• <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 ?

    Differences between two methods to derive a nonlinear Schr?dinger equation and their application scopes

    2024-02-29 09:19:02YuXiChen陳羽西HengZhang張恒andWenShanDuan段文山
    Chinese Physics B 2024年2期
    關(guān)鍵詞:張恒文山

    Yu-Xi Chen(陳羽西), Heng Zhang(張恒), and Wen-Shan Duan(段文山)

    College of Physics and Electronic Engineering,Northwest Normal University,Lanzhou 730070,China

    Keywords: dusty plasmas,nonlinear waves,particle-in-cell simulation

    1.Introduction

    Wave phenomena are a very common physical phenomenon that exist in many different systems, ranging from microscopic to macroscopic scales, from molecules to astronomical bodies.[1–6]Typical examples include plasma waves,[7–23]optical communications,[24–27]collective motion of particles in granular matter,[28,29]optical and acoustic wave phenomena in Bose–Einstein condensates,[30–36]etc.

    Wave phenomena can be divided into two types: linear waves and nonlinear waves.The nonlinear waves usually can be approximately described by KdV equation[37–43]and NLSE.[44–50]These two equations can describe many different types of nonlinear wave phenomena.

    It is worth noting that there are multiple methods available to obtain NLSE, such as the reductive perturbation method,[51–53]Krylov–Bogoliubov–Mitropolsky (KBM)method,[54–56]canonical transformation method,[57]inverse scattering transform method,[58]and so on.

    The present paper is focused on two different methods to obtain the NLSE:One is to indirectly derive the NLSE,which is first to derive a KdV equation and then derive the NLSE step by step from the KdV equation,[59–64]while the other is to directly derive the NLSE from the original equation.[44–50,65,66]Although both methods can describe nonlinear waves, it is currently unknown whether the nonlinear waves described by these two methods are the same and which method is more accurate.Therefore,this is a problem that requires further study.

    We investigated this problem based on dusty plasma and obtained the following results: First, the dispersion relation and group velocity of the envelope waves obtained by the two methods are different.Second,we used PIC numerical simulation method[67–74]to verify the two methods and found that both methods are correct for small amplitude envelope waves.Third, we investigated the dependence of wave amplitude on the perturbation parameterεε′for the first method and the dependence of wave amplitude on the perturbation parameterεfor the second method.The perturbation parameterε′actually stands for the quantity of ?fd/fd0,where ?fdrepresents the perturbed amplitude of the perturbations, whilefd0represent its corresponding amplitude at equilibrium state.For small amplitude, ?fd/fd0is small enough, i.e.,ε′is small enough.The parameterεactually stands for the quantity ofλD/λ,whereλDrepresent the Debye length,whileλrepresents the wavelength of the envelope waves.For long wave length approximation,εis small enough.[70,73–76]Our results show that as the perturbation parameters increases,the envelope waves amplitude gradually increases.In addition, we found that as the envelope wave amplitude increases,the deviation between numerical and analytical results gradually increases.We determined the applicable scope of each method based on the deviation between numerical and analytical results and found that the method of directly deriving NLSE from the original equations has a wider applicable scope than that of step-by-step deriving NLSE from KdV equation.Lastly,we investigated the dependence of envelope width on wave amplitude numerically and analytically.

    2.Theoretical model

    We now study dust acoustic waves in two-temperatureion dusty plasma which contain negatively charged dust particles, free electrons, and two different kinds of free ions.One kind of ions is high-temperature, while the other is lowtemperature.Charge neutrality condition at equilibrium isnil0+nih0=Zd0nd0+ne0, wherenα0is the number density of unperturbed particles of speciesα.α=il,ih,e,and d represent the low-temperature ion, the high-temperature ion, the free electron,and the dust particles,respectively.Zd0is the unperturbed number of charges residing on the dust grain measured in the unit of electron charge.Suppose that the dusty plasma is unmagnetized and collisionless.We now consider one-dimensional dust acoustic wave propagating in thexdirection.The dimensionless equations of motion of the system can be given as follows:[77]

    wherendandudrefer to the number density and the velocity of dust fluid.φis the electrostatic potential.γ′=γTd/Zd0Teff.ne=νesβ1φ,nil=μle-sφ, andnih=μhe-sβ2φare the number densities of electrons,lower temperature ions,and higher temperature ions respectively,ν=ne0/(Zd0nd0),μl =nil0/(Zd0nd0),μh=nih0/(Zd0nd0),β1=Til/Te,β2=Til/Tih, ands= 1/(vβ1+μ1+μhβ2).Te,Til,Tih, andTdare the temperatures of the electrons, the low temperature ions,the high temperature ions,and the dust particles respectively.Teffis the effective temperature which satisfy 1/Teff=(ne0/Te+nil0/Til+nih0/Tih)/Zd0nd0.

