玻色—爱因斯坦凝聚系统中的混沌冲击波和涡旋
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摘要
自从玻色-爱因斯坦凝聚现象在稀薄原子气体中实现以来,这个领域的相关研究引起了人们广泛的关注。玻色-爱因斯坦凝聚具有非常奇特的性质,不仅为量子理论的研究提供了一个可靠的宏观量子系统,而且在原子激光、量子计算、精密测量等领域有着非常光明的应用前景。近几十年来,玻色-爱因斯坦凝聚的理论和实验研究都取得了非常大的进展,关于玻色-爱因斯坦凝聚系统中的许多非线性结构在很多理论预测和实验实现上都进行了广泛的研究,比如暗孤子、亮孤子、塌缩波、涡旋、冲击波等等,这些都成为了研究的热点。本文以平均场理论框架下的Gross-Pitaevskii方程为主要模型,利用理论分析和数值模拟相结合的方法,探讨了不同外势中一维开放和二维玻色-爱因斯坦凝聚系统中冲击波和涡旋的动力学行为,得到了系统在一定参数条件下的精确解,并提出了抑制凝聚原子混沌运动,以及产生和控制涡旋运动的方法,得到了一些有意义的结论。
全文共分为六章。第一章为绪论部分,简要介绍了玻色-爱因斯坦凝聚的实验实现和相关概念,以及描述玻色-爱因斯坦凝聚体的平均场理论和Gross-Pitaevskii方程。从平均场理论出发,研究了玻色-爱因斯坦凝聚系统中混沌冲击波和涡旋的动力学特征。最后简要介绍了玻色-爱因斯坦凝聚的应用前景及其研究意义。
第二章,我们考虑非平衡热云对凝聚原子的补充效应,研究时间周期驱动的反谐振子势中一维开放和吸引的玻色-爱因斯坦凝聚系统。利用直接微扰法和定态Gross-Pitaevskii方程的精确冲击波解得到了系统的混沌微扰解和参数空间的Melnikov同宿混沌区域。基于解析分析和数值模拟的方法研究了补充强度对系统混沌运动的影响,发现调节补充强度可以抑制混沌的出现。对于固定边界的“不传播”冲击波,凝聚原子数随着补充强度的增加而快速地增加；而对于固定波前密度的亚稳态自由边界的冲击波,随着补充强度的增加,凝聚原子数非周期振荡地衰减。
第三章,我们研究了时间周期驱动的二维简谐势或反谐势中玻色-爱因斯坦凝聚系统中的相位效应,得到了含时Gross-Pitaevskii方程描述冲击波行为的形式精确解。我们利用数值方法研究了原子密度的分布。发现当相位可分离变量时,原子密度呈圆对称分布,而当相位不可分离变量时,原子密度呈轴对称分布,从而发现了一种新的效应——趋轴效应。通过解析和数值的分析,发现混沌可以抑制解的逃逸,以及玻色爱因斯坦凝聚体的塌缩和爆破。我们的结果也为控制二维玻色-爱因斯坦凝聚体的方向输运提供了可行的方法。
第四章,我们研究了周期驱动的二维光学晶格中玻色-爱因斯坦凝聚系统中涡旋的动力学行为,得到了一定参数区域下Gross-Pitaevskii方程精确的Floquet解,这个参数区域可以分为相跳变区域和相连续区域。当参数取在相跳变区域中,这个精确解可以描述多涡旋的时空演化。研究发现当驱动强度和光格高度比值较小时,随着时间的演化涡旋基本保持不动,呈均匀分布。随着该比值的增加,在等效力的作用下涡旋沿着一些固定的圆形轨道周期地靠近和分离,形成涡旋偶极子和涡旋四极子。当该比值超过某一临界值时,涡旋周期性地出现和消失。当参数取在相连续区域中,精确Floquet态中的凝聚体周期地演化,但没有零密度点和涡旋特征。最后我们数值地研究了不同区域中该精确解的稳定性,发现在相跳变区域的大部分区间在小的初始微扰下该解都是稳定的,但是小的参数微扰会导致该解稳定性的失去。在相跳变区域的一个小的子区域,该解是不稳定的。然而,在相连续区域不管是小的初始微扰或小的参数微扰该解始终是保持稳定的,即此时该解是结构稳定的。我们的结果为产生和控制涡旋的运动提供了一种有效的方法。
第五章,我们研究了二维简谐振子势中相互作用与空间相关或无关的玻色-爱因斯坦凝聚系统中涡旋的动力学行为。发现通过设计不同的激光控制势可以得到不同形式的涡旋解,并研究了定态涡旋解和非定态涡旋解的分布以及涡旋核的运动轨道。对于精确的定态涡旋,其涡旋核都保持静止。然而,对于精确的非定态涡旋,我们发现其中有稳定的涡旋团簇,也有不稳定的涡旋团簇。当涡旋都沿着某一封闭的轨道周期性地运动,这意味着该涡旋团簇是稳定的。而对于不稳定的涡旋团簇,随着时间的演化,涡旋会周期性地出现和消失,或是运动到无穷远处,这意味着涡旋团簇是不稳定的。我们的结果为产生不同的涡旋结构提供了有效的方法。
第六章,对本文的工作进行了总结与归纳,并对玻色-爱因斯坦凝聚体系统这一研究领域的发展前景作了简要的展望。
Since the realization of Bose-Einstein condensation in a dilute atomic gas, the relevant investigations have attracted enormous attention. It not only provides per-feet macroscopic quantum systems to investigate many fundamental problems in quantum mechanics, but also has a wide range of applications such as in atom laser, quantum computation and the exactitude measure. In recent years, great progress has been made in the theoretical and experimental studies of Bose-Einstein con-densation. There are also many investigations about the nonlinear structures in Bose-Einstein condensates, such as dark soliton, bright soliton, collapsing wave, vortices and shock wave, which are hot research topics nowadays. In the frame-work of mean-field theory, based on the theoretical analysis and numerical method we discuss the dynamic behavior of shock wave and vortices in one-dimensional and two-dimensional Bose-Einstein condensation existing in various external po-tentials. We obtain some exact solutions of the system under certain parameter conditions and propose a scheme to suppress chaos, to generate and control the movement of vortices. Some meaningful results are discovered in our study.
This thesis consists of the following six parts. In the first chapter, we give a brief introduction about the experimental realization of Bose-Einstein conden-sation and relevant fundamental concepts, the mean-field theory and the Gross-Pitaevskii equation. From the mean-field theory, we investigate the dynamic char-acteristic of shock wave and vortices in Bose-Einstein condensation. Finally, a brief introduction on the application prospects and research significances of Bose-Einstein condensation is also provided.
In the second chapter, we investigate a one-dimensional open Bose-Einstein condensation with attractive interaction, by considering the effect of feeding from nonequilibrium thermal cloud and applying the time-periodic inverted-harmonic potential. Using the direct perturbation method and the exact shock wave solution of the stationary Gross-Pitaevskii equation, we obtain the chaotic perturbed solu-tion and the Melnikov chaotic regions. Based on the analytical and the numerical methods, the influence of the feeding strength on the chaotie motion is revealed. In the case of "nonpropagated" shock wave with fixed boundary, the number of condensed atoms increases faster as the feeding strength increases. However, for the free boundary the metastable shock wave with fixed front density oscillates its front position and atomic number aperiodically, and their amplitudes decay with the increase of the feeding strength.
In chapter three, we investigate the phase effects of a periodically driven Bose-Einstein condensate held in a spatially two-dimensional harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross-Pitaevskii equation is found, which depicts the shock wave with chaotic or periodic ampli-tude and phase. The atomic densities are illustrated numerically, and the circularly symmetric distributions in the separable phase and the axially symmetric Bose-Einstein condensate clusters in the inseparable phase are shown. It is demonstrated that the periodic driving may lead to chaos for both phases, which plays a role in avoiding the escape of the solution and restraining the Bose-Einstein conden-sate collapse and blast. The results suggest a method for controlling the directed transports of the two-dimensional Bose-Einstein condensate.
In chapter four, we investigate vortex dynamics of a periodically driven Bose-Einstein condensate confined in a spatially two-dimensional optical lattice. An exact Floquet solution of the Gross-Pitaevskii equation is obtained for a certain pa-rameter region which can be divided into the phase-jumping and phase-continuing regions. In the former region, for a small ratio of driving strength to optical lat-tice depth the vortices keep nearly unmoved. With the increase of the ratio, the vortices undergo an effective interaction and periodically evolve along some fixed circular orbits that leads the vortex dipoles and quadrupoles to produce and break alternatively. There is a critical ratio in the phase-jumping region beyond which the vortices generate and melt periodically. In the phase-continuing region, the condensate in the exact Floquet state evolves periodically without zero-density nodes. The stability and instability of the exact solution are illustrated numer-ically. It is demonstrated that the exact solution is stable for both parameter regions, except for a subregion of the phase-jumping region in which stability of the condensate is lost. However, in the phase-jumping region stability of the solu-tion can be destroyed by a small parameter perturbation. In the phase-continuing region the solution is structurally stable under a small parameter perturbation. The results suggest a scheme for creating and controlling matter-wave vortices.
In chapter five, we investigate vortex dynamics behavior of Bose-Einstein con-densate in a spatially two-dimensional harmonic potential with spatial dependent or independent interaction. The stationary and non-stationary vortex solutions can be obtained by designing various laser potentials which can be experimentally realized. For the stationary vortex solutions, vortices remain unmoved. The non-stationary vortex solutions contain stable and unstable vortices clusters. When vortices move periodically along some fixed and closed orbits, vortices clusters are stable. However, for the unstable clusters, vortices disappear and appear period-ically with the evolution of time. The results suggest a scheme for creating and controlling matter-wave vortices.
In chapter six, we give a brief summary of the work and a outlook of the applications of the Bose-Einstein condensate system.
