• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    Inverse Scattering Transform of the Coupled Sasa–Satsuma Equation by Riemann–Hilbert Approach?

    2018-01-24 06:23:03JianPingWu吳建平andXianGuoGeng耿獻國2SchoolofScienceZhengzhouUniversityofAeronauticsZhengzhou450046China
    Communications in Theoretical Physics 2017年5期

    Jian-Ping Wu(吳建平)and Xian-Guo Geng(耿獻國)2School of Science,Zhengzhou University of Aeronautics,Zhengzhou 450046,China

    2School of Mathematics and Statistics,Zhengzhou University,100 Kexue Road,Zhengzhou 450001,China

    1 Introduction

    The nonlinear Schr?dinger(NLS)equation is wellknown to be an important integrable system in mathematical physics.There are many physical contexts where the NLS equation appears.For example,the NLS equation describes the weakly nonlinear surface wave in deep water.More importantly,the NLS equation models the soliton propagation in optical fibers where only the group velocity dispersion and the self-phase modulation effects are considered.However,for ultrashort pulse in optical fibers,the effects of the third-order dispersion,the selfsteepening,and the stimulated Raman scattering should be taken into account.Due to these effects,the dynamics of the ultrashort pulses can be described by the Sasa–Satsuma higher-order nonlinear Schr?dinger equation[1?4]

    whereq=q(x,t)is a complex-valued function.Moreover,to describe the propagations of two optical pulse envelopes in birefringent fibers,some coupled Sasa–Satsuma equations were also proposed and studied.[5?9]Particularly,a coupled form of the Sasa–Satsuma equation reads[5]

    whereq1=q1(X,T),q2=q2(X,T)are two complex functions of variablesX,T.As done in Ref.[5],in order to analyze Eq.(2),it is more convenient to rewrite it in the following form

    by use of the gauge,Galilean and scale transformations In this paper,we refer to Eq.(3)as the coupled Sasa–Satsuma equation.The complete integrability of Eq.(3)was established and soliton solutions were obtained using Darboux–B?cklund transformation.[5]In addition,the coupled Sasa–Satsuma equation has also been investigated via various methods such as Darboux transformation,Painlevé singularity analysis,Hirota method,[6?9]and so on.

    Recently,there are many investigations on solutions of nonlinear evolution equations.[10?16]It is also known that the inverse scattering transform is a powerful approach to derive soliton solutions.However,since Eq.(3)involves a 5×5 matrix spectral problem,[5]the inverse scattering transform for this equation is rather complicated to dealwith.To our knowledge,the research in this direction has not been conducted before.The aim of the present paper is to study the multi-soliton solutions of Eq.(3)by utilizing the inverse scattering transform via Riemann–Hilbert(RH)approach.[17?25]

    This paper is arranged as follows.In Sec.2,starting from the Lax pair we give direct scattering transform of Eq.(3).Then an RH problem is formulated and solved in the reflectionless cases.In Sec.3,the inverse scattering transform for Eq.(3)is established.In Sec.4,we construct multi-soliton solutions of Eq.(3).Moreover,we will give some interesting figures describing the corresponding soliton characteristics,including breather types,single-hump solitons,double-hump solitons,and two-bell solitons.

    2 Riemann–Hilbert Problem

    In this section,we give direct scattering transform of the coupled Sasa–Satsuma equation(3).According to Ref.[5],the Lax pair for Eq.(3)is

    where Ψ = Ψ(x,t;ζ)is a column vector function of the complex spectral parameterζ,and

    where

    For the sake of convenience,we extend Ψ in Eq.(4)to a matrix and then introduce a new matrix spectral functionJ=J(x,t;ζ)defined by Ψ =JeiζΛx+4iζ3Λt.Then the Lax pair(4)can be rewritten as

    Now let us construct two matrix solutionsJ±=of Eq.(5a)under the asymptotic conditions

    Here each[J±]ldenotes thel-th column ofJ±,respectively.The symbol I is the 5×5 identity matrix,and the subscripts ofJrepresent which end of thex-axis the boundary conditions are set.The matrix solutionsJ±are uniquely determined by the Volterra integral equations

    It is easy to find that[J+]1,[J+]2,[J+]3,[J+]4,[J?]5allow analytic extensions to the upper halfζ-plane C+.On the other hand,[J?]1,[J?]2,[J?]3,[J?]4,[J+]5are analytically extendible to the lower halfζ-plane C?.

    In what follows,we investigate the properties ofJ±.Indeed,from the fact thatQis traceless we know detJ±are independent ofx.Therefore we have detJ±=1 forζ∈ R.Moreover,J±eiζΛxare related by a scattering matrixS(ζ)=(skj)5×5

    which implies that

    Then it follows from the analytic property ofJ?thats55can be analytically extended to C+,whereasskj(1≤k,j≤ 4)allow analytic extensions to C?.In general,sk5,s5j(1≤k,j≤4)cannot be extended offthe realζ-axis.

