亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Exact Solutions of Atmospheric(2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations?

        2016-05-09 08:54:34PingLiu劉萍YaXiongWang王亞雄BoRen任博andJinHuaLi李金花
        Communications in Theoretical Physics 2016年12期
        關鍵詞:劉萍字樣秒針

        Ping Liu(劉萍),Ya-Xiong Wang(王亞雄),Bo Ren(任博),and Jin-Hua Li(李金花)

        1College of Electron and Information Engineering,University of Electronic Science and Technology of China Zhongshan Institute,Zhongshan 528402,China

        2School of Physical Electronics,University of Electronic Science and Technology of China,Chengdu 610054,China

        3Institute of Nonlinear Science,Shaoxing University,Shaoxing 312000,China

        4School of Physics and Optoelectronic Engineering,Nanjing University of Information Science and Technology,Nanjing,China

        1 Introduction

        Atmospheric gravity waves are fluctuations in the dynamic equilibrium of the gravity and buoyancy of the atmosphere.They can break,produce wind shear and typhoon,and then play a very important factor on disastrous weather and climate.The formation and propagation of atmospheric gravity waves are essential in forecasting the damage caused by rainstorm.[1?4]

        The basic controlling laws of atmospheric movement are mass conservation equations,momentum conservation equations,energy conservation equations,and water mass conservation equations,and other gas and aerosol mass conservation equations.The motion of atmospheric gravity waves are mesoscale atmospheric motion,whose governing equations can be simplified from the basic governing equations of atmosphere as follows,[5]

        where u and w are wind velocities in x-and z-directions,ρ means the density,T0denotes background temperature,p is the perturbation pressure,T′denotes the perturbation temperature, γ and γdare the moist lapse rate and the dry adiabatic lapse rate,g is the gravitational constant,t means the time,x and z denote the coordinates in latitudinal and vertical directions.The calculation on symmetries shows that the equations are invariant under the Galilean transformations,the scaling transformations,and the space-time translations.[6]Equations(1a)–(1b)are the kinematic equations in latitudinal and vertical directions,respectively,which are the momentum conservation equations in fluid.Equation(1c)is continuity equation,which is a mass conservation equation in fluid.Equation(1d)is the thermodynamic equation,which is an energy conservation equation in fluid.

        On the(2+1)-dimensional nonlinear INHB equations,numerical simulation[7?10]and analytical analysis[1,11?13]are two commonly used methods.The interactions of atmospheric conoidal waves with a critical level are examined by using numerical simulation in Ref.[7].The authors of Ref.[8]investigated properties of solitary waves propagating in a two-layer fluid by comparing numeri-cal simulation experiments and theory,and found that stability or instability of the flow occurs in approximate with the theorem of Miles and Howard.Researchers made some effective attempts on traveling wave solutions of the(2+1)-dimensional nonlinear INHB equations.A general practice is that INHB equations are approximated to some other equations with the help of some assumptions.By Introducing traveling wave transformation and truncated expansion,the authors of Refs.[1,11]reduced the INHB equations into some known soliton equatons,such as KdV equation,KdV-mKdV equation and KdV-mKdV-GmKdV equation,and then obtained some known periodic or solitary wave solutions of these soliton equatons.In Ref.[12],the INHB equations degenerate to the KdV equation and Benjamin–Davis–Ono(BDO)equation by introduced to some boundary conditions,and then solitary wave of the KdV equation and algebraic solutions of the BDO equations are applied to describe the INHB equations.By introducing different scales of varying time and space variables,the authors of Ref.[13]reduce the INHB equations to a type of nonlinear Schr?dinger equation,and then demonstrate the solitary wave evolutions of the gravity waves envelope.

        Lie group symmetry theory is a very powerful method to solve partial differential equations(PDEs). Many methods on symmetries of PDEs have been proposed,such as the non-classical symmetry group,[14]the classical symmetry group,[15?17]residual symmetry,[18?19]and some other symmetry methods.[20?22]Tanh function expansion method is also a simple and useful method and is often used to research for exact solutions for PDEs.[23?25]In this paper,we will combine the tanh function method and the symmetry method to research the exact solutions of the INHB equations.