    All physical quantities in Eqs.(1)–(3) are normalized ones.They are normalized as follows:ndis normalized bynd0,nil,nih, andneare all byZd0nd0,ZdbyZd0,xby the Debye lengthλD=(kBε0Teff/nd0Zd0e2)2,tby the inverse of dust plasma frequencyω-1=(ε0md/nd0Zd02e2)1/2,udby the dust-acoustic speedcd=(kBzd0Teff/md)1/2,φbykBTeff/e,kBis boltzmann constant,ε0is vacuum permittivity.[77]

    3.Derivation of NLSE for the system

    To study nonlinear waves in a nonlinear system, two main methods are commonly used.One is to derive a KdV equation for localized waves using the reductive perturbation technique in the limit of small but finite amplitude and long wavelength.[37–43]The other method is to derive an NLSE when there are background waves present.[44–50]

    There are also two different methods to derive an NLSE.One method involves first deriving a KdV equation and then obtaining an NLSE from the KdV equation.[59–64]The other method is to derive the NLSE from the original equation of motion,[44–50,65,66]for example, by obtaining an NLSE from hydrodynamical equations.A fundamental question is whether the NLSE and its solutions obtained from these two different methods are the same, and which one is more accurate.This paper aims to address these questions.

    3.1.Derivation of NLSE from the CKdV equation

    In this section, we will derive the NLSE from a coupled KdV (CKdV) equation for this system.Then we will compare its solutions with numerical results.Additionally,we will discuss the application scope of the analytical solution.

    First, we derive a CKdV equation.By using the following expansionsξ=ε′(x-v0t),τ=ε′3t,nd=1+ε′2nd1+ε′4nd2+···,ud=ε′2ud1+ε′4ud2+···,andφ=ε′2φ1+ε′4φ2+···,whereε′is a small parameter,v0is the velocity of the dust acoustic KdV solitary waves.We then can obtainnd1=-φ1,ud1=-v0φ1,v02=1+γ′,and the KdV equation

    Generally,we can use the KdV equation to approximately describe nonlinear waves in dusty plasmas.However, when the coefficients in the KdV equation are zero, the equation is no longer applicable.To address this specific condition of nonlinear wave behavior, we derived a new equation called the CKdV equation.The CKdV equation is an extension of the KdV equation that overcomes the limitation of zero coefficients.These coefficients can be tuned according to specific physical conditions, allowing the CKdV equation to more accurately describe the nonlinear wave in dusty plasmas,including phenomena that cannot be covered by the KdV equation.[77]whereX=ε(ξ-Vτ),T=ε2τ,Vis the group velocity.Substituting Eq.(6)into the CKdV equation of Eq.(5),we obtain the dispersion relationω=-Bk3,group velocityV=-3Bk2,φ(1,0)=0,φ(1,l)=0 for|l|>1,φ(2,2)=(A/6Bk2)[φ(1,1)]2,φ(2,0)=(A/V)|φ(1,1)|2and NLSE as follows:

    whereP=-3Bk,Q=(A2/6Bk)-Ck.For an NLSE, whenPQ>0, the equation possesses bright envelope soliton solutions.Conversely,whenPQ<0,the equation possesses dark envelope soliton solutions.

    In the present paper,we only consider the case ofPQ>0,i.e.,bright envelope soliton solutions.For this case,there are modulation instability which has been well studied in the previous investigations.[78–81]Therefore,in this paper,we do not do further research on modulation instability.Then,the envelope wave solution of Eq.(7)is as follows:

    3.2.Derivation of NLSE from original hydrodynamical equations

    In this section, we will use a different method for deriving the NLSE from the hydrodynamical equations given in Eqs.(1)–(3).We will also compare the numerical results with the analytical ones.Moreover, we will infer the application scope of the analytical solution.Later,we will also provide a comparison between the two methods employed for deriving the NLSE.

    We use the following transformations

    whereξ=ε(x-Vt),τ=ε2t,Vis the group velocity.Substituting these expansions into Eqs.(1)–(3), we haven(1,1)=-(k2/ω2)φ(1,1),u(1,1)=-(k/ω)φ(1,1),and the dispersion relationω2=k2/(k2+1),n(1,l)=u(1,l)=φ(1,l)=0 when|l|>1,n(1,0)=u(1,0)=φ(1,0)=0,the group velocity

    It is noted from Eq.(14)that the group velocity and phase velocity expressions in the laboratory coordinates are as follows:VG=V+cgε,VP=(ω+εKV+ε2?)/(k+εK).Relations among physical quantities aren=1-(k2/ω2)φ(x,t),u=-(k/ω)φ(x,t).

    4.Comparisons between two different envelope waves obtained from two different methods

    We have obtained two different expressions for envelope nonlinear waves, given by Eqs.(9) and (14), which were obtained from two different methods.One method involved first deriving a CKdV equation and then obtaining the NLSE,while the other method involved directly obtaining the NLSE from the hydrodynamical equations.

    In this section, we will compare the differences between the two solutions and identify the differences.Additionally,we will determine the application scope of the two methods.

    4.1.Comparisons the dispersion relation and the group velocity of the two methods

    Firstly,in order to understand the differences in the envelope waves obtained from the two methods,we compared the dispersion relation and the group velocity of the two methods in Figs.1(a)and 1(b),respectively.

    Fig.1.(a) The dispersion relation obtained from two different methods.(b) The group velocity of the envelope wave obtained from two methods.