    Using the analytic properties ofJ±,we can construct a matrix functionP1=P1(x,ζ)which is analytic forζ∈ C+

    Moreover,we have the large-ζasymptotic behavior ofP1

    In order to obtain an RH problem for Eq.(3),we have to construct an analytic matrixP2in C?.To this end,we shall consider the adjoint equation of Eq.(5a)

    It is easy to check that the matrix inverses ofJ±satisfy Eq.(13).Let us denote

    and the large-ζasymptotic behavior ofP2can be shown to be

    In addition,it is easy to find thatare related by the scattering matrixR(ζ)≡ (rkj)5×5=S?1(ζ)

    Moreover,similar to the scattering coeffcientsskjabove,we can show thatr55allows an analytical extension to C?,whereasrkj(1≤k,j≤4)have analytic extensions to C+.In addition,rk5,r5j(1≤k,j≤4)are only defined on the realζ-axis.

    Summarizing the above results,we have constructed two matrix functionsP1andP2,which are analytic in C+and C?,respectively.Now we denote the limit ofP1from the left-hand side of the realζ-axis asP+,and the limit ofP2from the right-hand side of the realζ-axis asP?.Consequently,we obtain an RH problem for the coupled Sasa–Satsuma equation(3)

    where the matrixG=G(x,ζ)is as follows

    and the canonical normalization condition for this RH problem is

    In order to investigate the inverse scattering transform for Eq.(3),let us solve the RH problem(18).To this end,we assume that the RH problem(18)is irregular.Here the irregularity means both detP1and detP2have certain zeros in their analytic domains.Recalling the definitions ofP1andP2,we have

    To specify these zeros,we notice that there is a symmetry relation forQin Eq.(5a)

    where?means the Hermitian conjugate.Therefore from Eq.(13),we get

    Then it follows that

    which implies the following relations

    Furthermore,from the definitions ofP1andP2,the following property also holds

    To solve the RH problem(18),we have to consider one more symmetry relation

    where

    From Eq.(27)we obtain that

    which leads to

    Obviously,Eq.(29)implies that

    Furthermore,we point out that Eq.(28)also yields a property forP1itself

    Therefore,from Eqs.(20),(21),and(25c),we find that ifζjis a zero of detP1,thenis a zero of detP2.Moreover,in view of Eq.(30c),we know thatis also a zero of detP1.Thus we can investigate the zeros of detP1in two cases.Firstly,we assume that detP1has a total number of 2Nsimple zerosζj(1 ≤j≤ 2N)satisfyingwhich are all in C+.Correspondingly,detP2possesses 2Nsimple zeros(1≤j≤2N)satisfyingwhich are all in C?.Obviously,the zeros of detP1and detP2always appear in quadruples in this case.The second case is that detP1possesses onlyNsimple zerosζj(1 ≤j≤N)in C+,where eachζjis pure imaginary.Then detP2hasNsimple zerosin C?,whereThe scattering data we need to solve the RH problem(18)consists of the continuous scattering data{s15,s35}as well as the discrete scattering data{ζj,ζj,vj,vj}.Herevjandare nonzero column and row vectors,respectively,satisfying

    Next we deducevjandvjsuch that multi-soliton solutions can be obtained for the coupled Sasa–Satsuma equation(3).For the first type of zeros,we obtain a special relation by using Eqs.(26)and(32)

    On the other hand,from(31)we obtain a particular column vector relation

    Now we shall get the vectorsvj(1≤j≤N).For this purpose,we take thex-derivative ofP1(ζj)vj=0.Then utilizing Eq.(5a),we obtain without loss of generality that

    wherevj,0is independent ofx.Therefore,using Eqs.(33)–(35),all the vectorsvjandcan be determined explicitly.Note that,to derive soliton solutions for Eq.(3),we choose the matrixGin(18)to be the 5×5 identity matrix.This is guaranteed by settings15=s35=0,which corresponds to the reflectionless case.Consequently,the unique solution for the special RH problem is

    whereM=(mkj)2N×2Nis a matrix whose entries aremkj=vkvj/(ζj?ζk).For the second type of zeros,the corresponding vectorsvj,(1≤j≤N)can be derived as

    wherevj,0is independent ofx.Using these vectors,the RH problem(18)in the reflectionless case can also be solved exactly

    whereM=(mkj)N×Nis a matrix with entriesmkj=

    3 Inverse Scattering Transform

    In this section,we give the inverse scattering transform of the coupled Sasa–Satsuma equation(3),from which we recover the potentialsu,vby using the scattering data.In fact,we can expandP1(ζ)as

    Then substituting it into Eq.(5a),and then comparingO(1)terms gives

    which implies thatu,vcan be obtained as

    4 Multi-Soliton Solutions

    To derive solutions for the coupled Sasa–Satsuma equation(3),we have to investigate the temporal evolutions of the scattering data.From(5b)and(9),and noticing the decaying properties ofuandv,we arrive at

    which leads to the temporal evolutions

    In addition,using Eq.(5b)we obtain thatTherefore,for the first type of zeros,we obtain