全文共分为六章。第一章为绪论部分,简要介绍了玻色-爱因斯坦凝聚的实验实现和相关概念,以及描述玻色-爱因斯坦凝聚体的平均场理论和Gross-Pitaevskii方程。从平均场理论出发,研究了玻色-爱因斯坦凝聚系统中混沌冲击波和涡旋的动力学特征。最后简要介绍了玻色-爱因斯坦凝聚的应用前景及其研究意义。
第二章,我们考虑非平衡热云对凝聚原子的补充效应,研究时间周期驱动的反谐振子势中一维开放和吸引的玻色-爱因斯坦凝聚系统。利用直接微扰法和定态Gross-Pitaevskii方程的精确冲击波解得到了系统的混沌微扰解和参数空间的Melnikov同宿混沌区域。基于解析分析和数值模拟的方法研究了补充强度对系统混沌运动的影响,发现调节补充强度可以抑制混沌的出现。对于固定边界的“不传播”冲击波,凝聚原子数随着补充强度的增加而快速地增加；而对于固定波前密度的亚稳态自由边界的冲击波,随着补充强度的增加,凝聚原子数非周期振荡地衰减。
第三章,我们研究了时间周期驱动的二维简谐势或反谐势中玻色-爱因斯坦凝聚系统中的相位效应,得到了含时Gross-Pitaevskii方程描述冲击波行为的形式精确解。我们利用数值方法研究了原子密度的分布。发现当相位可分离变量时,原子密度呈圆对称分布,而当相位不可分离变量时,原子密度呈轴对称分布,从而发现了一种新的效应——趋轴效应。通过解析和数值的分析,发现混沌可以抑制解的逃逸,以及玻色爱因斯坦凝聚体的塌缩和爆破。我们的结果也为控制二维玻色-爱因斯坦凝聚体的方向输运提供了可行的方法。
第四章,我们研究了周期驱动的二维光学晶格中玻色-爱因斯坦凝聚系统中涡旋的动力学行为,得到了一定参数区域下Gross-Pitaevskii方程精确的Floquet解,这个参数区域可以分为相跳变区域和相连续区域。当参数取在相跳变区域中,这个精确解可以描述多涡旋的时空演化。研究发现当驱动强度和光格高度比值较小时,随着时间的演化涡旋基本保持不动,呈均匀分布。随着该比值的增加,在等效力的作用下涡旋沿着一些固定的圆形轨道周期地靠近和分离,形成涡旋偶极子和涡旋四极子。当该比值超过某一临界值时,涡旋周期性地出现和消失。当参数取在相连续区域中,精确Floquet态中的凝聚体周期地演化,但没有零密度点和涡旋特征。最后我们数值地研究了不同区域中该精确解的稳定性,发现在相跳变区域的大部分区间在小的初始微扰下该解都是稳定的,但是小的参数微扰会导致该解稳定性的失去。在相跳变区域的一个小的子区域,该解是不稳定的。然而,在相连续区域不管是小的初始微扰或小的参数微扰该解始终是保持稳定的,即此时该解是结构稳定的。我们的结果为产生和控制涡旋的运动提供了一种有效的方法。
第五章,我们研究了二维简谐振子势中相互作用与空间相关或无关的玻色-爱因斯坦凝聚系统中涡旋的动力学行为。发现通过设计不同的激光控制势可以得到不同形式的涡旋解,并研究了定态涡旋解和非定态涡旋解的分布以及涡旋核的运动轨道。对于精确的定态涡旋,其涡旋核都保持静止。然而,对于精确的非定态涡旋,我们发现其中有稳定的涡旋团簇,也有不稳定的涡旋团簇。当涡旋都沿着某一封闭的轨道周期性地运动,这意味着该涡旋团簇是稳定的。而对于不稳定的涡旋团簇,随着时间的演化,涡旋会周期性地出现和消失,或是运动到无穷远处,这意味着涡旋团簇是不稳定的。我们的结果为产生不同的涡旋结构提供了有效的方法。
第六章,对本文的工作进行了总结与归纳,并对玻色-爱因斯坦凝聚体系统这一研究领域的发展前景作了简要的展望。
Since the realization of Bose-Einstein condensation in a dilute atomic gas, the relevant investigations have attracted enormous attention. It not only provides per-feet macroscopic quantum systems to investigate many fundamental problems in quantum mechanics, but also has a wide range of applications such as in atom laser, quantum computation and the exactitude measure. In recent years, great progress has been made in the theoretical and experimental studies of Bose-Einstein con-densation. There are also many investigations about the nonlinear structures in Bose-Einstein condensates, such as dark soliton, bright soliton, collapsing wave, vortices and shock wave, which are hot research topics nowadays. In the frame-work of mean-field theory, based on the theoretical analysis and numerical method we discuss the dynamic behavior of shock wave and vortices in one-dimensional and two-dimensional Bose-Einstein condensation existing in various external po-tentials. We obtain some exact solutions of the system under certain parameter conditions and propose a scheme to suppress chaos, to generate and control the movement of vortices. Some meaningful results are discovered in our study.
This thesis consists of the following six parts. In the first chapter, we give a brief introduction about the experimental realization of Bose-Einstein conden-sation and relevant fundamental concepts, the mean-field theory and the Gross-Pitaevskii equation. From the mean-field theory, we investigate the dynamic char-acteristic of shock wave and vortices in Bose-Einstein condensation. Finally, a brief introduction on the application prospects and research significances of Bose-Einstein condensation is also provided.
In the second chapter, we investigate a one-dimensional open Bose-Einstein condensation with attractive interaction, by considering the effect of feeding from nonequilibrium thermal cloud and applying the time-periodic inverted-harmonic potential. Using the direct perturbation method and the exact shock wave solution of the stationary Gross-Pitaevskii equation, we obtain the chaotic perturbed solu-tion and the Melnikov chaotic regions. Based on the analytical and the numerical methods, the influence of the feeding strength on the chaotie motion is revealed. In the case of "nonpropagated" shock wave with fixed boundary, the number of condensed atoms increases faster as the feeding strength increases. However, for the free boundary the metastable shock wave with fixed front density oscillates its front position and atomic number aperiodically, and their amplitudes decay with the increase of the feeding strength.
In chapter three, we investigate the phase effects of a periodically driven Bose-Einstein condensate held in a spatially two-dimensional harmonic or inverted-harmonic potential. A formally exact solution of the time-dependent Gross-Pitaevskii equation is found, which depicts the shock wave with chaotic or periodic ampli-tude and phase. The atomic densities are illustrated numerically, and the circularly symmetric distributions in the separable phase and the axially symmetric Bose-Einstein condensate clusters in the inseparable phase are shown. It is demonstrated that the periodic driving may lead to chaos for both phases, which plays a role in avoiding the escape of the solution and restraining the Bose-Einstein conden-sate collapse and blast. The results suggest a method for controlling the directed transports of the two-dimensional Bose-Einstein condensate.
In chapter four, we investigate vortex dynamics of a periodically driven Bose-Einstein condensate confined in a spatially two-dimensional optical lattice. An exact Floquet solution of the Gross-Pitaevskii equation is obtained for a certain pa-rameter region which can be divided into the phase-jumping and phase-continuing regions. In the former region, for a small ratio of driving strength to optical lat-tice depth the vortices keep nearly unmoved. With the increase of the ratio, the vortices undergo an effective interaction and periodically evolve along some fixed circular orbits that leads the vortex dipoles and quadrupoles to produce and break alternatively. There is a critical ratio in the phase-jumping region beyond which the vortices generate and melt periodically. In the phase-continuing region, the condensate in the exact Floquet state evolves periodically without zero-density nodes. The stability and instability of the exact solution are illustrated numer-ically. It is demonstrated that the exact solution is stable for both parameter regions, except for a subregion of the phase-jumping region in which stability of the condensate is lost. However, in the phase-jumping region stability of the solu-tion can be destroyed by a small parameter perturbation. In the phase-continuing region the solution is structurally stable under a small parameter perturbation. The results suggest a scheme for creating and controlling matter-wave vortices.
In chapter five, we investigate vortex dynamics behavior of Bose-Einstein con-densate in a spatially two-dimensional harmonic potential with spatial dependent or independent interaction. The stationary and non-stationary vortex solutions can be obtained by designing various laser potentials which can be experimentally realized. For the stationary vortex solutions, vortices remain unmoved. The non-stationary vortex solutions contain stable and unstable vortices clusters. When vortices move periodically along some fixed and closed orbits, vortices clusters are stable. However, for the unstable clusters, vortices disappear and appear period-ically with the evolution of time. The results suggest a scheme for creating and controlling matter-wave vortices.
In chapter six, we give a brief summary of the work and a outlook of the applications of the Bose-Einstein condensate system.
引文
[1]S. N. Bose. Plancks gesetz und licht quanten hypothese [J]. Z. Phys.,1924,26: 178.
[2]M. H. Anderson, J. R. Ensher, M. R. Matthews. C. E. Wieman and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor [J]. Science, 1995,269:198.
[3]C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet. Evidence of Bose-Einstein Condensation in an atomie gas with attractive interactions [J]. Phys. Rev. Lett.,1995,75:1687.
[4]K. B. Davis, M.0. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle. Bose-Einstein Condensation in a gas of sodium atoms [J]. Phys. Rev. Lett.,1995,75:3969.
[5]F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari. Theory of Bose-Einstein condensation in trapped gases [J]. Rev. Mod. Phys.,1999,71:463.
[6]江开军.铷原子玻色-爱因斯坦凝聚的实验研究[D].武汉：中国科学院武汉物理与数学研究所,2005.
[7]周蜀渝,龙全,周善钰,付海翔,王育竹.玻色-爱因斯坦凝聚在中国科学院上海光机所实现[J].物理,2002,31：481.
