August 2013 Published: 17 February 2014
_______________________________________________________________________
b 4 H 2 v, y y 2 2 g 2v c 3 a v 2 v h2 , 2
For
(1.10)
h1 h2 g 2 0, a
c c 2 12 g1 1 ,b , c 2 12 g1 0, 6 c 2 a
(1.1)
which were derived to model the interaction of water waves. Here , , b are constants. Liu et al. in [13] based on a subtle balance method, a given function expansion which was applied to Eq. (1.1), a series of periodic solutions, solitary wave solutions and singular solutions are obtained by aid of symbolic computation. For b 3,
British Journal of Mathematics & Computer Science
4(8): ……., 2014
SCIENCEDOMAIN international

EXACT SOLITARY WAVE SOLUTIONS TO A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS USING DIRECT ALGEBR

and u s 2 ( x, t ) =
r
³ ´ 6 x − ct sech √ +δ , α γ
Ann . o f Diﬀ . Eqs. 24 :2(2008) , 184-192
EXACT SOLITARY WAVE SOLUTIONS TO A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS USING DIRECT ALGEBRAIC METHOD ∗
n n
whe re a i ( i = 1 , 2 , · · · , 9) are re al con stan ts, an d u ( x, t ) is a sc alar fu nc tion d eﬁ ne d for all ( x, t) ∈ R × I , in wh ich t de note s a variab le in th e in terval I = [0 , ∞ ). It is assum ed that u ( x, t ) is continu ous for all valu es of its re spe ctive argu me nts, and th at variou s partial de rivatives of u e xist and are continu ous with re sp e ct to th e in de pe nd en t variab le s x and t . The m ain pu rpose of this work is to ﬁ nd exac t hy pe rb olic -ty p e solitary wave solu tion s to e quation (1.1) wh en n = 1 or n = 2. Th e solutions to (1.1) c an b e investigated u sin g a direc t alge braic m etho d, wh ic h was introdu ce d an d d isc usse d in [12-14]. Furth ermore, we show how th e d issipative term s (sec ond ord er te rms) in (1.1) aﬀec t of the solution s to th is class. In an earlier pap er [2], u sing a direc t alge braic me thod , th e solitary wave s u s 1 ( x, t ) and u s 2 ( x, t ) of p ulse- typ e solutions to (1.1) are ob tain ed , wh ere us 1 (x, t) =

Meromorphic traveling wave solutions of the Kuramoto-Sivashinsky equation

2
other meromorphic solutions form a one-parametric family w(z−z0), z0 ∈ C. Furthermore, (i) Elliptic solutions exist only if b2 = 16µν. They are of order 3 and have one triple pole per parallelogram of periods. (ii) All exponential solutions have the form P (tan kz), where P is a polynomial of degree at most 3 and k ∈ C. (iii) Non-constant rational solutions occur if and only if b = µ = A = 0 and they have the form w(z) = 120ν(z − z0)−3, z0 ∈ C.
arXiv:nlin/0504053v1 [nlin.SI] 25 Apr 2005
Meromorphic traveling wave solutions of the Kuramoto–Sivashinsky equation
Alexandre Eremenko∗ February 8, 2008
We will see that the proof of Theorem 1 is of very general character, and applies to many other equations which have the uniqueness property of formal Laurent solutions stated above. In [4] the author proved a similar result about the generalized Briot–Bouquet equation F (w(k), w) = 0, where F is a polynomial in two variables and k is odd. If k is even, the equation does not have the uniqueness property, as stated above. However, the conjecture that all meromorphic solutions of all generalized Briot–Bouquet equations belong to the class W is plausible, and recently Tuen Wai Ng informed the author that he made a progress towards this conjecture.

-类广义mKdV型方程的行波解

-类广义mKdV型方程的行波解摘要：在降阶法的基础上，通过变量代换的方法，研究了一类广义mKdV方程，得到了方程不同物理性质的行波解。

关键词：mKdV方程孤立子行波解1 引言近年来,大量非线性发展方程,如KdV方程,mKdV方程,Burgers方程,Boussinesq方程,ZK方程等被广泛地研究。

为了获得这些非线性方程的确切解,特别是孤立子解,涌现了tanh方法,齐次平衡法,正余弦拟设法等多种方法。

在Taogetusang[2]所做工作的基础上,本文将对广义的mKdV方程进行讨论。

要说明的是,在下面的计算中我们引入如下定义的q型双曲函数(q为常数,且)3 结论本文所讨论的非线性方程Eq.(1)尽管此前已被研究过,但我们仍然希望能找到新的方法去获得它的确切解。