    Now for the first type of zeros of detP1,we setvj,0=(αj,βj,γj,μj,1)Tto be complex constant vectors.Then utilizing Eqs.(43)and(46)–(47),from Eq.(42)we obtain anN-soliton solution formula for Eq.(3)

    whereM=(mkj)2N×2Nwith

    Now we are interested in the simplest situation that occurs whenN=1 in Eq.(50).To illustrate the single-soliton solution explicitly,we shall specify the corresponding parameters in Eq.(50).Firstly,we setβ1=α?1andμ1=γ?1in Eq.(50).Then by denotingζ1=ξ1+iη1(ξ1/=0,η1>0),a breather-type solution for the coupled Sasa–Satsuma equation(3)is obtained from Eq.(50)

    where

    Its breather-type behavior is plotted in Fig.1.

    Fig.1 The breather-type solution via(51) with the parameters

    Secondly,we chooseβ1=μ1=0 in(50),then another type of soliton solution for the coupled Sasa–Satsuma equation(3)is obtained

    where

    We remark that the single-soliton solution(52)features that it can be single-humped or double-humped,which is similar to the case for the Sasa–Satsuma equation.[1,20]By choosing appropriate parameters,we plot the single-hump and double-hump soliton solutions in Figs.2–3,respectively.

    Fig.2 The single-hump soliton via Eq.(52)with the parameters

    Fig.3 The double-hump soliton via Eq.(52)with the parameters

    Fig.4 Collisions between two single-hump solitons via Eq.(50)with N =2,and(α1,β1,γ1,μ1)=(1,0,0,0),(α2,β2,γ2,μ2)=(1,0,1,0),ζ1=0.4+0.5i,ζ2=0.7+0.8i.

    Now we shall investigate the case forN=2 in Eq.(50).To illustrate the corresponding two-soliton interactions,we first choose the parameters in Eq.(50)as(α1,β1,γ1,μ1)=(1,0,0,0),(α2,β2,γ2,μ2)=(1,0,1,0),ζ1=0.4+0.5i,ζ2=0.7+0.8i.Then a type of polarization-changing collision[2]between two single-hump solitons will be obtained.The polarization-changing collisions lead to the enhancement of intensity in one component of|u|or|v|,and the suppression of intensity in the other component,as shown in Fig.4.This figure also shows that one component of|v|even becomes zero after collision.Moreover,for the caseN=2 in Eq.(50),another type of two-soliton interaction can be obtained by choosing proper parameters.For example,we set(α1,β1,γ1,μ1)=(1,1,1,1),(α2,β2,γ2,μ2)=(1,0,2,0),ζ1=0.5+0.5i,ζ2=1+i.Under these parameters,the interactions of a single-hump soliton and a breather can be obtained,as can be seen in Fig.5.This figure demonstrates that a single-hump soliton changes into a breather when interacting with a breather.

    Fig.5 Interactions between a single-hump soliton and a breather via Eq.(50)with N=2,and(α1,β1,γ1,μ1)=(1,1,1,1),(α2,β2,γ2,μ2)=(1,0,2,0),ζ1=0.5+0.5i,ζ2=1+i.

    Fig.6 The two-bell soliton solution Eq.(55)with the parameters α1=1,γ1=1+i,α2=i,γ2=0.5i,ζ1=0.3i,ζ2=0.5i.

    In what follows,we shall turn to the second type of zeros of detP1.In this case,we assume that detP1has onlyNsimple zerosλj(1 ≤j≤N)in C+,whereλjis pure imaginary.In this case,we obtain another kind ofN-soliton solution.Settingto be complex constant vectors,anotherN-soliton solution formula for Eq.(3)follows from(42)by using Eqs.(44),(48)–(49)

    whereM=(mkj)N×Nwith

    WhenN=1,Eq.(53)gives a bell-soliton solution for the coupled Sasa–Satsuma equation(3)

    WhenN=2,Eq.(53)reduces to a two-bell soliton solution of the coupled Sasa–Satsuma equation(3)

    whereM=(mkj)2×2with

    5 Conclusions

    In this paper,we have obtained two kinds of multisoliton solution formulae for the coupled Sasa–Satsuma equation(3).One is Eq.(50)and the other is Eq.(53).These two forms ofN-soliton solutions correspond to two different types of zero structures of the RH problem.In addition,based on the twoN-soliton solution formulae,we obtain some interesting soliton solutions which include breather-type solutions,single-hump soliton solutions,double-hump soliton solutions,and two-bell soliton solutions.

    [1]N.Sasa and J.Satsuma,J.Phys.Soc.Jpn.60(1991)409.

    [2]T.Xu and X.M.Xu,Phys.Rev.E 87(2013)032913.