[8]陈帅,周小计,杨帆,夏林,马修泉,熊炜,王义遒,陈徐宗.北京大学玻色-爱因斯坦凝聚的实现及研究进展[J].量子光学学报,2004,10：4.
[9]S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag and R. Grimm. Bose-Einstein Condensation of molecules [J]. Science, 2003,302:2101.
[10]M. Greiner, C. A. Regal and D. S. Jin. Emergence of a molecular Bose-Einstein condensate from a Fermi gas [J]. Nature,2003,426:537.
[11]M. W. Zwierlcin, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadz-ibabic and W. Ketterle. Observation of Bose-Einstein Condensation of molecules [J], Phys. Rev. Lett.,2003,91:250401.
[12]C. A. Regal, M. Greiner and D. S. Jin. Observation of resonance condensation of Fermionic atom pairs [J]. Phys. Rev. Lett.,2004,92:040403.
[13]M. Bartenstein, A. Altmeyer, S. Riedl. S. Jochim, C. Chin, J. Hecker Denschlag and R. Grimm. Crossover from a molecular Bose-Einstein Condensate to a degen-erate Fermi gas [J]. Phys. Rev. Lett.,2004,92:120401.
[14]M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupaeh, A. J. Kerman and W. Ketterle. Condensation of pairs of Fermionic atoms near a Feshbach Resonance [J]. Phys. Rev. Lett.,2004,92:120403.
[15]熊宏伟,吕宝龙,刘淑娟,詹明生.超冷费米气体研究的新进展[J].物理学进展,2005,25：296.
[16]徐克尊.高等原子分子物理学[M].北京：科技出版社,2002：69.
[17]F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari. Theory of Boso-Einstein condensation in trapped gases [J]. Rev. Mod. Phys.,1999,71:463.
[18]A. J. Leggett. Bose-Einstein condensation in the alkali gases:Some fundamental concepts [J]. Rev. Mod. Phys.,2001,73:307.
[19]N. N. Bogoliubov. On the theory of superfluidity [J]. J. Phys. USSR,1947,11:23.
[20]E. P. Gross. Structure of a quantized vortex in Boson Systems [J]. Nuovo Cimento, 1961,20:454.
[21]E. P. Gross. Hydrodynamics of a superfluid condensate [J]. J. Math. Phys.,1963, 4:195.
[22]L. P. Pitaevskii. Vortex lines in an imperfect Bose gas [J]. Zh. Eksp. Teor. Fiz., 1961,40:646.
[23]V. M. Pcrez-Garcia, H. Michinel and H. Herrero. Bose-Einstein solitons in highly asymmetric traps [J]. Phys. Rev. A,1998,57:3837.
[24]K. E. Strecker, G. B. Partridge, A. G. Truscott and R. G. Hulet. Formation and propagation of matter-wave soliton trains [J]. Nature,2002,417:150.
[25]L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin and C. Salomon. Formation of a matter-wave bright soliton [J]. Science, 2002,296:1290.
[26]S. Burger, K. Bongs. S. Dettmer, W. Ertmer and K. Sengstock. Dark solitons in Bose-Einstein Condensates [J]. Phys. Rev. Lett.,1999,83:5198.
27] J. Denschlag, J. E. Simsarian, D. L. Feder, Charles W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Hclmerson, W. P. Reinhardt, S. L. R.olston, B. I. Schneider and W. D. Phillips. Generating solitons by phase engineering of a Bose-Einstein Condensate [J]. Science,2000,287:97.
[28]C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet. Evidence of Bose-Einstein Condensation in an atomic gas with attractive interactions [J]. Phys. Rev. Lett.,1995,75:1687.
[29]Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell and C. E. Wieman. Dynamics of collapsing and exploding Bose-Einstein condensates [J]. Nature,2001,412:295.
[30]F. Sols. Vortex matter in atomic Bose-Einstein condensates [J]. Physica C:Super-conductivity,2002,369:125.
[31]V. M. Perez-Garcia, V. V. Konotop and V. A. Brazhnyi. Feshbach resonance induced shock waves in Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2004,92: 220403.
[32]P. J. Chen. Growth and decay of waves in solids [M]. Berlin:Springer,1973:303.
[33]J. Serrin. Mathematical principles of classical fluid dynamics [M]. Berlin:Springer, 1959:125.
[34]J. Sinoller and B. Temple. Shock-wave cosmology inside a black hole [J]. Proc. Natl Acad. Sci. USA,2003,100:11216.
[35]R. N. C. Pfeifer and T. A. Nieminen. Visualization of Cerenkov radiation and the fields of a moving charge [J]. Eur. J. Phys.,2006,27:521.
[36]R. Quintanilla and B. Straughan. A note on discontinuity waves in type III ther-moelasticity [J]. Proc. R. Soc. A,2004,460:1369.
[37]A. Morro. Jump relations and discontinuity waves in conductors with memory [J]. Math. Comput. Modelling,2006,43:138.
[38]B. D. Coleman and D. C. Newman. Implications of a nonlinearity in the theory of second sound in solids [J]. Phys. Rev. B,1988,37:1492.
[39]D. Jou, J. Casas-Vazquez and G. Lebon. Extended irreversible thermodynamics [M]. Berlin:Springer,2001:41.
[40]C. I. Christov and P. M. Jordan. Heat conduction paradox involving second-sound propagation in moving media [J]. Phys. Rev. Lett.,2005,94:154301.
[41]Z. Dutton, M. Budde, C. Slowe and L. V. Hau. Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein Conden-sate [J]. Science,2001,293:663.
[42]T. P. Simula, P. Engels, I. Coddington, V. C. Schweikhard, E. A. Cornell and R. J. Ballagh. Observations on sound propagation in rapidly rotating Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2005,94:080404.
[43]L. Onsager. Statistical Hydrodynamics [J]. Nuovo Cimento Suppl.,1949,6:279.
[44]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell. Vortices in a Bose-Einstein Condensate [J]. Phys. Rev. Lett.,1999,83: 2498.
[45]J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle. Observation of vortex lattices in Bose-Einstein Condensates [J]. Science,2001,292:476.
[40]D. S. Rokhsar. Vortex stability and persistent currents in trapped Bose gases [J]. Phys. Rev. Lett.,79:2164.
[47]E. Hodby, G. Hechenblaikner, S.A. Hopkins, O. M. Marago and C. J. Foot. Vortex nucleation in Bose-Einstein Condensates in an oblate, purely magnetic potential [J]. Phys. Rev. Lett.,2001,88:010405.
[48]B. Jackson, J. F. McCann and C. S. Adams. Vortex formation in dilute inhomo-geneous Bose-Einstein Condensates [J]. Phys. Rev. Lett.,1998,80:3903.
[49]L. Dobrek, M. Gajda, M. Lewenstein, K. Sengstock, G. Birkl and W. Ertmer. Optical generation of vortices in trapped Bose-Einstein condensates [J]. Phys. Rev. A,1999,60:R3381.
[50]A. E. Leanhardt, A. Gorlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard and W. Ketterle. Imprinting vortices in a Bose-Einstein Condensate using topological phases [J]. Phys. Rev. Lett.,2002,89:190403.
[51]T. Isoshima, M. Nakahara, T. Ohmi and K. Machida. Creation of a persistent current and vortex in a Bose-Einstein condensate of alkali-metal atoms [J]. Phys. Rev. A,2000,61:063610.
[52]B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark and E. A. Cornell. Watching dark solitons decay into vortex rings in a Bose-Einstein Condensate [J]. Phys. Rev. Lett.,2001,86:2926.
[53]S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Gorlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard and W. Ketterle. Observation of vortex phase singularities in Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2001,87:080402.
[54]E. L. Bolda. D. F. Walls. Detection of vorticity in Bose-Einstein Condensed gases by matter-wave interference [J]. Phys. Rev. Lett.,1998,81:5477.
[55]J. Tempere, J. T. Devreese. Fringe pattern of interfering Bose-Einstein Condensate with a vortex [J]. Solid State Commu.,1998,108:993.
[50]Q. Chen, J. Stajic, S. Tan and K. Levin. BCS-BEC crossover:From high tem-perature.superconductors to ultracold superfluids [J]. Physics Reports,2005,412: 1.
[57]B. Borca, J. W. Dunn. V. Kokoouline and C. H. Greene. Atom-molecule laser fed by stimulated three-body recombination [J]. Phys. Rev. Lett.,2003,91:070404.
[58]S. A. Haine, J. J. Hope, N. P. Robins and C. M. Savage. Stability of continuously pumped atom lasers [J]. Phys. Rev. Lett.,2002,88:170403.
[59]G. Cennini, G. Ritt, C. Geckeler and M. Weitz. All-optical realization of an atom laser [J]. Phys. Rev. Lett.,2003,91:240408.
[60].T. K. Pachos and P. L. Knight. Quantum computation with a one-dimensional optical lattice [J]. Phys. Rev. Lett.,2003,91:107902.
[61]R. Ionicioiu and P. Zanardi. Quantum-information processing in bosonic lattices [J]. Phys. Rev. A,2002,66:050301 (R).
[62]C. J. Pet hick and H. Smith. Bose-Einstein Condensation in dilute gases [M]. Cam-bridge:Cambridge University Press,2002.
[63]M. Inguscio, S. Stringari and C. E. Wieman. Bose-Einstein Condensation in atomic gases [M]. Amsterdam:IOS Press,1999.
[64]L. Pitaevskii and S. Stringari. Bose-Einstein Condensation [M]. Oxford:Oxford University Press,2003.
[65]J. O. Andersen. Theory of the weakly interacting Bose gas[J]. Rev. Mod. Phys., 2004,76:599.
[66]A. Griffin, D. W. Snoke and S. Stringari. Bose-Einstein Condensation [M], Cam-bridge:Cambridge University Press,1995.