事实上,我们通过引入的新变量,借助降阶法,得到了包括孤立子解在内的新解。

应该说本文的方法更加简单有效。

参考文献[1] A.H.Khater, W.Maliet, D.K.Callebaut, E.S.Kamel, Travelling wave solutions of equations in(1+1) and (2+1) dimensions, Journal of Computational and Applied Mathematics, 140(2002)469-477.[2] Taogetusang, Sirendaoreji, New exact wave solutions to generalized mKdV equation and gener-alized Zakharov-Kuzentsov equation, Chinese Physics, 15(06)(2006)1143-1148.[3] A.M.Wazwwaz, Variants of the two-dimensional Boussinesq equation with com-pactons,solitons,and periodic solutions, Computers and Mathematics with Applications,49(2005)295-301.。

Exact Solutions of the Generalized Equal Width Wave Equation

+∞ −∞
(u )2 dξ = k2 , as ξ −→ +∞ ,
(13)
By substituting (14) into (13) and using the relation (11), we obtain the analytical expression of the following important square integral,
2
Derivation of the Exact Solutions
We concentrate on ﬁnding an exact solitary wave solution of the form u(x, t) = u(x − x0 − Ct) . (3)
This corresponds to a traveling-wave propagating with steady celerity C . We are interested in solutions depending only on the moving coordinate ξ = x − x0 − Ct as, u(x, t) = u(x − x0 − Ct) ≡ u(ξ ) . (4) Substituting into (2), the function u(ξ ) satisﬁes a third order nonlinear ordinary diﬀerential equation (ODE), −Cu + aup u − δu + µCu =0, (5)
Abstract. The equal width wave (EW) equation is a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. The EW-Burgers equation models the propagation of nonlinear and dispersive waves with certain dissipative eﬀects. In this work, we derive exact solitary wave solutions for the general form of the EW equation and the generalized EW-Burgers equation with nonlinear terms of any order. We also derive analytical expressions of three invariants of motion for solitary wave solutions of the generalized EW equation.

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.。

Travelling Wave Solutions for a coupled KdV equations

−ac + 3αa2 + (6αab − bc − 3)v 2 + 3αb2 v 4 + 2bα(vv ′′ + (v ′ )2 ) = g1 , −cv + 3αv (a + 1 bv 2 ) + αv ′′ = g2 , 3
(1.4)
where g1 and g2 are integral constants. Thus, we have two Hamiltonian systems dv dy g1 + ac − 3αa2 + (bc − 6αab + 3)v 2 − 3αb2 v 4 − 2bαy 2 = y, = , dξ dξ 2bαv (1.5)
(Grant Nos.
2011jjA10047, 2012GXNSFAA053003), China. Corresponding author: Shengqiang Tang, E-mail address: tangsq@
1
Traveling Wave Solutions for a coupled KdV equations
−cu′ + 6αuu′ − 6vv ′ + αu′′′ = 0, −cv ′ + 3αuv ′ + αv ′′′ = 0,
(1.3)
where ”′” is the derivative with respect to ξ . To remove the coupled relationship of (1.3), let u = a + bv 2 , substituting it into (1.3) and integrating once, we obtain

EXACT_PERIODIC-WAVE_SOLUTIONS_FOR_(2+1)-DIMENSIONAL_BOUSSINESQ_EQUATION_AND_(3+1)-DIMENSIONAL_KP_EQU

tain de rivatives (this is true for the e quation s c on side red h ere). In th is p roce ss we take the integration con stants to b e ze ro. T he n ext cruc ial step is to exp ress th e solu tions of th e resu ltin g O DE b y the Jac obi e lliptic- fun ction meth od in Ref. [12], u (ξ ) c an be ex pre sse d as a ﬁn ite p ower se rie s of J acob i ellip tic sine fu nc tion , sn ξ , i.e., the an satz u( ξ ) =
d cn ξ = −sn ξ d n ξ , dξ d dξ dn ξ = −m 2 sn ξ cn ξ . (8)
In this artic le, for Jacob i e llip tic fu nc tions, we u se the n otation sn ξ, cn ξ, d n ξ w ith argu me nt ξ an d mo du lu s parame ter m (0 < m < 1). T he param eter n in Eq. (4) will b e ﬁ xe d b y balan cing th e h ighe st ord er of de rivative term an d the n online ar term in th e non lin ear OD E Eq. (3) by u sin g Eq. (5). Su bstituting Eq . (4) (w ith ﬁx ed valu e of n ) in to the re du ce d non linear O DE (3) an d equ ating the c oeﬃ cients of variou s p owers of sn ξ to ze ro we get a se t of algeb raic e quation s for aj , k , l , s , an d ω . S olving th em c onsistently we obtain re lation s am ong the p arWave Solut ions for (2+1)-Dimen sional Bou ssines q Equ at ion an d ( 3+ 1)-D im ens ional K P Equ at ion∗

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