    [3]T.Xu,M.Li,and L.Li,Europhys.Lett.109(2015)30006.

    [4]J.J.C.Nimmo and H.Yilmaz,J.Phys.A:Math.Theor.48(2015)425202.

    [5]K.Nakkeeran,K.Porsezian,P.Shanmugha Sundaram,and A.Mahalingam,Phys.Rev.Lett.80(1998)1425.

    [6]X.Lü,Commun.Nonlinear Sci.Numer.Simulat.19(2014)3969.

    [7]L.C.Zhao,Z.Y.Yang,and L.M.Ling,J.Phys.Soc.Jpn.83(2014)104401.

    [8]M.N.Vinoj and V.C.Kuriakose,Phys.Rev.E 62(2000)8719.

    [9]A.Mahalingam and K.Porsezian,J.Phys.A:Math.Gen.35(2002)3099.

    [10]M.J.Ablowitz,B.Prinari,and A.Trubatch,Discrete and Continuous Nonlinear Schr?dinger Systems,Cambridge University Press,Cambridge(2004).

    [11]R.Hirota,The Direct Methods in Soliton Theory,Cambridge University Press,Cambridge(2004).

    [12]X.Lü,W.X.Ma,Y.Zhou,and C.M.Khalique,Comput.Math.Appl.71(2016)1560.

    [13]X.Lü,W.X.Ma,J.Yu,F.H.Lin,and C.M.Khalique,Nonlinear Dyn.82(2015)1211.

    [14]X.Lü and W.X.Ma,Nonlinear Dyn.85(2016)1217.

    [15]X.Lü,S.T.Chen,and W.X.Ma,Nonlinear Dyn.86(2016)523.

    [16]X.Lü and L.M.Ling,Chaos 25(2015)123103.

    [17]M.J.Ablowitz and A.S.Fokas,Complex Variables:Introduction and Applications,Cambridge University Press,Cambridge(2003).

    [18]M.J.Ablowitz and P.A.Clarkson,Solitons,Nonlinear Evolution Equations and Inverse Scattering,Cambridge University Press,Cambridge(1991).

    [19]L.D.Faddeev and L.A.Takhtajan,Hamiltonian Methods in the Theory of Solitons,Springer,Berlin(1987).

    [20]J.K.Yang,Nonlinear Waves in Integrable and Nonintegrable Systems,SIAM,Philadelphia(2010).

    [21]V.S.Shchesnovich and J.K.Yang,J.Math.Phys.44(2003)4604.

    [22]D.S.Wang,Y.Q.Ma,and X.G.Li,Commun.Nonlinear Sci.Numer.Simulat.19(2014)3556.

    [23]D.S.Wang,D.J.Zhang,and J.K.Yang,J.Math.Phys.51(2010)023510.

    [24]B.L.Guo and L.M.Ling,J.Math.Phys.53(2012)073506.

    [25]X.G.Geng and J.P.Wu,Wave Motion 60(2016)62.