[67]G. B. Whitham. Linear and nonlinear waves [M]. New York:Wiley,1999.
[68]R. Courant and K. O. Friedrichs. Supersonic flow and shock Waves [M]. Berlin: Springer-Verla,1948.
[69]F. V. Shugaev and L. S. Shtemenko. Propagation and reflection of shock waves [M]. Singapore:World Scientific,1998.
[70]V. T. Gurovich, A. Grinenko and Y. E. Krasik. Semianalytical solution of the problem of converging shock waves [J]. Phys. Rev. Lett.,2007,99:124503.
[71]A. B. Shvartsburg, L. Stenflo and P. K. Shukla. Breakdown of waves described by exact solutions of the Thomas-Fermi model [J]. Eur. Phys. J. B,2002,28:71.
[72]V. A. Brazhnyi, A. M. Kamchatnov and V. V. Konotop. Hydrodynamic flow of expanding Bose-Einstein condensates [J]. Phys. Rev. A,2003,68:035603.
[73]I. Kulikov and M. Zak. Shock waves in a Bose-Einstein condensate [J]. Phys. Rev. A,2003,67:063605.
[74]B. Damski. Formation of shock waves in a Bose-Einstein condensate [J]. Phys. Rev. A,2004,69:043610.
[75]W. H. Hai, Q. Q. Zhu and S. G. Rong. Chaotic shock waves of a Bose-Einstein condensate [J]. Phys. Rev. A,2009; 79:023603.
[76]Y. Castin and R. Dum. Bose-Einstein Condensates in time dependent traps [J]. Phys. Rev. Lett.,1996,77:5315.
[77]Z. X. Liang, Z. D. Zhang and W. M. Liu. Dynamics of a bright soliton in Bose-Ein-stein Condensates with time-dependent atomic scattering length in an expulsive parabolic potential [J]. Phys. Rev. Lett.,2005,94:050402.
[78]F. K. Abdullaev and R. Galimzyanov. The dynamics of bright matter wave solitons in a quasi one-dimensional Bose-Einstein condensate with a rapidly varying trap [J]. J. Phys. B,2003,36:1099.
[79]G. S. Chong and W. H. Hai. Dynamical evolutions of matter-wave bright solitons in an inverted parabolic potential [J]. J. Phys. B,2007,40:211.
[80]Z. Zhou, H. Y. Yu and J. R. Yan. Dynamics of vector solitons in two-component Bose-Einstein condensates with time-dependent interactions and harmonic poten-tial [J]. Chin. Phys. B,2010,19:010304.
[81]C. A. Sackctt, H. T. C.Stoof and R. G. Hulct. Growth and collapse of a Bosc-Einstciu Condensate with attractive interactions [J]. Phys. Rev. Lett.,1998,80: 2031.
[82]Y. Kagan, A. E. Muryshev and G. V. Shlyapnikov. Collapse and Bose-Einstein Condensation in a trapped Bose gas with negative scattering length [J]. Phys. Rev. Lett.,1998,81:933.
[83]A. Gammal, T. Frederico, L. Tomio and P. Chomaz. Liquid-gas phase transition in Bose-Einstein condensates with time evolution [J]. Phys. Rev. A,2000,61: 051602(R).
[84]Y. Li and W. H. Hai. Three-body recombination in two coupled Bose-Einstein condensates [J]. J. Phys. A.2005,38:4105.
[85]A. X. Mu, X. Y. Zhou and J. K. Xuc. The dynamics of triple-well trapped Bose-Einstein condensates with atoms feeding and loss effects [J]. Chin. Phys. B,2008, 17:764.
[86]Hai W H, X. L. Zhang, W. L. Huang and G. S. Chong. Chaotic solution of the rf-driven josephson system with quadratic damping [J]. Int. J. Bifurcation and Chaos,2001,11:2263.
[87]G. X. Huang. Second harmonic generation of propagating collective excitations in Bose-Einstein condensates [J]. Chin. Phys.,2004,13:1866.
[88]W. H. Hai, Y. W. Duan and L. X. Pan. An analytical study for controlling unstable periodic motion in magneto-elastic chaos [J]. Phys. Lett. A,1997,234:198.
[89]W. H. Hai, S. G. R.ong and Q. Q. Zhu. Discrete chaotic states of a Bose-Einstein condensate [J]. Phys. Rev. E,2008,78:066214.
[90]J. S. Fang. Nonlinear transport of Bose-Einstein condensates in a double barrier potential [J]. Chin. Phys. B,2008,17:3996.
[91]V. K. Melnikov. On the stability of a center for time-periodic perturbations [J]. Trans. Moscow Math. Soc.,1963,12:1.
[92]Q. Q. Zhu, W. H. Hai and S. G. Rong. Transition probability from matter-wave soliton to chaos [J]. Phys. Rev. E,2009,80:016203.
[93]S. Parthasarathy. Homoclinic bifurcation sets of the parametrically driven duffing oscillator [J]. Phys. Rev. A,1992,46:2147.
[94]M. A. Hoofer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels and V. Schweikhard. Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics [J]. Phys. Rev. A,2006,74:023623.
[95]H. Saito and M. Ueda. Intermittent implosion and pattern formation of trapped Bose-Einstein Condensates with an attractive interaction [J]. Phys. Rev. Lett., 2001,86:1406.
[96]M. P. Brenner, S. Hilgenfeldt and D. Lohse. Single-bubble sonoluminescence [J]. Rev. Mod. Phys.,2002,74:425.
[97]E. J. Reed, M. Soljacic and J. D. Joannopoulos. Color of shock waves in photonic crystals [J]. Phys. Rev. Lett.,2003,90:203904.
[98]T. Brunet, J. L. Thomas, R.. Marchiano and F. Coulouvrat. Experimental obser-vation of azimuthal shock waves on nonlinear acoustical vortices [J]. New. J. Phys, 2009,11:013002.
[99]R. A. Austin, D. L. McDowell and D. J. Benson. Numerical simulation of shock wave propagation in spatially-resolved particle systems [J]. Modelling Simul. Mater. Sci. Eng.,2006,14:537.
[100]I. Christov and P. M. Jordan. Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation [J]. New. J. Phys,2008,10:043027.
[101]Y. Deng, W. Hai, S. Rong and H. Zhong. Chaotic, matter shock wave of an open system [J]. Chin. Phys. B,2010,19:120502.
[102]S. Horluck, M. V. Hecke and P. Dimon. Shock waves in two-dimensional granular flow:Effects of rough walls and polydispersity [J]. Phys. Rev. E,2003,67:021304.
[103]A. M. Kamchatnov, A. Gammal and R. A. Kraenkcl. Dissipationless shock waves in Bosc-Einstein condensates with repulsive interaction between atoms [J]. Phys. Rev. A,2004,69:063605.
[104]S. Ghosh. Dust acoustic shock waves in two-component dusty plasma [J]. New J. Phys.,2003,5:142.
[105]M. A. Hoefer and B. Ilan. Theory of two-dimensional oblique dispersive shock waves in supersonic flow of a superfluid [J]. Phys. Rev. A,2009,80:061601(R).
[106]X. Y. Tang and P. K. Shukla. Symmetry analysis and similarity electrostatic waves in a nonuniform dusty magnetoplasma [J]. J. Phys. A:Math. Theor.,2007,40: 5921.
[107]P. A. M. Dirac. Relativity and quantum mechanics|J]. Fields and Quanta,1972, 3:139.
[108]K. Boaz. M. Peter and W. Eli. The spectrum of cosmic rays escaping from rela-tivistic shocks [J]. J. Cosmol. Astropart. Phys.,2010,10:012.
[109]G. Lu and W. Hai. Quantum tunneling switch in a planar four-well system [J]. Phys. Rev. A,2011,83:053424.
[110]F. K. Abdullaev, R. M. Galimzyanov, M. Brtka and R. A. Kraenkel. Resonances in a trapped 3D Bose-Einstein condensate under periodically varying atomic scat-tering length [J]. J. Phys. B,2004,37:3535.
[111]J. K. Xue. Controllable compression of bright soliton matter waves [J]. J. Phys. B,2005,38:3841.
[112]S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn and W. Ketterle. Observation of Feshbach resonances in a Bose-Einstein condensate [J]. Nature,1998,392:151.
[113]M. Bartuccelli, P. Christiansen, N. Pedersen and M. Soerensen. Prediction of chaos in a Josephson junction by the Melnikov-function technique [J]. Phys. Rev. B,1986, 33:4686.
[114]H. Lignier, C. Sias, D. Cianipini, Y. Singh, A. Zcncsini,O. Morsch and E. Ari-mondo. Dynamical control of matter-wave tunneling in periodic potentials [J]. Phys. Rev. Lett.,2007,99:220403.
[115]S. Wimberger, R. Mannella, O. Morsch and F. Arimondo. Resonant nonlinear quantum transport for a periodically kicked Bose condensate [J]. Phys. Rev. Lett., 2005,94:130404.
[116]R. Dum, A. Sanpera, K. A. Suominen, M. Brewczyk, M. Kus, K. Rzazewski and M. Lewenstein. Wave packet dynamics with Bose-Einstein Condensates [J]. Phys. Rev. Lett.,1998,80:3899.
[117]J. L. Mateos. Chaotic transport and current reversal in deterministic ratchets [J]. Phys. Rev. Lett.,2000,84:258.
[118]T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina and M. Weitz. Directed transport of atoms in a Hamiltonian quantum ratchet [J]. Science,2009, 326:1241.
[119]R. J. Donnelly. Quantized vortices in Helium Ⅱ [M]. Cambridge:Cambridge Uni-versity Press,1991.
[120]L. M. Pismen. Vortices in nonlinear fields [M]. Oxford:Oxford University Press, 1999.