        The paper is organized as follows.The tanh function is applied and 12 expansion equations governed various tanh function terms are determined in the next section.In Sec.3,the symmetries of the expansion equations are researched.Section 4 is devoted to similarity reductions and similarity solutions of the expansion equations.Exact solutions of reduction equations are discussed in Sec.5.In Sec.6,Non-traveling wave solutions of the INHB equations are obtained by symmetry method.Traveling wave solutions of the INHB equations are obtained and some evolutions of atmospheric physical quantities with space and time are graphed in Sec.7.Summary and discussion are proposed in the final section.

        2 Tanh Function Expansions of the INHB Equations

        For the INHB equations,we can make the following truncated tanh function expansions

        where{r,u0,u1,w0,w1,p0,p1,s0,s1}are functions of x and t.Substituting Formula(2)into Eq.(1),and vanishing the coefficients of all the powers of tanh(r),we obtain 12 over-determined equations for nine undetermined functions

        where subscripts x,z,and t represent partial differentiations.From the solutions of{u0,u1,w0,w1,p0,p1,s0,s1}and substituting these solutions into(2),we then can obtain the exact solutions of the INHB equations.It is very difficult to directly solve Eq.(3).In the next three sections,we will solve Eq.(3)with the help of symmetry group theory.

        3 Symmetries of Expansion Equations

        We will now try to solve Eq.(3)by means of symmetry method.A symmetry of a nonlinear partial differential equation is defined as a solution of its linearized system.The nine lie point symmetry components{σi,i=1,2,3,4,5,6,7,8,9}can be supposed to have the forms

        Substituting Formula(4)into Eq.(5),and identifying the coefficients of various order terms on variables{r,u0,u1,w0,w1,p0,p1,s0,s1},we obtain an over-determined set of equations for the unknown functions.Solving the determinant equations,we obtain

        where F1,F2,F3are functions of t,and{Ci,i=1,2,3,4,5,6,7}are arbitrary constants.Then symmetry components{σi,i=1,2,3,4,5,6,7,8,9}described by(4)are transformed to

        4 Similarity Solutions and Reduction Equations

        After determining the in finitesimal generators,the similarity variables can be found by solving the characteristic equations

        Because the procedure to get the symmetry reductions is standard,we just write down the final reduction results.The most general invariant variables can be written as

        Due to the existence of arbitrary functions F1(t),F2(t)and F3(t)in(7),the similarity solutions for{u0,u1,w0,w1,p0,p1,s0,s1}and the corresponding reduction equations are very lengthy,so we will discuss the simplified conditions,for example,F1(t)=F2(t)=F3(t)=0.Then we obtain the following three nontrivial types of reduction equations and similarity solutions.

        Case 1C26=0,C5C6+C726=0.

        In this case,the invariant variables ξ and η in(9)degenerate to

        The similarity solutions can be written as

        where

        The reduction variables R ≡ R(ξ,η),U1≡ U1(ξ,η),W1≡ W1(ξ,η),P1≡ P1(ξ,η),S1≡ S1(ξ,η),which are arbitrary functions of ξ and η.The similarity solutions of u0,w0,p0and s0are very lengthy,so we will not write them down here.The reduction equations can be written as

        Case 2C26=0,C5C6+C72=0.

        The similarity solutions in this case are generated as

        where the{U0,U1,W0,W1,P0,P1,S0,S1}are functions of group-invariant variables{ξ,η},which are determined by

        The corresponding reduction equations are

        Case 3C2=0.

        On condition of C2=0,the group invariant variables read

        When C5C6+C726=0,the reduction equations are very complicated,and we will write only down the reduction equations on condition C5C6+C72=0.Then C5can be substituted by C5=?C72/C6,and the reduction equations are generated by

        where R ≡ R(ξ,η),U0≡ U0(ξ,η),U1≡ U1(ξ,η),W0≡ W0(ξ,η),W1≡ W1(ξ,η),P0≡ P0(ξ,η),P1≡ P1(ξ,η),S0≡ S0(ξ,η),and S1≡ S1(ξ,η)are all arbitrary functions of ξ and η.