    As shown in Fig.1,two different methods have significant different behaviors for dispersion relation and group velocity.The wave frequency of the envelope wave obtained by the first method is smaller than that obtained by the second method for the same wave number.Moreover, the frequencies increase with the increase of the wave number.The group velocity of the envelope wave obtained by the first method is larger than that obtained by the second method for the same wave number.These results suggest that the envelope wave obtained by the two methods are significantly different.The purpose of the present paper is to verify the correctness of these two different methods.If both the two methods are correct, we will further compare the two methods and determine their respective application scope.

    4.2.Comparisons the numerical results and the analytical result between two methods

    In this section, we aim to compare the differences between the two solutions using the PIC numerical method,and then determine the application scope of each solution.

    4.2.1.Particle-in-cell method

    Particle-in-cell (PIC) simulation is commonly used for numerical work in plasma physics and particle dynamics because it provides an effective way to model the behavior of charged particles in a self-consistent electromagnetic field.PIC simulations are particularly well-suited for studying phenomena such as plasma waves,particle acceleration,and interactions between particles and fields.So, we use PIC method to simulate the envelope waves in this paper.

    During the simulation process, the dust grains are represented by a limited number of “super-particles” (SPs), while both electrons and ions are treated as Boltzmann-distributed fluids.Each SP is assigned a weight factor, denoted byS,which represents the number of real particles it represents.Initially, the SPs are uniformly distributed in the simulation space,and their initial weight parameters S and velocities are determined from the initial conditions.[69,70]

    To carry out the simulation,the simulation region is partitioned into several grid cells using the PIC method.As the dust particles move along their trajectories,they constantly exchange information with the background grid.At each time step,the positions and velocities of the SPs are weighted to all the grids,allowing for the calculation of the charge densityρg(or electric current densityJg).Once the charge densityρgis obtained, numerical solutions of either Maxwell’s equations(for the electromagnetic model) or the Poisson–Boltzmann equation (for the electrostatic model) are used to derive the electric field at each grid.In the electrostatic model,the magnetic field is assumed to be zero.[71–73]

    Subsequently,the electric field imposed on each SP is determined, driving each SP according to Newton’s equation.This equation can be numerically solved using the leap-frog algorithm.Finally,the new positions and velocities of each SP are determined,and the simulation process repeats until completion.The summary of a computational cycle of the PIC method is shown in Fig.2.[71,72]

    Fig.2.The summary of a computational cycle of the PIC method.

    4.2.2.The initial conditions for the numerical simulation of the envelope wave

    The simulation parameters we use are as follows:the spatial step is ?x=0.5, the time step is ?t=0.01, the number of grid cells isNx= 30000, the number of super particles contained in per cell is 100, the total length of thexaxis isLx=?xNx,x0=3000.We choose periodic boundary conditions.The other parameters areTe= 5 eV,Til= 0.3 eV,Tih=5 eV,Td=200 K,Zd0=1000,nd0=1.0×1012m-3,nil0=1.0×1014m-3,nih0=1.0×1015m-3,k=0.1,K=0.1,a=1.

    The initial conditions of the first method are given from Eq.(9)as follows:

    The initial conditions of the second method are given from Eq.(14)as follows:

    4.2.3.Comparisons between two methods

    Figure 3 displays the envelope wave obtained by the first method at different timetthrough PIC numerical simulations.The numerical results exhibit excellent agreement with the analytical solutions derived from Eq.(9),as shown in Fig.4.

    Fig.3.The PIC simulation results of the first method at different time t=0,t=2555.59/ω-1,t=5111.18/ω-1,where εε′=0.01,A=0.37.

    Likewise,the numerical results of the envelope wave obtained by the second method at different timetis shown in Fig.5.The numerical results obtained through PIC simulations also show good agreement with the analytical solutions derived from Eq.(14),as depicted in Fig.6.

    Fig.4.The comparisons between PIC simulation results and the analytical ones of the first method at different time t =0,t =2555.59/ω-1,t=5111.18/ω-1,where εε′=0.01,A=0.37.

    It is evident from Figs.3 and 4 that the analytical result of the envelope wave obtained by the first method seems valid,as the numerical results show that it can propagate stably.This suggests that the NLSE obtained by the first method is correct.Similarly,figures 5 and 6 indicate that the analytical result of the envelope wave obtained by the second method is also valid,and therefore,the NLSE obtained by the second method is correct as well.

    While figures 4 and 6 show good agreement between the numerical and analytical results for both methods, it is unclear if this agreement holds for larger amplitude (φm) envelope waves.In order to understand it,we compare the numerical results with that of the analytical ones for larger amplitude waves for both cases by varying the parameter ofεsinceεindirectly stands for the wave amplitude (It is noted from Eqs.(9) and (11) that the amplitudes of the envelope waves contain the parameter ofε), while keeping the other parameters constants.

    Fig.5.The PIC simulation results of the second method at different time t=0,t=2537.59/ω-1,t=5075.19/ω-1,where ε =0.01.

    Fig.6.The comparisons between PIC simulation results and the analytical ones of the second method at different time t=0,t=2537.59/ω-1,t=5075.19/ω-1,where ε =0.01.