    a级片在线免费高清观看视频| 一本色道久久久久久精品综合| 深夜精品福利| 乱人伦中国视频| 欧美精品亚洲一区二区| 国产在视频线精品| 91老司机精品| 亚洲国产av影院在线观看| 交换朋友夫妻互换小说| 91精品国产国语对白视频| 欧美变态另类bdsm刘玥| 国产日韩欧美视频二区| 国产精品99久久99久久久不卡| 精品国内亚洲2022精品成人 | 老司机午夜福利在线观看视频 | 久热爱精品视频在线9| 国产人伦9x9x在线观看| 日本撒尿小便嘘嘘汇集6| 91国产中文字幕| 叶爱在线成人免费视频播放| 在线永久观看黄色视频| www.自偷自拍.com| 欧美精品亚洲一区二区| 大片免费播放器 马上看| 高清在线国产一区| 菩萨蛮人人尽说江南好唐韦庄| 伊人亚洲综合成人网| 国产免费现黄频在线看| 成年动漫av网址| 男人舔女人的私密视频| 手机成人av网站| 啦啦啦在线免费观看视频4| 夫妻午夜视频| 精品久久久久久电影网| 如日韩欧美国产精品一区二区三区| 国产在线免费精品| 久久午夜综合久久蜜桃| 久久久久久久久久久久大奶| 成人亚洲精品一区在线观看| 天堂中文最新版在线下载| 国产成人精品无人区| 国产免费现黄频在线看| 成年动漫av网址| 亚洲精品在线美女| 国产亚洲精品一区二区www | 久久久久久人人人人人| 丰满人妻熟妇乱又伦精品不卡| 亚洲色图 男人天堂 中文字幕| 大香蕉久久成人网| 国产一区二区三区在线臀色熟女 | 午夜福利一区二区在线看| 老汉色∧v一级毛片| 国产亚洲av高清不卡| 王馨瑶露胸无遮挡在线观看| 高清av免费在线| 黄色片一级片一级黄色片| 十八禁人妻一区二区| 男女下面插进去视频免费观看| www.av在线官网国产| 国产精品久久久av美女十八| 亚洲精品在线美女| 精品福利观看| 一本综合久久免费| 欧美久久黑人一区二区| 欧美中文综合在线视频| e午夜精品久久久久久久| 青草久久国产| 亚洲一码二码三码区别大吗| 在线看a的网站| 国产精品.久久久| 精品人妻在线不人妻| 国产野战对白在线观看| 免费在线观看影片大全网站| 久久国产精品影院| av电影中文网址| 9191精品国产免费久久| 亚洲av电影在线观看一区二区三区| 欧美黄色淫秽网站| 老司机午夜十八禁免费视频| 这个男人来自地球电影免费观看| 欧美xxⅹ黑人| 国产免费现黄频在线看| 亚洲成av片中文字幕在线观看| 精品一品国产午夜福利视频| 12—13女人毛片做爰片一| 亚洲中文字幕日韩| 97人妻天天添夜夜摸| 国产老妇伦熟女老妇高清| 一本综合久久免费| 欧美日韩亚洲高清精品| 久久精品亚洲av国产电影网| 美女脱内裤让男人舔精品视频| 热99久久久久精品小说推荐| 乱人伦中国视频| 激情视频va一区二区三区| 欧美日韩av久久| 手机成人av网站| 侵犯人妻中文字幕一二三四区| 一区二区三区乱码不卡18| 一区二区三区乱码不卡18| 国产成人欧美在线观看 | 久久久久久久大尺度免费视频| 悠悠久久av| 久久久久久免费高清国产稀缺| 婷婷成人精品国产| 久久精品国产a三级三级三级| 在线观看人妻少妇| 亚洲av成人不卡在线观看播放网 | 天堂中文最新版在线下载| 亚洲一码二码三码区别大吗| 色精品久久人妻99蜜桃| 国产色视频综合| 一进一出抽搐动态| 日韩大片免费观看网站| 亚洲第一欧美日韩一区二区三区 | 国产精品av久久久久免费| 午夜福利在线免费观看网站| 成在线人永久免费视频| 一级片免费观看大全| 丰满人妻熟妇乱又伦精品不卡| 色精品久久人妻99蜜桃| 精品国产乱码久久久久久男人| 成年美女黄网站色视频大全免费| 在线精品无人区一区二区三| 老司机影院毛片| av超薄肉色丝袜交足视频| 视频在线观看一区二区三区| 热99re8久久精品国产| 好男人电影高清在线观看| 亚洲国产看品久久| 一边摸一边做爽爽视频免费| 国产一级毛片在线| 欧美av亚洲av综合av国产av| 国产精品国产av在线观看| 91精品三级在线观看| 日韩欧美免费精品| 亚洲欧美精品自产自拍| 人成视频在线观看免费观看| 国产福利在线免费观看视频| 中国美女看黄片| 波多野结衣一区麻豆| 操美女的视频在线观看| √禁漫天堂资源中文www| 一本久久精品| 日韩一区二区三区影片| 欧美人与性动交α欧美软件| 性色av一级| 国产无遮挡羞羞视频在线观看| 国产精品1区2区在线观看. | 一二三四在线观看免费中文在| 国产精品1区2区在线观看. | 久热这里只有精品99| 精品国产一区二区三区四区第35| 