        The corresponding similarity solutions are determined by

        5 Exact Solutions of Reduction Equations

        From similarity solutions of the researched systems and exact solutions of reduction equations,one can obtain some exact solutions of the researched systems.By this means,we can obtain some exact solutions of the expansion equations(3),and then obtain some exact solutions of the INHB equations.Three cases of similarity solutions and reduction equations are introduced in Sec.4.Obviously,different reduction equations will lead to different exact solutions of the expansion equations.

        Now,we will focus on the first case of reduction equations,because it is the most general case in the three cases.Solving Eqs.(15)–(29),we obtain two types of non-trivial solutions.

        Solution 1

        where C9is an arbitrary constant,F5,F6,F7,F8are functions of η,which solve

        Solution 2

        where F13is a function of η,and F9,F10,F11,F12,F14,and F15are functions of ξ and these functions solve the following equations:

        There are some parameters bounded by constraint equations in Solution 1 and Solution 2.We can obtain the exact solutions of the first type of reduction equations only by solving these parameters.It is still very difficult to obtain general solutions of these constraint equations.So we will try to obtain some special exact solutions of the constraint equations.The method of solving the constraint equations of Solution 1 is the same as those of Solution 2,and we will only discuss Solution 1.

        Plugging Formula(69)into Eqs.(59)–(62),one can obtain that

        Substituting formulas(69)–(72)into Formulas(58a)–(58i),we obtain the following exact solution of the reduction equations(15)–(29)

        6 Non-traveling Wave Solutions of the INHB Equations

        In the last section,we have obtained some exact solutions of reduction equations(15)–(29).Combining similarity solutions and the exact solutions of reduction equations,one can obtain the exact solutions of the original equations.Exact solutions of reduction equations discussed in the previous section are related to the first type of reduction equations,where the variables are{ξ,η}in(10).Because{ξ,η}in(10)are non-traveling wave transformation variables,the exact solutions obtained with the help of them are necessarily non-traveling wave solutions.To obtain traveling wave solutions of the expansion equations,one should make further transformations for the reduction equations or make directly traveling wave transformation for the expansion equations,which will be discussed in the next section.

        In this section,we will try to obtain exact non-traveling wave solutions of the INHB equations by symmetry method.Plugging formulas(73)–(81)into(11a)–(11e)and noticing the auxiliary variables(12)–(14),we can obtain the solutions of{r,u0,u1,w0,w1,p0,p1,s0,s1}.The substitution of{r,u0,u1,w0,w1,p0,p1,s0,s1}into Eq.(2)leads to the solution of the INHB equations as follows:

        where

        and{ξ,η}are expressed as(10).

        Above solution for{u,w,p,T′}in the INHB equations are non-traveling wave solutions.In the next section,we will discuss traveling wave solutions of the INHB equations.

        打開床頭邊的手機,手機上清晰的顯示著2014年1月20日,01:20:33字樣。殷明看著這日期,恍然想起明天,不,是今天將在市中心有場面試。艱難地坐在床上,看著手機上的時間一分一秒流逝,電子表面的秒針嘲笑般地看著他,慢吞吞地一下一下轉(zhuǎn)動著。

        7 Traveling Wave Solutions of the INHB Equations

        To search for traveling wave solutions on gravity waves is one important subject.We can make traveling wave transformation to{r,u0,u1,w0,w1,p0,p1,s0,s1}as follows

        where k1,k2,and k3are arbitrary constants.

        Equation(3)is then transformed to

        Integrating Eq.(88b)once over δ,we can rewrite Eq.(88b)as

        where C16are arbitrary integral constant.Then

        Integrated once over δ,Eq.(88c)can be rewritten as

        where C17is an arbitrary integral constant,and will be chosen as zero in the rest of this paper.Thencan be expressed as

        which leads Eq.(88a)to be zero.

        Substituting Formulas(90)and(92)into Eqs.(88d)–(88l),we obtain

        Solving Eqs.(93a)–(93i)leads to

        where C19–C24are arbitrary constants.

        Combining Eqs.(2),(87)and(94)–(102),we obtain the exact solution for the INHB equations

        Fig.1 The evolutions of the velocities in latitudinal and vertical directions in Formulas(103)–(104).The atmospheric parameters are g=9.81 m/s2,ρ =1.29 kg/m3,γd=0.01 ?C/m,γ =0.005 ?C/m,T0=279 K.