[121]Y. Castin, R. Dum. Bose-Einstein condensates with vortices in rotating traps [J]. Eur. Phys. J. D,1999,7:399.
[122]R. Dum, J. I. Cirac, M. Lewenstein and P. Zoller. Creation of dark solitons and vortices in Bose-Einstein condensates [J]. Phys. Rev. Lett.,1998,80:2972.
[123]L. Wu, L. Li, J. F. Zhang, D. Mihalache, B. A. Malomed and W. M. Liu. Exact so-lutions of the Gross-Pitaevskii equation for stable vortex modes in two-dimensional Bose-Einstein condensates [J]. Phys. Rev. A,2010,81:061805(R).
[124]L. H. Wen, H. Xiong and B. Wu. Hidden vortices in a Bose-Einstein condensate in a rotating double-well potential [J]. Phys. Rev. A,2010,82:053627.
[125]J. E. Williams. M. J. Holl. Preparing topological states of a Bose-Einstein Con-densate [J]. Nature,1999,401:568.
[126]K. W. Madison, F. Chevy, W. Wohllcben and J. Dalibard. Vortex formation in a stirred Bose-Einstein Condensate [J]. Phys. Rev. Lett.,2000,84:806.
[127]C. Raman, J. R. Abo-Shaeer. J. M. Vogels, K. Xu and W. Ketterle. Vortex nu-cleation in a stirred Bose-Einstein Condensate [J]. Phys. Rev. Lett.,2001.87: 210402.
[128]G. Andrelczyk. M. Brewczyk, L. Dobrek, M. Gajda and M. Lewenstein. Landau and dynamical instabilities of the superflow of Bose-Einstein condensates in optical lattices [J]. Phys. Rev. A,2001,64:043601(R).
[129]F. Dalfovo and S. Stringari. Bosons in anisotropic traps:Ground state and vortices [J]. Phys. Rev. A,1996,53:2477.
[130]M. Tsubota, K. Kasamatsua and M. Ueda. Dynamics of vortex lattice formation in a rotating Bose-Einstein condensate [J]. Physica B,2003,329:21.
[131]J. W. Reijnders and R.. A. Duine. Pinning and collective modes of a vortex lattice in a Bose-Einstein condensate [J]. Phys. Rev. A,2005,71:063607.
[132]P. G. Kevrekidis, R. Carretero-Gonzalez, D. J. Frantzeskakis and I. G. Kevrekidis. Vortices in Bose-Einstein Condensates:some recent developments [J]. Mod. Phys. Lett. B.2004,18:1481.
[133]K. J. H. Law, P. G. Kevrekidis, B. P. Anderson, R. Carretero-Gonzalez and D. J. Frantzeskakis. Structure and stability of two-dimensional Bose-Einstein con-densates under both harmonic and lattice confinemen [J]. J. Phys. B,2008,41: 195303.
[134]M. Modugno, L. Pricoupenko and Y. Castin. Bose-Einstein condensates with a bent vortex in rotating traps [J]. Eur. Phys. J. D,2003,22:235.
[135]R. Driben and B. A. Malomcd. Stabilization of two-dimensional solitons and vor-tices against supercritical collapse by lattice potentials [J]. Eur. Phys. J. D,2008, 50:317.
[136]H. C. Lee and T. F. Jiang. Band structures of a slowly rotating dipolar Bose-Einstein condensate with a quantized vortex along a one-dimensional optical lattice [J]. Eur. Phys. J. D,2010,58:311.
[137]J. P. Martikainen and H. T. C. Stoof. Quantum fluctuations of a vortex in an optical lattice [J]. Phys. Rev. Lett.,2003,91:240403.
[138]M. Snoek and H. T. C. Stoof. Vortex-lattice melting in a one-dimensional optical lattice [J]. Phys. Rev. Lett.,2006,96:230402.
[139]K. Kasamatsu and M. Tsubota. Dynamical vortex phases in a Bose-Einstein Con-densate driven by a rotating optical lattice [J]. Phys. Rev. Lett.,2006,97:240404.
[140]W. Hai, C. Lee and Q. Zhu. Exact Floquet states of a driven condensate and their stabilities [J]. J. Phys. B,2008,41:095301.
[141]W. Hai, G. Chong, Q. Xie and J. Lu. Optical operation of ultracold atomic quasi-clusters [J]. Eur. Phys. J. D,2004,28:267.
[142]W. Hai, Y. Li, B. Xia and X. Luo. Exact solutions of a two-component BEC interacting with a lattice potential [J]. Europhys. Lett.,2005,71:28.
[143]H. Rabitz, R. de Vivie-R.iedle, M. Motzkus and K. Kompa. Whither the future of controlling quantum phenomena? [J]. Science,2000,288:824.
[144]H. Fielding, M. Shapiro and T. Baumert. Coherent control [J]. J. Phys. B,2008, 41:070201.
[145]W. Hai, Q. Xie and S. Rong. Macroscopic quantum control of exact many-body coherent states [J]. Eur. Phys. J. D,2011,61:431.
[146]B. A. Malomed and Y. A. Stepanyants. The inverse problem for the Gross-Pitaevskii equation [J]. Chaos,2010,20:013130.
[147]T. Maytcevarunyoo and B. A. Malomcd. Stability limits for gap solitons in a Bose-Einstcin condensate trapped in a time-modulated optical lattice [J]. Phys. Rev. A,2006,74:033616.
[148]G. Burlak and B. A. Malomed. Dynamics of matter-wave solitons in a time-modulated two-dimensional optical lattice [J]. Phys. Rev. A,2008,77:053606.
[149]M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch and I. Bloch. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms [J]. Nature,2002,415:39.
[150]J. C. Bronski, L. D. Carr, B. Deconinck and J. N. Kutz. Bose-Einstein Conden-sates in standing waves:the cubic nonlinear Schrodinger equation with a periodic potential [J]. Phys. Rev. Lett.,2001,86:1402.
[151]A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon and H. Walthcr. Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium[J]. Phys. Rev. E,1999,60:6111.
[152]N. R. Cooper, N. K. Wilkin and J. M. F. Gunn. Quantum phases of vortices in rotating Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2001,87:120405.
[153]L. C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Perez-Garcia and D. Mihalache. Globally linked vortex clusters in trapped wave fields [J]. Phys. Rev. E,2002,66:036612.
[154]L. C. Crasovan, V. Vekslerchik, V. M. Perez-Garcia, J. P. Torres, D. Mihalache and L. Torner. Stable vortex dipoles in nonrotating Bose-Einstein condensates [J]. Phys. Rev. A,2003,68:063609.
[155]J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz and K. Promislow. Stability of repulsive Bose-Einstein condensates in a periodic potential [J]. Phys. Rev. E, 2001,63:036612.
[156]J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz and K. Promislow. Stability of attractive Bose-Einstein condensates in a periodic potential [J]. Phys. Rev. E,2001,64:056615.
[157]J. J. Garcia-Ripoll, G. Molina-Terriza, V. M. Pcrez-Garcia and L. Torner. Struc-tural instability of vortices in Bose-Einstein Condensates [J]. Phys. Rev. Lett., 2001,87:140403.
[158]B. Wu and Q. Niu. Landau and dynamical instabilities of the superflow of Bose-Einstein condensates in optical lattices [J]. Phys. Rev. A,2001,64:061603(R).
[159]A. L. Fetter and A. A. Svidzinsky. Vortices in a trapped dilute Bose-Einstein condensate [J]. J. Phys.:Condens. Matter,2001,13:R135.
[160]J. Y. Choi, W. J. Kwon, M. Lee, H. Jeong, K. An and Y. Shin. Imprinting Skyrmion spin textures in spinor Bose-Einstein condensates [J]. New J. Phys., 2012,14:053013.
[161]R. Barnett, G. Refael, M. A. Porter and H. P. Buchler. Vortex lattice locking in rotating two-component Bose-Einstein condensates [J]. New J. Phys.,2008,10: 043030.
[162]R. Barnett, E. Chen and G. Refael. Vortex synchronization in Bose-Einstein con-densates:a time-dependent Gross-Pitaevskii equation approach [J]. New J. Phys., 2010,12:043004.
[163]I. Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa. Motion of vortex lines in quantum mechanics [J]. Phys. Rev. A,2000,61:032110.
[164]A. Klein, D. Jaksch, Y. Zhang and W. Bao. Dynamics of vortices in weakly inter-acting Bose-Einstein condensates [J]. Phys. Rev. A,2007,76:043602.
[165]V. M. Perez-Garcia and M. A. Garcfa-March. Symmetry-assisted vorticity control in Bose-Einstein condensates [J]. Phys. Rev. A,2007,75:033618.
[166]O. V. Borovkova, Y. V. Kartashov, L. Torner and B. A. Malomed. Bright solitons from defocusing nonlinearities [J]. Phys. Rev. E,2011,84:035602(R).
[167]S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R.. Carretero-Gonzalez and P. Schmelcher. Emergence and stability of vortex clusters in Bose-Einstein con- densates:A bifurcation approach near the linear limit [J]. Physica D,2011,240: 1449.
[2]M. H. Anderson, J. R. Ensher, M. R. Matthews. C. E. Wieman and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor [J]. Science, 1995,269:198.
[3]C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet. Evidence of Bose-Einstein Condensation in an atomie gas with attractive interactions [J]. Phys. Rev. Lett.,1995,75:1687.
[4]K. B. Davis, M.0. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle. Bose-Einstein Condensation in a gas of sodium atoms [J]. Phys. Rev. Lett.,1995,75:3969.
[5]F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari. Theory of Bose-Einstein condensation in trapped gases [J]. Rev. Mod. Phys.,1999,71:463.
[6]江开军.铷原子玻色-爱因斯坦凝聚的实验研究[D].武汉：中国科学院武汉物理与数学研究所,2005.