    Fig.7.The comparisons between PIC simulation results and the analytical ones at different time t,where φm=0.005.(a)The result of the first method.(b)The result of the second method.

    The waveforms for the numerical and analytical results of the different amplitude envelope waves at different time for two different methods are shown in Figs.7 (φm=0.005), 8(φm=0.015), and 9 (φm=0.025).By examining these figures, we can determine whether the numerical and analytical results are in good agreement for larger amplitude envelope waves.Notice from Figs.7 and 8 that both the numerical and analytical results are in good agreement for larger amplitude waves.However,in Fig.9,the difference between the numerical and analytical results is more obvious for both methods.It appears that as the amplitude of the envelope wave increases the differences between the numerical results and analytical ones become more pronounced.

    To gain further insight into the differences between the PIC numerical results and the analytical ones for both methods,we present the dependence of the wave amplitude(φm)onεε′orεfor the two cases in Fig.11.It can be observed from Fig.11,that the amplitude of the envelope wave increases with increasing ofεε′orε.Asεε′orεincreases,i.e.,wave amplitude increases, the differences between the analytical results and the numerical ones obtained from both methods become larger.In other word, the analytical results are valid if the amplitude is small enough.Based on these results, the application scope of both methods can be determined from Fig.10.Specifically,the first method can be applied whenφm<0.01,while the second method can be applied whenφm<0.015.

    Fig.8.The comparisons between PIC simulation results and the analytical ones at different time t,where φm=0.015.(a)The result of the first method.(b)The result of the second method.

    Fig.9.The comparisons between PIC simulation results and the analytical ones at different time t,where φm=0.025.(a)The result of the first method.(b)The result of the second method.

    Fig.10.Comparisons between the numerical results and the analytical ones.(a)The dependence of the wave amplitude(φm)on the parameter εε′ for the first methods.(b) The dependence of the wave amplitude(φm)on the parameter ε for the second methods.

    Fig.11.Comparisons between the numerical results and the analytical ones of the dependence of the width of envelope waves on the wave amplitude: (a)for the first method and(b)for the second method.

    To further understand the differences between the two methods, the dependence of the width of the envelope waves(W)on the wave amplitude(φm)is shown numerically and analytically in Fig.11.It seems from Fig.11 that the deviation of the wave width between the analytical results and the numerical ones of the first method is obvious when the wave amplitude is larger than 0.01, while the deviation of the second method is still not significant even when the wave amplitude is around 0.02.Therefore, the application scope of the envelope wave obtained from the second method is wider than that from the first method.

    5.Discussion and conclusion

    The present paper chooses a dusty plasma as an example to numerically and analytically study the differences between two different methods of obtaining NLSE.It is found that the envelope waves from the two methods have different dispersion relations, different group velocities.Specifically,the wave frequency of the first method is lower than that of the second one for the same wave number,while the group velocity of the first method is larger than that of the second method for the same wave number.It is noted that the two methods are completely different.

    Additionally, the application scopes of two different methods are shown.The results indicate that the application scope of the envelope wave obtained from the second method is wider than that of the first method.It is suggest that both methods to derive NLSE are correct in the regime of their application scope.In other words, if the amplitude of the envelope solitary wave is smaller that a critical values(the critical values are different for two different methods), the analytical results are valuable to describe the real solutions of the envelopes waves.If the amplitude of the envelope solitary wave is larger than this critical value,the neglected higher order terms in deriving the NLSE play an important role which should not be neglected.

    In conclusion,although both methods are valuable within the range of their respective application scopes,the two envelope wave solutions obtained from the two different methods are completely different.For other systems,both methods may also be used to derive the NLSE and obtain an envelope wave or other nonlinear waves such as Rogue waves,but their solutions are possibly different.

    Acknowledgements

    Project supported by the National Natural Science Foundation of China (Grant Nos.11965019 and 42004131) and the Foundation of Gansu Educational Committee (Grant No.2022QB-178).