        Fig.2 The evolutions of perturbation pressure and perturbation temperature in Formulas(105)–(106).The atmospheric parameters are similar to Fig.1.

        Figure 1 describes the evolutions for the velocities in latitudinal and vertical directions,which are governed by Formulas(103)–(104).The atmospheric parameters are g=9.81 m/s2,ρ =1.29 kg/m3,γd=0.01?C/m,γ =0.005?C/m,T0=279 K. C19satisfies C19=?C22tanh(C18)+C16ρ(C16?k3)/k12and the other parameters are C18=2,k1=k2=0.1,k3=0.005,C20=C21=C22=1,C23=10,C16=0,C24=10.Figure 1 demonstrates that the wind velocity evolutions in latitudinal and vertical directions can both satisfy periodic functions.The evolution of perturbation pressure with time and the evolution of perturbation temperature with time are proposed in Fig.2,which demonstrates the periodic evolutions of perturbation pressure and perturbation temperature with time and space.

        On traveling wave solutions,the difference between this article and the literatures listed in Sec.1 is very obvious.The INHB equations are approximated to some other equations,and the traveling wave solutions of the approximate equation are applied to described to traveling waves solutions in the literatures.In this paper,we have not make any assumptions,and the traveling wave solutions are really exact solutions of the INHB equations.

        8 Summary and Discussion

        Atmospheric gravity waves play a very important role in typhoon,rainstorm,and other disastrous weather.Due to complexity,numeric simulations are often applied to atmospheric dynamical equations,while analytical analyses are much less discussed.Exact solutions and symmetries of the(2+1)-dimensional INHB equations are researched in this paper.

        Tanh function expansion is applied to the INHB equations and 12 over-determined equations are determined by the expansion coefficients.Some references show that some meaningful exact solutions can be obtained by tanh function expansion method for some consistent tanh expansion(CTE)solvable systems.[23?25]Unfortunately,the INHB equations are not CTE solvable,and the auxiliary coefficient variables{r,u0,u1,w0,w1,p0,p1,s0,s1}can not be expressed intuitively as the auxiliary variable r.In order to obtain some exact solutions of the 12 coefficient equations,we introduce symmetry method to the 12 coefficient equations.Three types of non-trivial similarity solutions and the corresponding symmetry reduction equations are proposed.Combining exact solutions of reduction equations and the similarity solutions,we obtain some non-traveling wave solutions of the INHB equations.Traveling wave solutions are discussed with the help of traveling wave transformation in the paper.Some graphes on the evolutions of the latitudinal wind velocity and vertical wind velocity,perturbation pressure and perturbation temperature with space and time are demonstrated,which show directly the periodicity of some important atmospheric physical quantities with space and time.

        There are still many open problems on climate and weather.The(2+1)-dimensional INHB equations,as a valuable model of atmospheric gravity waves,should be further studied.

        Acknowledgements

        The authors would like to thank ProfessorSen-Yue Lou and Professor Bao-Qing Zeng for their valuable discussion.

        [1]Z.L.Li,J.Phys.A-Math.Theor.41(2008)145206.

        [2]Z.L.Li,Appl.Math.Comput.217(2010)1398.

        [3]J.R.Qing,S.M.Chen,D.W.Li,and B.Liang,Chin.Phys.B 21(2012)089401.

        [4]P.Liu,Z.L.Li,and S.Y.Lou,Appl.Math.Mech.31(2010)1383.

        [5]James C.McWilliams,Fundamentals of Geophysical Fluid Dynamics,Cambridge University Press,London(2006).

        [6]P.Liu and X.N.Gao,Commun.Theor.Phys.53(2010)609.

        [7]E.D.Shyllingstad,J.Atmos.Sci.48(1991)1613.

        [8]J.Grue,A.Jensen,P.Reusas,and J.K.Sveen,J.Fluid Mech.380(1999)257.

        [9]W.D.Zhao,J.Zhao,X.Z.Han,and Z.J.Xu,Chin.J.Liquid Cryst.Disp.29(2014)281.