精品一区二区三卡| 国产精品香港三级国产av潘金莲| 久久亚洲精品不卡| 国产欧美日韩一区二区三 | 99国产精品一区二区蜜桃av | 悠悠久久av| 美女午夜性视频免费| 在线观看免费视频网站a站| 亚洲国产看品久久| 国产不卡av网站在线观看| 在线天堂中文资源库| 国产亚洲欧美在线一区二区| 欧美老熟妇乱子伦牲交| 天天影视国产精品| 国产欧美日韩一区二区三 | 国产一区二区三区在线臀色熟女 | 国产在线视频一区二区| 美女脱内裤让男人舔精品视频| 免费在线观看黄色视频的| 欧美日韩亚洲国产一区二区在线观看 | 免费少妇av软件| 99re6热这里在线精品视频| 叶爱在线成人免费视频播放| a级毛片在线看网站| 亚洲精华国产精华精| 中国美女看黄片| av片东京热男人的天堂| 三级毛片av免费| 高清在线国产一区| 欧美日韩国产mv在线观看视频| 丁香六月欧美| 国产亚洲精品一区二区www | 日韩三级视频一区二区三区| 亚洲中文av在线| 精品福利永久在线观看| 黄色a级毛片大全视频| 亚洲欧美日韩高清在线视频 | 欧美国产精品一级二级三级| 久久综合国产亚洲精品| 精品一区二区三区av网在线观看 | 亚洲人成电影观看| 欧美亚洲 丝袜 人妻 在线| 91麻豆av在线| 中文字幕高清在线视频| 亚洲欧美一区二区三区久久| 曰老女人黄片| 久久久久久免费高清国产稀缺| 国产精品1区2区在线观看. | a级片在线免费高清观看视频| 日韩,欧美,国产一区二区三区| 99国产综合亚洲精品| 亚洲人成电影观看| 日韩制服骚丝袜av| 亚洲人成电影免费在线| 制服诱惑二区| 一进一出抽搐动态| 免费不卡黄色视频| 在线观看免费日韩欧美大片| 老鸭窝网址在线观看| 91字幕亚洲| 午夜激情av网站| 精品国产超薄肉色丝袜足j| 99国产极品粉嫩在线观看| 涩涩av久久男人的天堂| 亚洲视频免费观看视频| 老汉色av国产亚洲站长工具| 亚洲av欧美aⅴ国产| 91国产中文字幕| 欧美日韩亚洲国产一区二区在线观看 | 亚洲精品久久久久久婷婷小说| 大码成人一级视频| xxxhd国产人妻xxx| 不卡av一区二区三区| 自线自在国产av| 久热这里只有精品99| 久久精品人人爽人人爽视色| 亚洲精品中文字幕一二三四区 | 99国产综合亚洲精品| 国产精品久久久久久人妻精品电影 | 日韩精品免费视频一区二区三区| 精品人妻熟女毛片av久久网站| 电影成人av| 人人妻人人澡人人看| 成人三级做爰电影| 美国免费a级毛片| 国产高清视频在线播放一区 | 亚洲成国产人片在线观看| 丰满人妻熟妇乱又伦精品不卡| 丁香六月欧美| 国产亚洲av高清不卡| 久久国产精品男人的天堂亚洲| 日本精品一区二区三区蜜桃| 国产亚洲精品久久久久5区| 亚洲精品日韩在线中文字幕| 各种免费的搞黄视频| 久久久久久久久免费视频了| 国产精品99久久99久久久不卡| 久久人人爽人人片av| 99国产极品粉嫩在线观看| 亚洲精品一二三| 男女之事视频高清在线观看| 亚洲人成电影免费在线| 亚洲伊人久久精品综合| 国产成人精品久久二区二区免费| 激情视频va一区二区三区| 无遮挡黄片免费观看| 岛国毛片在线播放| 久久久久国内视频| 日本猛色少妇xxxxx猛交久久| 搡老乐熟女国产| 一区二区三区乱码不卡18| 激情视频va一区二区三区| 国产一区有黄有色的免费视频| 99国产精品免费福利视频| 久久精品国产亚洲av高清一级| 亚洲一卡2卡3卡4卡5卡精品中文| 欧美一级毛片孕妇| av国产精品久久久久影院| 国产老妇伦熟女老妇高清| 久久久久久人人人人人| 午夜福利一区二区在线看| 亚洲av日韩精品久久久久久密| 亚洲人成电影观看| 久久久国产一区二区| 国产成人a∨麻豆精品| 黑人巨大精品欧美一区二区mp4| 国产欧美日韩一区二区三区在线| 日韩 欧美 亚洲 中文字幕| 亚洲av日韩精品久久久久久密| 麻豆av在线久日| 亚洲免费av在线视频| 99国产精品99久久久久| 美女高潮喷水抽搐中文字幕| 男女边摸边吃奶| 50天的宝宝边吃奶边哭怎么回事| 老司机亚洲免费影院| 国产亚洲av片在线观看秒播厂| 人人妻人人澡人人爽人人夜夜| 69精品国产乱码久久久| 亚洲精品久久午夜乱码| 黄色 视频免费看| 精品乱码久久久久久99久播| 亚洲中文av在线| 男女边摸边吃奶| 操出白浆在线播放| 国产免费视频播放在线视频| 亚洲一区中文字幕在线| 免费黄频网站在线观看国产| 国产伦理片在线播放av一区| 国产男女超爽视频在线观看| 