[7]周蜀渝,龙全,周善钰,付海翔,王育竹.玻色-爱因斯坦凝聚在中国科学院上海光机所实现[J].物理,2002,31：481.
[8]陈帅,周小计,杨帆,夏林,马修泉,熊炜,王义遒,陈徐宗.北京大学玻色-爱因斯坦凝聚的实现及研究进展[J].量子光学学报,2004,10：4.
[9]S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag and R. Grimm. Bose-Einstein Condensation of molecules [J]. Science, 2003,302:2101.
[10]M. Greiner, C. A. Regal and D. S. Jin. Emergence of a molecular Bose-Einstein condensate from a Fermi gas [J]. Nature,2003,426:537.
[11]M. W. Zwierlcin, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadz-ibabic and W. Ketterle. Observation of Bose-Einstein Condensation of molecules [J], Phys. Rev. Lett.,2003,91:250401.
[12]C. A. Regal, M. Greiner and D. S. Jin. Observation of resonance condensation of Fermionic atom pairs [J]. Phys. Rev. Lett.,2004,92:040403.
[13]M. Bartenstein, A. Altmeyer, S. Riedl. S. Jochim, C. Chin, J. Hecker Denschlag and R. Grimm. Crossover from a molecular Bose-Einstein Condensate to a degen-erate Fermi gas [J]. Phys. Rev. Lett.,2004,92:120401.
[14]M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupaeh, A. J. Kerman and W. Ketterle. Condensation of pairs of Fermionic atoms near a Feshbach Resonance [J]. Phys. Rev. Lett.,2004,92:120403.
[15]熊宏伟,吕宝龙,刘淑娟,詹明生.超冷费米气体研究的新进展[J].物理学进展,2005,25：296.
[16]徐克尊.高等原子分子物理学[M].北京：科技出版社,2002：69.
[17]F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari. Theory of Boso-Einstein condensation in trapped gases [J]. Rev. Mod. Phys.,1999,71:463.
[18]A. J. Leggett. Bose-Einstein condensation in the alkali gases:Some fundamental concepts [J]. Rev. Mod. Phys.,2001,73:307.
[19]N. N. Bogoliubov. On the theory of superfluidity [J]. J. Phys. USSR,1947,11:23.
[20]E. P. Gross. Structure of a quantized vortex in Boson Systems [J]. Nuovo Cimento, 1961,20:454.
[21]E. P. Gross. Hydrodynamics of a superfluid condensate [J]. J. Math. Phys.,1963, 4:195.
[22]L. P. Pitaevskii. Vortex lines in an imperfect Bose gas [J]. Zh. Eksp. Teor. Fiz., 1961,40:646.
[23]V. M. Pcrez-Garcia, H. Michinel and H. Herrero. Bose-Einstein solitons in highly asymmetric traps [J]. Phys. Rev. A,1998,57:3837.
[24]K. E. Strecker, G. B. Partridge, A. G. Truscott and R. G. Hulet. Formation and propagation of matter-wave soliton trains [J]. Nature,2002,417:150.
[25]L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin and C. Salomon. Formation of a matter-wave bright soliton [J]. Science, 2002,296:1290.
[26]S. Burger, K. Bongs. S. Dettmer, W. Ertmer and K. Sengstock. Dark solitons in Bose-Einstein Condensates [J]. Phys. Rev. Lett.,1999,83:5198.
27] J. Denschlag, J. E. Simsarian, D. L. Feder, Charles W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Hclmerson, W. P. Reinhardt, S. L. R.olston, B. I. Schneider and W. D. Phillips. Generating solitons by phase engineering of a Bose-Einstein Condensate [J]. Science,2000,287:97.
[28]C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet. Evidence of Bose-Einstein Condensation in an atomic gas with attractive interactions [J]. Phys. Rev. Lett.,1995,75:1687.
[29]Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell and C. E. Wieman. Dynamics of collapsing and exploding Bose-Einstein condensates [J]. Nature,2001,412:295.
[30]F. Sols. Vortex matter in atomic Bose-Einstein condensates [J]. Physica C:Super-conductivity,2002,369:125.
[31]V. M. Perez-Garcia, V. V. Konotop and V. A. Brazhnyi. Feshbach resonance induced shock waves in Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2004,92: 220403.
[32]P. J. Chen. Growth and decay of waves in solids [M]. Berlin:Springer,1973:303.
[33]J. Serrin. Mathematical principles of classical fluid dynamics [M]. Berlin:Springer, 1959:125.
[34]J. Sinoller and B. Temple. Shock-wave cosmology inside a black hole [J]. Proc. Natl Acad. Sci. USA,2003,100:11216.
[35]R. N. C. Pfeifer and T. A. Nieminen. Visualization of Cerenkov radiation and the fields of a moving charge [J]. Eur. J. Phys.,2006,27:521.
[36]R. Quintanilla and B. Straughan. A note on discontinuity waves in type III ther-moelasticity [J]. Proc. R. Soc. A,2004,460:1369.
[37]A. Morro. Jump relations and discontinuity waves in conductors with memory [J]. Math. Comput. Modelling,2006,43:138.
[38]B. D. Coleman and D. C. Newman. Implications of a nonlinearity in the theory of second sound in solids [J]. Phys. Rev. B,1988,37:1492.
[39]D. Jou, J. Casas-Vazquez and G. Lebon. Extended irreversible thermodynamics [M]. Berlin:Springer,2001:41.
[40]C. I. Christov and P. M. Jordan. Heat conduction paradox involving second-sound propagation in moving media [J]. Phys. Rev. Lett.,2005,94:154301.
[41]Z. Dutton, M. Budde, C. Slowe and L. V. Hau. Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein Conden-sate [J]. Science,2001,293:663.
[42]T. P. Simula, P. Engels, I. Coddington, V. C. Schweikhard, E. A. Cornell and R. J. Ballagh. Observations on sound propagation in rapidly rotating Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2005,94:080404.
[43]L. Onsager. Statistical Hydrodynamics [J]. Nuovo Cimento Suppl.,1949,6:279.
[44]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman and E. A. Cornell. Vortices in a Bose-Einstein Condensate [J]. Phys. Rev. Lett.,1999,83: 2498.
[45]J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle. Observation of vortex lattices in Bose-Einstein Condensates [J]. Science,2001,292:476.
[40]D. S. Rokhsar. Vortex stability and persistent currents in trapped Bose gases [J]. Phys. Rev. Lett.,79:2164.
[47]E. Hodby, G. Hechenblaikner, S.A. Hopkins, O. M. Marago and C. J. Foot. Vortex nucleation in Bose-Einstein Condensates in an oblate, purely magnetic potential [J]. Phys. Rev. Lett.,2001,88:010405.
[48]B. Jackson, J. F. McCann and C. S. Adams. Vortex formation in dilute inhomo-geneous Bose-Einstein Condensates [J]. Phys. Rev. Lett.,1998,80:3903.
[49]L. Dobrek, M. Gajda, M. Lewenstein, K. Sengstock, G. Birkl and W. Ertmer. Optical generation of vortices in trapped Bose-Einstein condensates [J]. Phys. Rev. A,1999,60:R3381.
[50]A. E. Leanhardt, A. Gorlitz, A. P. Chikkatur, D. Kielpinski, Y. Shin, D. E. Pritchard and W. Ketterle. Imprinting vortices in a Bose-Einstein Condensate using topological phases [J]. Phys. Rev. Lett.,2002,89:190403.
[51]T. Isoshima, M. Nakahara, T. Ohmi and K. Machida. Creation of a persistent current and vortex in a Bose-Einstein condensate of alkali-metal atoms [J]. Phys. Rev. A,2000,61:063610.
[52]B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark and E. A. Cornell. Watching dark solitons decay into vortex rings in a Bose-Einstein Condensate [J]. Phys. Rev. Lett.,2001,86:2926.
[53]S. Inouye, S. Gupta, T. Rosenband, A. P. Chikkatur, A. Gorlitz, T. L. Gustavson, A. E. Leanhardt, D. E. Pritchard and W. Ketterle. Observation of vortex phase singularities in Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2001,87:080402.
[54]E. L. Bolda. D. F. Walls. Detection of vorticity in Bose-Einstein Condensed gases by matter-wave interference [J]. Phys. Rev. Lett.,1998,81:5477.
[55]J. Tempere, J. T. Devreese. Fringe pattern of interfering Bose-Einstein Condensate with a vortex [J]. Solid State Commu.,1998,108:993.
[50]Q. Chen, J. Stajic, S. Tan and K. Levin. BCS-BEC crossover:From high tem-perature.superconductors to ultracold superfluids [J]. Physics Reports,2005,412: 1.
[57]B. Borca, J. W. Dunn. V. Kokoouline and C. H. Greene. Atom-molecule laser fed by stimulated three-body recombination [J]. Phys. Rev. Lett.,2003,91:070404.
[58]S. A. Haine, J. J. Hope, N. P. Robins and C. M. Savage. Stability of continuously pumped atom lasers [J]. Phys. Rev. Lett.,2002,88:170403.
[59]G. Cennini, G. Ritt, C. Geckeler and M. Weitz. All-optical realization of an atom laser [J]. Phys. Rev. Lett.,2003,91:240408.
[60].T. K. Pachos and P. L. Knight. Quantum computation with a one-dimensional optical lattice [J]. Phys. Rev. Lett.,2003,91:107902.
[61]R. Ionicioiu and P. Zanardi. Quantum-information processing in bosonic lattices [J]. Phys. Rev. A,2002,66:050301 (R).
[62]C. J. Pet hick and H. Smith. Bose-Einstein Condensation in dilute gases [M]. Cam-bridge:Cambridge University Press,2002.
[63]M. Inguscio, S. Stringari and C. E. Wieman. Bose-Einstein Condensation in atomic gases [M]. Amsterdam:IOS Press,1999.