    猜你喜歡
    張恒文山
    詩與象
    保證書
    詩與學(xué)
    Investigation of the confinement of high energy non-neutral proton beam in a bent magnetic mirror
    Penguins Are in Danger
    Particle-in-cell simulation of ion-acoustic solitary waves in a bounded plasma?
    文竹
    文山肉丁
    幼兒100(2018年32期)2018-12-05 05:24:26
    山歌唱文山
    民族音樂(2017年6期)2017-04-19 02:18:19
    霧和霾的十大區(qū)別
    地理教育(2015年12期)2015-12-07 11:58:30
    少妇高潮的动态图| 91av网一区二区| 波多野结衣高清作品| 亚洲国产精品成人综合色| 乱系列少妇在线播放| 精品无人区乱码1区二区| 精品99又大又爽又粗少妇毛片| 亚洲天堂国产精品一区在线| 欧美激情在线99| 亚洲精品一区av在线观看| 国产精品一区二区性色av| 伦理电影大哥的女人| 插逼视频在线观看| 老熟妇乱子伦视频在线观看| 亚洲性久久影院| 人人妻人人澡人人爽人人夜夜 | 亚洲最大成人手机在线| 18禁裸乳无遮挡免费网站照片| 99热只有精品国产| 成人特级黄色片久久久久久久| 久久久久国产精品人妻aⅴ院| 中文字幕免费在线视频6| 国产日本99.免费观看| 三级毛片av免费| 午夜激情欧美在线| 亚洲高清免费不卡视频| 亚洲精品粉嫩美女一区| 精品久久国产蜜桃| 午夜福利在线观看免费完整高清在 | 午夜影院日韩av| 亚洲中文字幕日韩| 1000部很黄的大片| 国产成人精品久久久久久| 三级国产精品欧美在线观看| av.在线天堂| 亚洲人成网站在线播放欧美日韩| av天堂中文字幕网| 免费无遮挡裸体视频| 寂寞人妻少妇视频99o| 波野结衣二区三区在线| 色综合色国产| 特级一级黄色大片| 国产一区二区三区av在线 | 国产美女午夜福利| 天堂av国产一区二区熟女人妻| 国产午夜福利久久久久久| 国产私拍福利视频在线观看| 午夜免费激情av| 亚洲国产精品国产精品| 国产老妇女一区| 色噜噜av男人的天堂激情| 成人午夜高清在线视频| 色av中文字幕| 白带黄色成豆腐渣| 久久精品国产亚洲av香蕉五月| 色播亚洲综合网| 特大巨黑吊av在线直播| 少妇高潮的动态图| 国产精品久久久久久精品电影| 亚洲五月天丁香| 性欧美人与动物交配| 国产亚洲精品av在线| 中文字幕av成人在线电影| 日本 av在线| 国产欧美日韩精品一区二区| 国产91av在线免费观看| 国产精品伦人一区二区| 日本在线视频免费播放| 国产aⅴ精品一区二区三区波| 国产精品一区二区免费欧美| 亚洲人成网站在线观看播放| 尾随美女入室| 久久久久免费精品人妻一区二区| 中文字幕av在线有码专区| 亚洲欧美日韩东京热| 国产毛片a区久久久久| 伦理电影大哥的女人| 夜夜看夜夜爽夜夜摸| 日本-黄色视频高清免费观看| 在线观看66精品国产| 欧美区成人在线视频| 精品一区二区免费观看| 欧美xxxx黑人xx丫x性爽| 色尼玛亚洲综合影院| av福利片在线观看| 免费不卡的大黄色大毛片视频在线观看 | 在线看三级毛片| 午夜日韩欧美国产| 精品人妻视频免费看| 在线观看美女被高潮喷水网站| 久久99热6这里只有精品| 成人永久免费在线观看视频| 少妇人妻一区二区三区视频| 少妇猛男粗大的猛烈进出视频 | 伦理电影大哥的女人| 日韩制服骚丝袜av| 国产精华一区二区三区| 免费人成在线观看视频色| 中文字幕av成人在线电影| 最近视频中文字幕2019在线8| 精品无人区乱码1区二区| 欧美+日韩+精品| 精品人妻视频免费看| 69av精品久久久久久| 中文字幕免费在线视频6| 欧美最新免费一区二区三区| 亚洲av美国av| 国产精品一区www在线观看| 国产av麻豆久久久久久久| 日韩av不卡免费在线播放| 久久久精品94久久精品| 亚洲一区高清亚洲精品| 亚洲专区国产一区二区| 亚洲自偷自拍三级| 国产成人aa在线观看| 国产在线男女| 亚洲美女黄片视频| 俄罗斯特黄特色一大片| 久久草成人影院| 亚洲最大成人中文| 日韩欧美三级三区| 99热全是精品| 黄色一级大片看看| 99热网站在线观看| 国产真实乱freesex| 国产精品野战在线观看| 日产精品乱码卡一卡2卡三| 亚洲性夜色夜夜综合| 国产v大片淫在线免费观看| 国产精品一区二区三区四区免费观看 | 丝袜喷水一区| 国产av一区在线观看免费| 亚洲美女视频黄频| 日韩制服骚丝袜av| a级毛片a级免费在线| 国产精品三级大全| 听说在线观看完整版免费高清| 日日干狠狠操夜夜爽| 美女cb高潮喷水在线观看| 中出人妻视频一区二区| 亚洲av电影不卡..