国产色视频综合| 国产片内射在线| a级毛片黄视频| 一个人免费看片子| 亚洲精品国产精品久久久不卡| 美女大奶头黄色视频| 精品久久蜜臀av无| 咕卡用的链子| 91字幕亚洲| 18禁国产床啪视频网站| 一级片免费观看大全| 大码成人一级视频| 午夜91福利影院| 999精品在线视频| 国产不卡av网站在线观看| 黄频高清免费视频| av片东京热男人的天堂| 丁香六月欧美| 在线观看人妻少妇| 久久99一区二区三区| 搡老乐熟女国产| 777久久人妻少妇嫩草av网站| 亚洲少妇的诱惑av| 国产成人精品在线电影| 性少妇av在线| 少妇猛男粗大的猛烈进出视频| 亚洲精品国产av蜜桃| 国产伦理片在线播放av一区| 国产精品 国内视频| 精品久久蜜臀av无| 99九九在线精品视频| 99国产极品粉嫩在线观看| 欧美日韩亚洲综合一区二区三区_| 国产极品粉嫩免费观看在线| 久久久久久久久免费视频了| 欧美成人午夜精品| 国产伦理片在线播放av一区| tocl精华| 日韩大片免费观看网站| 午夜91福利影院| 国产亚洲精品第一综合不卡| 五月开心婷婷网| 女警被强在线播放| 黄色毛片三级朝国网站| www.自偷自拍.com| videosex国产| 人人妻人人澡人人看| 日日爽夜夜爽网站| 好男人电影高清在线观看| 久久女婷五月综合色啪小说| 又大又爽又粗| 伦理电影免费视频| 欧美 日韩 精品 国产| www.av在线官网国产| 免费在线观看黄色视频的| 午夜福利视频精品| 91成人精品电影| www.av在线官网国产| 欧美精品人与动牲交sv欧美| 国产片内射在线| 亚洲精品国产av成人精品| 大片免费播放器 马上看| 欧美 日韩 精品 国产| 亚洲久久久国产精品| 欧美精品一区二区大全| 欧美成狂野欧美在线观看| 人人澡人人妻人| 在线亚洲精品国产二区图片欧美| 国产在线免费精品| 三级毛片av免费| 亚洲精品国产av成人精品| 美女午夜性视频免费| 欧美精品人与动牲交sv欧美| 免费av中文字幕在线| 亚洲专区国产一区二区| h视频一区二区三区| 啪啪无遮挡十八禁网站| 日韩一卡2卡3卡4卡2021年| 免费在线观看日本一区| 国产精品一二三区在线看| 亚洲成av片中文字幕在线观看| 男人操女人黄网站| 久久青草综合色| 国产欧美日韩一区二区三区在线| 午夜福利免费观看在线| bbb黄色大片| 精品卡一卡二卡四卡免费| 日本91视频免费播放| 男女之事视频高清在线观看| 亚洲精品国产一区二区精华液| 在线十欧美十亚洲十日本专区| 啦啦啦免费观看视频1| av网站免费在线观看视频| 久久影院123| 亚洲欧美激情在线| 成年人免费黄色播放视频| 热99re8久久精品国产| av电影中文网址| h视频一区二区三区| 亚洲精品久久久久久婷婷小说| 亚洲精品国产色婷婷电影| 欧美日韩黄片免| 黑丝袜美女国产一区| 国产国语露脸激情在线看| 国产精品久久久久久精品电影小说| 久久狼人影院| 免费在线观看黄色视频的| 精品少妇内射三级| 国产野战对白在线观看| 欧美精品人与动牲交sv欧美| 91老司机精品| 亚洲国产成人一精品久久久| av有码第一页| 黑人巨大精品欧美一区二区蜜桃| 精品国产一区二区三区久久久樱花| 少妇人妻久久综合中文| 青草久久国产| 青草久久国产| 国产精品一二三区在线看| 69av精品久久久久久 | 国产成人影院久久av| 97人妻天天添夜夜摸| 69av精品久久久久久 | 999久久久国产精品视频| 国产成人影院久久av| 黑人巨大精品欧美一区二区mp4| 大型av网站在线播放| 无限看片的www在线观看| 国产精品二区激情视频| 欧美黄色淫秽网站| 免费在线观看影片大全网站| 久久天躁狠狠躁夜夜2o2o| 又黄又粗又硬又大视频| 人人妻人人澡人人爽人人夜夜| 在线av久久热| 老熟妇仑乱视频hdxx| 啦啦啦 在线观看视频| 国产日韩一区二区三区精品不卡| 国产主播在线观看一区二区| 成人三级做爰电影| 97在线人人人人妻| 精品久久久久久电影网| 欧美日韩精品网址| 久久精品人人爽人人爽视色| 久久人人爽av亚洲精品天堂| 啦啦啦免费观看视频1| 午夜精品久久久久久毛片777| 在线十欧美十亚洲十日本专区| 美女午夜性视频免费| 欧美97在线视频| 两性夫妻黄色片| 久久 成人 亚洲| 韩国精品一区二区三区| 亚洲国产精品成人久久小说| 免费在线观看视频国产中文字幕亚洲 | 午夜福利影视在线免费观看| 1024香蕉在线观看| 大片电影免费在线观看免费| 亚洲成人免费av在线播放| 