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

        ?

        Conserved Gross–Pitaevskii equations with a parabolic potential

        2022-11-10 12:15:02ShiminLiuandDajunZhang
        Communications in Theoretical Physics 2022年10期

        Shi-min Liu and Da-jun Zhang

        Department of Mathematics,Shanghai University,Shanghai 200444,People’s Republic of China

        Abstract An integrable Gross–Pitaevskii equation with a parabolic potential is presented where particle density |u|2 is conserved.We also present an integrable vector Gross–Pitaevskii system with a parabolic potential,where the total particle density is conserved.These equations are related to nonisospectral scalar and vector nonlinear Schr?dinger equations.Infinitely many conservation laws are obtained.Gauge transformations between the standard isospectral nonlinear Schr?dinger equations and the conserved Gross–Pitaevskii equations,both scalar and vector cases are derived.Solutions and dynamics are analyzed and illustrated.Some solutions exhibit features of localized-like waves.

        Keywords:Gross–Pitaevskii equation,gauge transformation,nonisospectral,conserved particle density

        1.Introduction

        2.The conserved GP equation(4)

        In this section,we investigate the conserved GP equation(4)and its integrability.

        Let us first explain how we are motivated by solutions of the nonconserved GP equation(2)and arrive at the conserved equation(4).Recalling the carrier wave(particle density)of the explicit 1SS of(2)[10],

        Figure 1.Shape and motion of 1SS for equation(4)with δ >0.(a)Stationary soliton |u|2 with δ=0.36,a1=0.5,b1=h1=0.(b)The 2D plot of(a)at t=0(red dashed curve),t=1(blue dashed curve),t=2(black dashed curve).

        Figure 2.Shape and motion of 1SS for equation(4)with δ <0.(a)A moving soliton|u|2 with δ=?0.36,a1=0.5,b1=0.5,h1=0.(b)The 2D plot of(a)at t=0(red dashed curve),t=10(blue dashed curve),t=20(black dashed curve).

        3.The conserved vector GP equation(5)

        3.1.Lax pair

        3.2.Conservation laws

        3.3.Gauge equivalence and NSS

        3.4.Dynamics for the conserved vector GP equation(5)

        In this subsection,we investigate dynamics for the conserved vector GP equation(5).

        3.4.1.Solutions for the caseδ >0.Taking N=1 in the formula(34)and combined with the transformation(31),it is straightforward to obtain 1SS for the conserved vector GP equation(5).Let us consider the two-component case,i.e.

        3.4.2.Solutions for the caseδ <0.In the case of δ <0,1SS for equation(5)with δ <0 can be written as

        Figure 3.Shape and motion of 1SS given by(38)for equation(5)with δ=0.36.(a)A moving wave runs like with (b)A moving wave runs likemoving wave runs like(d)A stationary wave with.

        4.Conclusions and remarks

        In this paper,we have introduced a way to obtain a conserved GP equation with a parabolic potential.This idea was explained and illustrated in section 2,where we started from the non-conserved GP equation(2)with a parabolic potential,by calculating its total particle number N(see equation(7))associated with 1SS,we were led to the transformation(9),and the resulting GP equation(4)turns out to be conserved in terms of total particle number N.The conserved GP equation(4)contains a time-dependent coefficient g(t)to measure inter-particle interactions.This idea was then extended to the vector GP equation in section 3 and the conserved version is given in equation(5).The dynamics of some solutions are illustrated.

        We remark that the conserved GP equation(4)has been included in[7]as one of the GP equations that are gauge equivalent to the standard NLS equation(see table 1 in[7]).In our paper,we corrected an inaccurate statement given in[7]about the periodic singularities appearing in the case δ <0.

        Figure 4.(a)1SS of |u1(x,t)| given by(45)for equation(5)with δ=?0.36,a1=1,b1=1,β1,1=1,β1,2=1.(b)Density plot of(a).

        The idea of this paper might be applied to the differential-difference GP models,which will be investigated elsewhere.

        Acknowledgments

        The authors are grateful to the referees for their invaluable comments.This project is supported by the NSF of China(Nos.11 875 040,12 126 352,12 126 343).

        国产亚洲sss在线观看| 天堂网www资源在线| 欧美饥渴熟妇高潮喷水水| 97色在线视频| 久久一二三四区中文字幕| 白白色发布会在线观看免费| 人妻少妇无码精品视频区 | 国产精品麻豆va在线播放| 久草中文在线这里只有精品| 欧美白人战黑吊| 午夜成人理论无码电影在线播放| 囯产精品无码一区二区三区| 久久青青草原亚洲av| www国产亚洲精品| 国产精品白丝喷水在线观看| 国产精品中文第一字幕| 口爆吞精美臀国产在线| 国产裸体美女永久免费无遮挡| 日韩电影一区二区三区| 久久与欧美视频| 日本美女中文字幕第一区| 亚洲精品色午夜无码专区日韩| 国产精品一区二区 尿失禁 | 男女性高爱潮免费观看| 26uuu欧美日本在线播放| 国产三级不卡视频在线观看| 成人做受黄大片| 日韩久久一级毛片| 国产喷白浆精品一区二区豆腐| 亚洲性无码av中文字幕| 欧美天欧美天堂aⅴ在线| 亚洲午夜无码视频在线播放 | 97丨九色丨国产人妻熟女| 久久精品人人做人人爽电影蜜月| 69国产成人综合久久精| 免费播放成人大片视频| 亚洲国产精品va在线看黑人| 欧美伊人亚洲伊人色综| 日本人妻高清免费v片| 精品国产麻豆免费人成网站| 久久天天躁狠狠躁夜夜爽蜜月|