在线观看| 久久久精品大字幕| 欧美人与善性xxx| 黄色一级大片看看| 国产亚洲精品久久久久久毛片| 亚洲美女黄片视频| 国产欧美日韩精品亚洲av| 亚洲欧美日韩卡通动漫| 麻豆国产av国片精品| 国产色爽女视频免费观看| 日韩一本色道免费dvd| 综合色丁香网| 国产精品一区二区免费欧美| 久久人人精品亚洲av| 欧美成人a在线观看| 亚洲在线自拍视频| 日产精品乱码卡一卡2卡三| 69av精品久久久久久| 国产高清三级在线| 午夜影院日韩av| 99国产精品一区二区蜜桃av| 亚洲精品456在线播放app| 国产精品99久久久久久久久| 人人妻人人澡欧美一区二区| 深爱激情五月婷婷| 国产中年淑女户外野战色| 国产精品不卡视频一区二区| 亚洲av中文av极速乱| 国产精品一区二区在线不卡| 久久久久国产精品人妻一区二区| 99热这里只有精品一区| 国产精品麻豆人妻色哟哟久久| 欧美高清成人免费视频www| 日本免费在线观看一区| 欧美日韩视频精品一区| 亚洲一级一片aⅴ在线观看| 成人影院久久| 国产精品久久久久成人av| 啦啦啦啦在线视频资源| tube8黄色片| 中文资源天堂在线| 国产精品久久久久久久电影| 精品久久久久久电影网| 亚洲国产色片| 欧美日本中文国产一区发布| 日韩三级伦理在线观看| 超碰97精品在线观看| 国产熟女欧美一区二区| 青春草亚洲视频在线观看| 免费黄网站久久成人精品| 免费观看在线日韩| 天美传媒精品一区二区| 亚洲第一av免费看| av在线观看视频网站免费| 91精品伊人久久大香线蕉| 精品一品国产午夜福利视频| 精品国产一区二区久久| 国产乱来视频区| 99精国产麻豆久久婷婷| 在线观看国产h片| 自线自在国产av| 国产免费一级a男人的天堂| 国产精品女同一区二区软件| 成人美女网站在线观看视频| 精品国产乱码久久久久久小说| 精品国产国语对白av| 人妻 亚洲 视频| 亚洲av成人精品一二三区| 亚洲欧美一区二区三区国产| 亚洲国产日韩一区二区| 一边亲一边摸免费视频| 亚洲av不卡在线观看| 制服丝袜香蕉在线| 久久人人爽av亚洲精品天堂| 观看免费一级毛片| 如日韩欧美国产精品一区二区三区 | 日韩伦理黄色片| 亚洲国产精品成人久久小说| 伦理电影大哥的女人| 国产亚洲午夜精品一区二区久久| 内地一区二区视频在线| 亚洲av欧美aⅴ国产| 五月玫瑰六月丁香| 婷婷色av中文字幕| 老司机影院毛片| a级毛片免费高清观看在线播放| av线在线观看网站| 国产精品无大码| 国产精品久久久久成人av| 99re6热这里在线精品视频| 亚洲av男天堂| 一级av片app| 欧美三级亚洲精品| 亚洲色图综合在线观看| 日韩欧美一区视频在线观看 | 国产亚洲av片在线观看秒播厂| av在线观看视频网站免费| 最近手机中文字幕大全| 久久久久国产网址| 久久ye,这里只有精品| 久久精品国产鲁丝片午夜精品| 只有这里有精品99| 下体分泌物呈黄色| 黄色日韩在线| 久久久久久久精品精品| 成人午夜精彩视频在线观看| 一本—道久久a久久精品蜜桃钙片| 国产精品一二三区在线看| 成年美女黄网站色视频大全免费 | 中文字幕免费在线视频6| 日韩一本色道免费dvd| 最后的刺客免费高清国语| 久久久久久久久久成人| 亚洲欧洲日产国产| 丁香六月天网| 国内揄拍国产精品人妻在线| 国产亚洲一区二区精品| 天天躁夜夜躁狠狠久久av| 精品国产国语对白av| 麻豆成人av视频| 啦啦啦在线观看免费高清www| 亚洲国产成人一精品久久久| 国产精品福利在线免费观看| 老司机影院成人| 欧美激情极品国产一区二区三区 | 黑人高潮一二区| 久热久热在线精品观看| 只有这里有精品99| 51国产日韩欧美| 欧美日韩一区二区视频在线观看视频在线| 免费观看在线日韩| 日产精品乱码卡一卡2卡三| 国产亚洲午夜精品一区二区久久| 亚洲欧美一区二区三区国产| 91精品一卡2卡3卡4卡| 亚洲精品久久久久久婷婷小说| 国产高清国产精品国产三级| 亚洲欧美日韩另类电影网站| 久久毛片免费看一区二区三区| 夜夜看夜夜爽夜夜摸| 国产亚洲91精品色在线| a 毛片基地| 精品亚洲成a人片在线观看| 亚洲av日韩在线播放| 九色成人免费人妻av| 五月天丁香电影| 精品视频人人做人人爽| 精品国产一区二区三区久久久樱花| 视频区图区小说| 精品卡一卡二卡四卡免费| 