[64]L. Pitaevskii and S. Stringari. Bose-Einstein Condensation [M]. Oxford:Oxford University Press,2003.
[65]J. O. Andersen. Theory of the weakly interacting Bose gas[J]. Rev. Mod. Phys., 2004,76:599.
[66]A. Griffin, D. W. Snoke and S. Stringari. Bose-Einstein Condensation [M], Cam-bridge:Cambridge University Press,1995.
[67]G. B. Whitham. Linear and nonlinear waves [M]. New York:Wiley,1999.
[68]R. Courant and K. O. Friedrichs. Supersonic flow and shock Waves [M]. Berlin: Springer-Verla,1948.
[69]F. V. Shugaev and L. S. Shtemenko. Propagation and reflection of shock waves [M]. Singapore:World Scientific,1998.
[70]V. T. Gurovich, A. Grinenko and Y. E. Krasik. Semianalytical solution of the problem of converging shock waves [J]. Phys. Rev. Lett.,2007,99:124503.
[71]A. B. Shvartsburg, L. Stenflo and P. K. Shukla. Breakdown of waves described by exact solutions of the Thomas-Fermi model [J]. Eur. Phys. J. B,2002,28:71.
[72]V. A. Brazhnyi, A. M. Kamchatnov and V. V. Konotop. Hydrodynamic flow of expanding Bose-Einstein condensates [J]. Phys. Rev. A,2003,68:035603.
[73]I. Kulikov and M. Zak. Shock waves in a Bose-Einstein condensate [J]. Phys. Rev. A,2003,67:063605.
[74]B. Damski. Formation of shock waves in a Bose-Einstein condensate [J]. Phys. Rev. A,2004,69:043610.
[75]W. H. Hai, Q. Q. Zhu and S. G. Rong. Chaotic shock waves of a Bose-Einstein condensate [J]. Phys. Rev. A,2009; 79:023603.
[76]Y. Castin and R. Dum. Bose-Einstein Condensates in time dependent traps [J]. Phys. Rev. Lett.,1996,77:5315.
[77]Z. X. Liang, Z. D. Zhang and W. M. Liu. Dynamics of a bright soliton in Bose-Ein-stein Condensates with time-dependent atomic scattering length in an expulsive parabolic potential [J]. Phys. Rev. Lett.,2005,94:050402.
[78]F. K. Abdullaev and R. Galimzyanov. The dynamics of bright matter wave solitons in a quasi one-dimensional Bose-Einstein condensate with a rapidly varying trap [J]. J. Phys. B,2003,36:1099.
[79]G. S. Chong and W. H. Hai. Dynamical evolutions of matter-wave bright solitons in an inverted parabolic potential [J]. J. Phys. B,2007,40:211.
[80]Z. Zhou, H. Y. Yu and J. R. Yan. Dynamics of vector solitons in two-component Bose-Einstein condensates with time-dependent interactions and harmonic poten-tial [J]. Chin. Phys. B,2010,19:010304.
[81]C. A. Sackctt, H. T. C.Stoof and R. G. Hulct. Growth and collapse of a Bosc-Einstciu Condensate with attractive interactions [J]. Phys. Rev. Lett.,1998,80: 2031.
[82]Y. Kagan, A. E. Muryshev and G. V. Shlyapnikov. Collapse and Bose-Einstein Condensation in a trapped Bose gas with negative scattering length [J]. Phys. Rev. Lett.,1998,81:933.
[83]A. Gammal, T. Frederico, L. Tomio and P. Chomaz. Liquid-gas phase transition in Bose-Einstein condensates with time evolution [J]. Phys. Rev. A,2000,61: 051602(R).
[84]Y. Li and W. H. Hai. Three-body recombination in two coupled Bose-Einstein condensates [J]. J. Phys. A.2005,38:4105.
[85]A. X. Mu, X. Y. Zhou and J. K. Xuc. The dynamics of triple-well trapped Bose-Einstein condensates with atoms feeding and loss effects [J]. Chin. Phys. B,2008, 17:764.
[86]Hai W H, X. L. Zhang, W. L. Huang and G. S. Chong. Chaotic solution of the rf-driven josephson system with quadratic damping [J]. Int. J. Bifurcation and Chaos,2001,11:2263.
[87]G. X. Huang. Second harmonic generation of propagating collective excitations in Bose-Einstein condensates [J]. Chin. Phys.,2004,13:1866.
[88]W. H. Hai, Y. W. Duan and L. X. Pan. An analytical study for controlling unstable periodic motion in magneto-elastic chaos [J]. Phys. Lett. A,1997,234:198.
[89]W. H. Hai, S. G. R.ong and Q. Q. Zhu. Discrete chaotic states of a Bose-Einstein condensate [J]. Phys. Rev. E,2008,78:066214.
[90]J. S. Fang. Nonlinear transport of Bose-Einstein condensates in a double barrier potential [J]. Chin. Phys. B,2008,17:3996.
[91]V. K. Melnikov. On the stability of a center for time-periodic perturbations [J]. Trans. Moscow Math. Soc.,1963,12:1.
[92]Q. Q. Zhu, W. H. Hai and S. G. Rong. Transition probability from matter-wave soliton to chaos [J]. Phys. Rev. E,2009,80:016203.
[93]S. Parthasarathy. Homoclinic bifurcation sets of the parametrically driven duffing oscillator [J]. Phys. Rev. A,1992,46:2147.
[94]M. A. Hoofer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels and V. Schweikhard. Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics [J]. Phys. Rev. A,2006,74:023623.
[95]H. Saito and M. Ueda. Intermittent implosion and pattern formation of trapped Bose-Einstein Condensates with an attractive interaction [J]. Phys. Rev. Lett., 2001,86:1406.
[96]M. P. Brenner, S. Hilgenfeldt and D. Lohse. Single-bubble sonoluminescence [J]. Rev. Mod. Phys.,2002,74:425.
[97]E. J. Reed, M. Soljacic and J. D. Joannopoulos. Color of shock waves in photonic crystals [J]. Phys. Rev. Lett.,2003,90:203904.
[98]T. Brunet, J. L. Thomas, R.. Marchiano and F. Coulouvrat. Experimental obser-vation of azimuthal shock waves on nonlinear acoustical vortices [J]. New. J. Phys, 2009,11:013002.
[99]R. A. Austin, D. L. McDowell and D. J. Benson. Numerical simulation of shock wave propagation in spatially-resolved particle systems [J]. Modelling Simul. Mater. Sci. Eng.,2006,14:537.
[100]I. Christov and P. M. Jordan. Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation [J]. New. J. Phys,2008,10:043027.
[101]Y. Deng, W. Hai, S. Rong and H. Zhong. Chaotic, matter shock wave of an open system [J]. Chin. Phys. B,2010,19:120502.
[102]S. Horluck, M. V. Hecke and P. Dimon. Shock waves in two-dimensional granular flow:Effects of rough walls and polydispersity [J]. Phys. Rev. E,2003,67:021304.
[103]A. M. Kamchatnov, A. Gammal and R. A. Kraenkcl. Dissipationless shock waves in Bosc-Einstein condensates with repulsive interaction between atoms [J]. Phys. Rev. A,2004,69:063605.
[104]S. Ghosh. Dust acoustic shock waves in two-component dusty plasma [J]. New J. Phys.,2003,5:142.
[105]M. A. Hoefer and B. Ilan. Theory of two-dimensional oblique dispersive shock waves in supersonic flow of a superfluid [J]. Phys. Rev. A,2009,80:061601(R).
[106]X. Y. Tang and P. K. Shukla. Symmetry analysis and similarity electrostatic waves in a nonuniform dusty magnetoplasma [J]. J. Phys. A:Math. Theor.,2007,40: 5921.
[107]P. A. M. Dirac. Relativity and quantum mechanics|J]. Fields and Quanta,1972, 3:139.
[108]K. Boaz. M. Peter and W. Eli. The spectrum of cosmic rays escaping from rela-tivistic shocks [J]. J. Cosmol. Astropart. Phys.,2010,10:012.
[109]G. Lu and W. Hai. Quantum tunneling switch in a planar four-well system [J]. Phys. Rev. A,2011,83:053424.
[110]F. K. Abdullaev, R. M. Galimzyanov, M. Brtka and R. A. Kraenkel. Resonances in a trapped 3D Bose-Einstein condensate under periodically varying atomic scat-tering length [J]. J. Phys. B,2004,37:3535.
[111]J. K. Xue. Controllable compression of bright soliton matter waves [J]. J. Phys. B,2005,38:3841.
[112]S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn and W. Ketterle. Observation of Feshbach resonances in a Bose-Einstein condensate [J]. Nature,1998,392:151.
[113]M. Bartuccelli, P. Christiansen, N. Pedersen and M. Soerensen. Prediction of chaos in a Josephson junction by the Melnikov-function technique [J]. Phys. Rev. B,1986, 33:4686.
[114]H. Lignier, C. Sias, D. Cianipini, Y. Singh, A. Zcncsini,O. Morsch and E. Ari-mondo. Dynamical control of matter-wave tunneling in periodic potentials [J]. Phys. Rev. Lett.,2007,99:220403.
[115]S. Wimberger, R. Mannella, O. Morsch and F. Arimondo. Resonant nonlinear quantum transport for a periodically kicked Bose condensate [J]. Phys. Rev. Lett., 2005,94:130404.
[116]R. Dum, A. Sanpera, K. A. Suominen, M. Brewczyk, M. Kus, K. Rzazewski and M. Lewenstein. Wave packet dynamics with Bose-Einstein Condensates [J]. Phys. Rev. Lett.,1998,80:3899.
[117]J. L. Mateos. Chaotic transport and current reversal in deterministic ratchets [J]. Phys. Rev. Lett.,2000,84:258.