亚洲国产av新网站| 性色av一级| 纵有疾风起免费观看全集完整版| 热99国产精品久久久久久7| 欧美xxⅹ黑人| 国产1区2区3区精品| 亚洲精品美女久久av网站| 成人18禁高潮啪啪吃奶动态图| 亚洲成人国产一区在线观看| 久久久精品94久久精品| 极品少妇高潮喷水抽搐| 欧美日韩视频精品一区| 性色av一级| 欧美亚洲 丝袜 人妻 在线| av免费在线观看网站| 亚洲av日韩在线播放| 老司机亚洲免费影院| 热99久久久久精品小说推荐| 国产精品 欧美亚洲| 在线永久观看黄色视频| 午夜久久久在线观看| 日本91视频免费播放| 日韩有码中文字幕| 日韩制服丝袜自拍偷拍| 别揉我奶头~嗯~啊~动态视频 | 日韩人妻精品一区2区三区| 精品卡一卡二卡四卡免费| 午夜福利免费观看在线| 大片免费播放器 马上看| 女性生殖器流出的白浆| 国产成人免费无遮挡视频| 国产欧美亚洲国产| 精品高清国产在线一区| 国产亚洲欧美在线一区二区| 啦啦啦在线免费观看视频4| 在线天堂中文资源库| 91精品国产国语对白视频| 国产三级黄色录像| 精品卡一卡二卡四卡免费| 国产精品av久久久久免费| 91九色精品人成在线观看| 久久人妻熟女aⅴ| 成年人午夜在线观看视频| 日韩人妻精品一区2区三区| 国产亚洲av高清不卡| 亚洲av日韩在线播放| 老司机亚洲免费影院| 久久中文字幕一级| 亚洲av片天天在线观看| 日韩熟女老妇一区二区性免费视频| 精品一区二区三区av网在线观看 | 亚洲自偷自拍图片 自拍| 亚洲av日韩精品久久久久久密| 国产区一区二久久| 99国产极品粉嫩在线观看| 亚洲av男天堂| 女人高潮潮喷娇喘18禁视频| 色婷婷av一区二区三区视频| 午夜免费成人在线视频| 国产麻豆69| www.自偷自拍.com| 黄色片一级片一级黄色片| 99久久人妻综合| 国产麻豆69| 日本91视频免费播放| 久久久久国产一级毛片高清牌| 国产淫语在线视频| 2018国产大陆天天弄谢| 亚洲国产看品久久| 国产亚洲午夜精品一区二区久久| 另类亚洲欧美激情| 欧美少妇被猛烈插入视频| 国产又色又爽无遮挡免| 在线观看一区二区三区激情| 免费不卡黄色视频| 亚洲伊人色综图| 久久精品国产a三级三级三级| 亚洲国产精品一区二区三区在线| 两性午夜刺激爽爽歪歪视频在线观看 | 亚洲七黄色美女视频| 9色porny在线观看| 精品亚洲成国产av| 桃红色精品国产亚洲av| 岛国在线观看网站| 91av网站免费观看| 色老头精品视频在线观看| 免费看十八禁软件| 麻豆乱淫一区二区| 精品免费久久久久久久清纯 | 女人高潮潮喷娇喘18禁视频| 最新在线观看一区二区三区| tube8黄色片| 一本—道久久a久久精品蜜桃钙片| 女人高潮潮喷娇喘18禁视频| 中文字幕色久视频| 成人国产一区最新在线观看| 又大又爽又粗| 999久久久精品免费观看国产| 欧美精品一区二区免费开放| 精品福利观看| 国产免费福利视频在线观看| 欧美乱码精品一区二区三区| 蜜桃在线观看..| 欧美黄色淫秽网站| 搡老岳熟女国产| 一级毛片电影观看| 人人妻人人爽人人添夜夜欢视频| √禁漫天堂资源中文www| 欧美老熟妇乱子伦牲交| 高潮久久久久久久久久久不卡| 久久久久精品人妻al黑| 高潮久久久久久久久久久不卡| 精品乱码久久久久久99久播| 午夜激情久久久久久久| 80岁老熟妇乱子伦牲交| 亚洲av电影在线观看一区二区三区| 午夜福利一区二区在线看| 黄色a级毛片大全视频| 久久久久精品人妻al黑| 热re99久久国产66热| 日本a在线网址| 建设人人有责人人尽责人人享有的| 如日韩欧美国产精品一区二区三区| 亚洲精品中文字幕在线视频| 免费高清在线观看视频在线观看| 久久久久久久大尺度免费视频| 日韩大片免费观看网站| 天天影视国产精品| 亚洲免费av在线视频| 啦啦啦免费观看视频1| 欧美另类一区| 美女高潮喷水抽搐中文字幕| 国产精品一区二区在线不卡| 成人亚洲精品一区在线观看| 黑人操中国人逼视频| 中文字幕人妻丝袜制服| 国产精品免费视频内射| bbb黄色大片| 美女中出高潮动态图| 黄色怎么调成土黄色| 黑丝袜美女国产一区| 男人爽女人下面视频在线观看| 中亚洲国语对白在线视频| 亚洲精品国产av蜜桃| 日韩制服骚丝袜av| 丝袜喷水一区| 亚洲av电影在线观看一区二区三区| 亚洲男人天堂网一区| 精品卡一卡二卡四卡免费| 精品福利永久在线观看| 亚洲精品国产区一区二| 日韩一卡2卡3卡4卡2021年| 丁香六月天网|