内地一区二区视频在线| 高清午夜精品一区二区三区| 一本色道久久久久久精品综合| 欧美日韩av久久| 色5月婷婷丁香| 有码 亚洲区| 国产色爽女视频免费观看| 日韩不卡一区二区三区视频在线| 久久久久久久大尺度免费视频| 日韩av不卡免费在线播放| 一本大道久久a久久精品| 三级经典国产精品| 久久久久人妻精品一区果冻| 精品视频人人做人人爽| 免费不卡的大黄色大毛片视频在线观看| 狠狠精品人妻久久久久久综合| 亚洲国产精品一区二区三区在线| 春色校园在线视频观看| 女性生殖器流出的白浆| 久久 成人 亚洲| a级片在线免费高清观看视频| 各种免费的搞黄视频| 久久久久久久国产电影| 91aial.com中文字幕在线观看| 欧美 亚洲 国产 日韩一| 黄色日韩在线| 日日爽夜夜爽网站| 九草在线视频观看| 亚洲欧美日韩卡通动漫| 看免费成人av毛片| 黄色怎么调成土黄色| 久热这里只有精品99| 亚洲综合色惰| 韩国高清视频一区二区三区| av有码第一页| 日本猛色少妇xxxxx猛交久久| 在线亚洲精品国产二区图片欧美 | 久久人妻熟女aⅴ| 久久影院123| 欧美成人精品欧美一级黄| 日本黄色日本黄色录像| 婷婷色综合大香蕉| 大片免费播放器 马上看| 国产免费又黄又爽又色| 一级,二级,三级黄色视频| 国产日韩一区二区三区精品不卡 | 国产 精品1| 日日摸夜夜添夜夜添av毛片| 精华霜和精华液先用哪个| 日日爽夜夜爽网站| 91成人精品电影| 麻豆成人av视频| 日本vs欧美在线观看视频 | 在线天堂最新版资源| www.色视频.com| 久久精品夜色国产| 午夜福利,免费看| 久久人人爽人人爽人人片va| 成年av动漫网址| freevideosex欧美| 性色avwww在线观看| 成人美女网站在线观看视频| 国产免费又黄又爽又色| 蜜臀久久99精品久久宅男| 日韩欧美精品免费久久| 丁香六月天网| 九九久久精品国产亚洲av麻豆| 六月丁香七月| 国产一区二区三区综合在线观看 | 伦理电影大哥的女人| 国产精品一区二区在线不卡| 中文字幕久久专区| 又大又黄又爽视频免费| 黄色视频在线播放观看不卡| 看十八女毛片水多多多| 精品久久国产蜜桃| 精品亚洲成国产av| 久久精品国产自在天天线| 国产69精品久久久久777片| 欧美精品一区二区大全| 国产成人a∨麻豆精品| 亚洲在久久综合| 免费av中文字幕在线| 欧美一级a爱片免费观看看| 精品国产一区二区久久| 91精品国产九色| 最新的欧美精品一区二区| 亚洲欧美中文字幕日韩二区| 熟女电影av网| 亚洲成人手机| a级一级毛片免费在线观看| 一本—道久久a久久精品蜜桃钙片| 婷婷色av中文字幕| 国产免费一级a男人的天堂| 久久99热这里只频精品6学生| 久久久久久久亚洲中文字幕| 亚洲精品乱码久久久久久按摩| 久久久久久久精品精品| 久久久a久久爽久久v久久| 欧美日韩在线观看h| 亚洲va在线va天堂va国产| 亚洲国产精品999| 国产欧美另类精品又又久久亚洲欧美| 免费在线观看成人毛片| 国产欧美日韩精品一区二区| 视频区图区小说| 亚洲精品第二区| 欧美最新免费一区二区三区| 久久午夜综合久久蜜桃| 一个人免费看片子| 久久99精品国语久久久| 亚洲欧美清纯卡通| 伊人久久精品亚洲午夜| 自拍欧美九色日韩亚洲蝌蚪91 | 欧美少妇被猛烈插入视频| 18+在线观看网站| 日韩强制内射视频| 精品人妻熟女毛片av久久网站| 一区二区三区四区激情视频| 国产男女超爽视频在线观看| 一本一本综合久久| 男人舔奶头视频| 亚洲精品成人av观看孕妇| 一区二区三区精品91| 亚洲精品456在线播放app| 国产欧美日韩综合在线一区二区 | 国产精品成人在线| 亚洲人成网站在线播| 少妇丰满av| 蜜桃久久精品国产亚洲av| 久久久久久伊人网av| 国产日韩欧美在线精品| 黑人猛操日本美女一级片| 亚洲国产精品999| 亚洲精品乱久久久久久| 精品久久久久久电影网| 久久99热6这里只有精品| 我的老师免费观看完整版| 欧美性感艳星| 男人添女人高潮全过程视频| 久久久久精品久久久久真实原创| 人妻少妇偷人精品九色| 国产精品一区二区三区四区免费观看| 成人国产av品久久久| 免费av不卡在线播放| a级毛片免费高清观看在线播放| 免费少妇av软件| 观看av在线不卡| 高清欧美精品videossex| 久久av网站| 日本猛色少妇xxxxx猛交久久| 乱人伦中国视频| 中文欧美无线码| 曰老女人黄片| .