[118]T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina and M. Weitz. Directed transport of atoms in a Hamiltonian quantum ratchet [J]. Science,2009, 326:1241.
[119]R. J. Donnelly. Quantized vortices in Helium Ⅱ [M]. Cambridge:Cambridge Uni-versity Press,1991.
[120]L. M. Pismen. Vortices in nonlinear fields [M]. Oxford:Oxford University Press, 1999.
[121]Y. Castin, R. Dum. Bose-Einstein condensates with vortices in rotating traps [J]. Eur. Phys. J. D,1999,7:399.
[122]R. Dum, J. I. Cirac, M. Lewenstein and P. Zoller. Creation of dark solitons and vortices in Bose-Einstein condensates [J]. Phys. Rev. Lett.,1998,80:2972.
[123]L. Wu, L. Li, J. F. Zhang, D. Mihalache, B. A. Malomed and W. M. Liu. Exact so-lutions of the Gross-Pitaevskii equation for stable vortex modes in two-dimensional Bose-Einstein condensates [J]. Phys. Rev. A,2010,81:061805(R).
[124]L. H. Wen, H. Xiong and B. Wu. Hidden vortices in a Bose-Einstein condensate in a rotating double-well potential [J]. Phys. Rev. A,2010,82:053627.
[125]J. E. Williams. M. J. Holl. Preparing topological states of a Bose-Einstein Con-densate [J]. Nature,1999,401:568.
[126]K. W. Madison, F. Chevy, W. Wohllcben and J. Dalibard. Vortex formation in a stirred Bose-Einstein Condensate [J]. Phys. Rev. Lett.,2000,84:806.
[127]C. Raman, J. R. Abo-Shaeer. J. M. Vogels, K. Xu and W. Ketterle. Vortex nu-cleation in a stirred Bose-Einstein Condensate [J]. Phys. Rev. Lett.,2001.87: 210402.
[128]G. Andrelczyk. M. Brewczyk, L. Dobrek, M. Gajda and M. Lewenstein. Landau and dynamical instabilities of the superflow of Bose-Einstein condensates in optical lattices [J]. Phys. Rev. A,2001,64:043601(R).
[129]F. Dalfovo and S. Stringari. Bosons in anisotropic traps:Ground state and vortices [J]. Phys. Rev. A,1996,53:2477.
[130]M. Tsubota, K. Kasamatsua and M. Ueda. Dynamics of vortex lattice formation in a rotating Bose-Einstein condensate [J]. Physica B,2003,329:21.
[131]J. W. Reijnders and R.. A. Duine. Pinning and collective modes of a vortex lattice in a Bose-Einstein condensate [J]. Phys. Rev. A,2005,71:063607.
[132]P. G. Kevrekidis, R. Carretero-Gonzalez, D. J. Frantzeskakis and I. G. Kevrekidis. Vortices in Bose-Einstein Condensates:some recent developments [J]. Mod. Phys. Lett. B.2004,18:1481.
[133]K. J. H. Law, P. G. Kevrekidis, B. P. Anderson, R. Carretero-Gonzalez and D. J. Frantzeskakis. Structure and stability of two-dimensional Bose-Einstein con-densates under both harmonic and lattice confinemen [J]. J. Phys. B,2008,41: 195303.
[134]M. Modugno, L. Pricoupenko and Y. Castin. Bose-Einstein condensates with a bent vortex in rotating traps [J]. Eur. Phys. J. D,2003,22:235.
[135]R. Driben and B. A. Malomcd. Stabilization of two-dimensional solitons and vor-tices against supercritical collapse by lattice potentials [J]. Eur. Phys. J. D,2008, 50:317.
[136]H. C. Lee and T. F. Jiang. Band structures of a slowly rotating dipolar Bose-Einstein condensate with a quantized vortex along a one-dimensional optical lattice [J]. Eur. Phys. J. D,2010,58:311.
[137]J. P. Martikainen and H. T. C. Stoof. Quantum fluctuations of a vortex in an optical lattice [J]. Phys. Rev. Lett.,2003,91:240403.
[138]M. Snoek and H. T. C. Stoof. Vortex-lattice melting in a one-dimensional optical lattice [J]. Phys. Rev. Lett.,2006,96:230402.
[139]K. Kasamatsu and M. Tsubota. Dynamical vortex phases in a Bose-Einstein Con-densate driven by a rotating optical lattice [J]. Phys. Rev. Lett.,2006,97:240404.
[140]W. Hai, C. Lee and Q. Zhu. Exact Floquet states of a driven condensate and their stabilities [J]. J. Phys. B,2008,41:095301.
[141]W. Hai, G. Chong, Q. Xie and J. Lu. Optical operation of ultracold atomic quasi-clusters [J]. Eur. Phys. J. D,2004,28:267.
[142]W. Hai, Y. Li, B. Xia and X. Luo. Exact solutions of a two-component BEC interacting with a lattice potential [J]. Europhys. Lett.,2005,71:28.
[143]H. Rabitz, R. de Vivie-R.iedle, M. Motzkus and K. Kompa. Whither the future of controlling quantum phenomena? [J]. Science,2000,288:824.
[144]H. Fielding, M. Shapiro and T. Baumert. Coherent control [J]. J. Phys. B,2008, 41:070201.
[145]W. Hai, Q. Xie and S. Rong. Macroscopic quantum control of exact many-body coherent states [J]. Eur. Phys. J. D,2011,61:431.
[146]B. A. Malomed and Y. A. Stepanyants. The inverse problem for the Gross-Pitaevskii equation [J]. Chaos,2010,20:013130.
[147]T. Maytcevarunyoo and B. A. Malomcd. Stability limits for gap solitons in a Bose-Einstcin condensate trapped in a time-modulated optical lattice [J]. Phys. Rev. A,2006,74:033616.
[148]G. Burlak and B. A. Malomed. Dynamics of matter-wave solitons in a time-modulated two-dimensional optical lattice [J]. Phys. Rev. A,2008,77:053606.
[149]M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch and I. Bloch. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms [J]. Nature,2002,415:39.
[150]J. C. Bronski, L. D. Carr, B. Deconinck and J. N. Kutz. Bose-Einstein Conden-sates in standing waves:the cubic nonlinear Schrodinger equation with a periodic potential [J]. Phys. Rev. Lett.,2001,86:1402.
[151]A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon and H. Walthcr. Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium[J]. Phys. Rev. E,1999,60:6111.
[152]N. R. Cooper, N. K. Wilkin and J. M. F. Gunn. Quantum phases of vortices in rotating Bose-Einstein Condensates [J]. Phys. Rev. Lett.,2001,87:120405.
[153]L. C. Crasovan, G. Molina-Terriza, J. P. Torres, L. Torner, V. M. Perez-Garcia and D. Mihalache. Globally linked vortex clusters in trapped wave fields [J]. Phys. Rev. E,2002,66:036612.
[154]L. C. Crasovan, V. Vekslerchik, V. M. Perez-Garcia, J. P. Torres, D. Mihalache and L. Torner. Stable vortex dipoles in nonrotating Bose-Einstein condensates [J]. Phys. Rev. A,2003,68:063609.
[155]J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz and K. Promislow. Stability of repulsive Bose-Einstein condensates in a periodic potential [J]. Phys. Rev. E, 2001,63:036612.
[156]J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz and K. Promislow. Stability of attractive Bose-Einstein condensates in a periodic potential [J]. Phys. Rev. E,2001,64:056615.
[157]J. J. Garcia-Ripoll, G. Molina-Terriza, V. M. Pcrez-Garcia and L. Torner. Struc-tural instability of vortices in Bose-Einstein Condensates [J]. Phys. Rev. Lett., 2001,87:140403.
[158]B. Wu and Q. Niu. Landau and dynamical instabilities of the superflow of Bose-Einstein condensates in optical lattices [J]. Phys. Rev. A,2001,64:061603(R).
[159]A. L. Fetter and A. A. Svidzinsky. Vortices in a trapped dilute Bose-Einstein condensate [J]. J. Phys.:Condens. Matter,2001,13:R135.
[160]J. Y. Choi, W. J. Kwon, M. Lee, H. Jeong, K. An and Y. Shin. Imprinting Skyrmion spin textures in spinor Bose-Einstein condensates [J]. New J. Phys., 2012,14:053013.
[161]R. Barnett, G. Refael, M. A. Porter and H. P. Buchler. Vortex lattice locking in rotating two-component Bose-Einstein condensates [J]. New J. Phys.,2008,10: 043030.
[162]R. Barnett, E. Chen and G. Refael. Vortex synchronization in Bose-Einstein con-densates:a time-dependent Gross-Pitaevskii equation approach [J]. New J. Phys., 2010,12:043004.
[163]I. Bialynicki-Birula, Z. Bialynicka-Birula and C. Sliwa. Motion of vortex lines in quantum mechanics [J]. Phys. Rev. A,2000,61:032110.
[164]A. Klein, D. Jaksch, Y. Zhang and W. Bao. Dynamics of vortices in weakly inter-acting Bose-Einstein condensates [J]. Phys. Rev. A,2007,76:043602.
[165]V. M. Perez-Garcia and M. A. Garcfa-March. Symmetry-assisted vorticity control in Bose-Einstein condensates [J]. Phys. Rev. A,2007,75:033618.
[166]O. V. Borovkova, Y. V. Kartashov, L. Torner and B. A. Malomed. Bright solitons from defocusing nonlinearities [J]. Phys. Rev. E,2011,84:035602(R).
[167]S. Middelkamp, P. G. Kevrekidis, D. J. Frantzeskakis, R.. Carretero-Gonzalez and P. Schmelcher. Emergence and stability of vortex clusters in Bose-Einstein con- densates:A bifurcation approach near the linear limit [J]. Physica D,2011,240: 1449.