        [10]F.Qian,T.Sun,J.Guo,and T.F.Wang,Chin.J.Liquid Crys.Disp.30(2015)317.

        [11]Z.L.Li,G.Fu,and J.Chen,Chaos,Solitons&Fractals 40 (2009)530.

        [12]J.W.Rottman and F.Einaudi,J.Atmos.Sci.50(1993)2116.

        [13]X.H.Xu and Y.H.Ding,Chin.J.Atmos.Sci.15(1991)58.

        [14]G.W.Bluman and S.Kumei,Symmetries and Di ff erential Equations,Spring-Verlag,New York(1989).

        [15]P.Olver,Applications of Lie Group to Di ff erential Equations,Spring-Verlag,New York(1986).

        [16]P.Liu,B.Q.Zeng,B.B.Deng,and J.R.Yang,Aip Adv.5(2015)087162.

        [17]P.Liu,B.Q.Zeng,and B.Ren,Commun.Theor.Phys.63(2015)413.

        [18]X.N.Gao,S.Y.Lou,and X.Y.Tang,J.High Energy Phys.5(2013)029.

        [19]P.Liu,B.Li,and J.R.Yang,Cent.Eur.J.Phys.12(2014)541.

        [20]W.X.Ma and T.C.Xia,Phys.Scr.87(2013)055003.

        [21]Z.J.Qiao,Commun.Math.Phys.239(2003)309.

        [22]B.Ren,X.N.Gao,Y.Jun,and P.Liu,Open Math.13(2015)502.

        [23]D.Yang,S.Y.Lou,and W.F.Yu,Commun.Theor.Phys.60(2013)387.

        [24]C.L.Chen and S.Y.Lou,Chin.Phys.Lett.30(2013)110202.

        [25]H.Pu and M.Jia,Commun.Theor.Phys.64(2015)623.

        猜你喜歡
        劉萍字樣秒針
        刻薄
        青年文學家(2024年2期)2024-03-09 04:22:35
        時間很小
        延河(2022年6期)2022-07-24 21:03:50
        秒針匆匆
        “合格”
        單堯堯、胡豐凡、劉萍、葉小輝作品
        做大自然的“翻譯官”
        ——淺談寫生的意義
        “從來不知道哪天會有投資”
        南方周末(2018-10-25)2018-10-25 20:00:21
        Molecular detection of microbial communities associated with Microcystis vs Synechococcus dominated waters in Tianjin, China*
        民政部未批準任何帶有“一帶一路”字樣的社會組織
        時代金融(2017年22期)2017-09-13 19:15:57
        穿越魔幻城堡
        国产精品国语对白露脸在线播放| 亚洲色图视频在线免费看 | 成人艳情一二三区| 国产av人人夜夜澡人人爽麻豆| 国产精品久久久久9999| 国产激情精品一区二区三区| 偷亚洲偷国产欧美高清| 丰满人妻一区二区三区免费 | 中文字幕av无码一区二区三区 | 青青草原综合久久大伊人| 91爱爱视频| 成年男女免费视频网站点播| 丰满少妇被猛烈进入高清播放| 88久久精品无码一区二区毛片| 国产精品爆乳在线播放| 亚洲一区有码在线观看| 精品无人区无码乱码毛片国产| 国产熟妇人妻精品一区二区动漫 | 欧美日韩亚洲tv不卡久久| 中文字幕亚洲好看有码| 亚洲精品美女久久久久99| 亚洲男同免费视频网站| 欧美成人精品a∨在线观看 | 国产精品区一区第一页| 精品国产亚洲一区二区在线3d| 日本久久精品国产精品| 免费亚洲一区二区三区av| 丰满人妻熟妇乱又伦精品软件| 97视频在线观看免费| 少妇高潮太爽了免费网站| 狂猛欧美激情性xxxx大豆行情 | 亚洲三区二区一区视频| 丝袜美腿亚洲综合在线播放| 看久久久久久a级毛片| 精品乱码久久久久久中文字幕| 天堂女人av一区二区| 国产91精品高潮白浆喷水| 亚洲av蜜桃永久无码精品| 久久青草国产精品一区| 久久爱91精品国产一区| 国产亚洲自拍日本亚洲|