国产精品久久| 国产成人精品婷婷| 乱人伦中国视频| 国产午夜精品久久久久久一区二区三区| 一本久久精品| 看免费成人av毛片| 男女国产视频网站| 精品99又大又爽又粗少妇毛片| 中文字幕人妻丝袜制服| 91精品伊人久久大香线蕉| 精品少妇黑人巨大在线播放| 97在线视频观看| 一级爰片在线观看| 99久久精品热视频| 国产精品伦人一区二区| 只有这里有精品99| 人妻系列 视频| 午夜日本视频在线| 精品人妻熟女av久视频| 国产成人精品无人区| 亚洲精品国产av成人精品| 色网站视频免费| 亚洲高清免费不卡视频| 亚洲中文av在线| 精品久久久久久久久av| 免费少妇av软件| 桃花免费在线播放| 国产深夜福利视频在线观看| 日韩中字成人| 久久精品久久久久久久性| 亚洲精品亚洲一区二区| 国产精品99久久99久久久不卡 | 91在线精品国自产拍蜜月| 成人综合一区亚洲| 黑人高潮一二区| 亚洲三级黄色毛片| 中文精品一卡2卡3卡4更新| 国产老妇伦熟女老妇高清| 亚洲成人av在线免费| 精品久久久精品久久久| 久久国产精品大桥未久av | 国产精品一区二区三区四区免费观看| 一区二区三区四区激情视频| 欧美+日韩+精品| 永久免费av网站大全| 观看美女的网站| 黄色一级大片看看| 国产精品久久久久久av不卡| 亚洲av二区三区四区| 18禁在线无遮挡免费观看视频| 岛国毛片在线播放| 少妇裸体淫交视频免费看高清| 久久亚洲国产成人精品v| 成人特级av手机在线观看| 赤兔流量卡办理| av一本久久久久| 80岁老熟妇乱子伦牲交| 日本猛色少妇xxxxx猛交久久| 日韩电影二区| 精品久久久久久久久av| 青春草视频在线免费观看| 国产国拍精品亚洲av在线观看| 又爽又黄a免费视频| 亚洲av.av天堂| av黄色大香蕉| 久久久久久久久大av| 99久久人妻综合| 日本vs欧美在线观看视频 | 成人国产av品久久久| 日本wwww免费看| 亚洲国产毛片av蜜桃av| 一级毛片久久久久久久久女| 全区人妻精品视频| 麻豆乱淫一区二区| 日日撸夜夜添| 日韩成人伦理影院| 99九九在线精品视频 | 成年av动漫网址| 日本猛色少妇xxxxx猛交久久| 日韩,欧美,国产一区二区三区| 亚洲精品国产av成人精品| 一区二区三区精品91| 极品少妇高潮喷水抽搐| 青春草亚洲视频在线观看| 最近中文字幕2019免费版| 国产黄色视频一区二区在线观看| 黄片无遮挡物在线观看| 欧美精品高潮呻吟av久久| 少妇裸体淫交视频免费看高清| 亚洲电影在线观看av| 欧美丝袜亚洲另类| 国产欧美另类精品又又久久亚洲欧美| 99re6热这里在线精品视频| 大香蕉97超碰在线| 国产午夜精品久久久久久一区二区三区| 日本91视频免费播放| 国产精品女同一区二区软件| 欧美三级亚洲精品| 91精品国产九色| 又粗又硬又长又爽又黄的视频| 日韩,欧美,国产一区二区三区| 成年人午夜在线观看视频| 如日韩欧美国产精品一区二区三区 | 国产精品国产三级国产专区5o| 亚洲精品日韩av片在线观看| 日本欧美视频一区| 亚洲,一卡二卡三卡| 亚洲av欧美aⅴ国产| 内地一区二区视频在线| 亚洲精品国产成人久久av| 免费av不卡在线播放| 久久精品国产亚洲网站| 国产极品粉嫩免费观看在线 | 国产精品一区二区三区四区免费观看| 黄色视频在线播放观看不卡| 国产伦精品一区二区三区视频9| 免费看不卡的av| 一本久久精品| 亚洲天堂av无毛| 久久婷婷青草| 嫩草影院入口| 自拍偷自拍亚洲精品老妇| 久久久久久久亚洲中文字幕| 亚洲色图综合在线观看| 日韩人妻高清精品专区| 97在线人人人人妻| 精品一区在线观看国产| 亚洲国产精品999| 热re99久久国产66热| 亚洲欧美一区二区三区黑人 | 自线自在国产av| 十分钟在线观看高清视频www | 亚洲精品日韩在线中文字幕| 日本黄色片子视频| av免费观看日本| 91成人精品电影| 亚洲欧洲日产国产| 高清视频免费观看一区二区| 国产成人免费无遮挡视频| 制服丝袜香蕉在线| 国产精品国产三级国产专区5o| 欧美精品一区二区免费开放| 极品教师在线视频| 国产精品偷伦视频观看了| 9色porny在线观看| 国产白丝娇喘喷水9色精品| 春色校园在线视频观看| 人妻夜夜爽99麻豆av| 妹子高潮喷水视频| 九九久久精品国产亚洲av麻豆| 男女无遮挡免费网站观看| 亚洲精品亚洲一区二区| 国产视频内射| 亚洲国产精品专区欧美| 国产在线男女| av在线播放精品| 在线看a的网站| 91精品国产九色| 亚洲自偷自拍三级| 久久久久久久久久成人| 插阴视频在线观看视频| 日韩欧美 国产精品| 一级av片app| 丰满人妻一区二区三区视频av| 蜜桃在线观看..| 一级毛片久久久久久久久女| 黄色日韩在线| 亚洲精品久久午夜乱码| av.在线天堂| 国产69精品久久久久777片| 我的女老师完整版在线观看|