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研究了(1+1)维的变系数Gross-Pitaevskii方程, 获得了该方程的精确畸形波解. 基于该精确畸形波解, 深入研究了非自治物质畸形波在随时间指数变化的相互作用下的传播动力学行为, 发现非自治畸形波除具有“来无影、去无踪”的不可预测特性外, 也可实现完全激发、抑制激发以及维持激发等操控. 研究表明, 畸形波操控的关键是对累积时间的最大值Tmax 与峰值位置T0 (或TI,TII)值大小关系的调节. 当Tmax > T0 (或TI,TII)时畸形波被快速地完全激发, 热原子团中的原子增加到凝聚体中. 当Tmax = T0 (或TI,TII) 时畸形波激发到最大振幅, 可以维持相当长的时间而不消失, 热原子团中的原子增加到凝聚体中. 当Tmax T0 (或TI,TII)时畸形波没有充足的时间来激发而被抑制甚至消失, 凝聚体中的原子减少. 这些结果在理论和实际应用上具有启迪意义.
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关键词:
- Gross-Pitaevskii方程 /
- 变系数 /
- 畸形波解 /
- 操控
We study a (1+1)-dimensional variable-coefficient Gross-Pitaevskii equation with parabolic potential. A similarity transformation connecting the variable-coefficient Gross-Pitaevskii equation with the standard nonlinear Schrödinger equation is constructed. According to this transformation and solutions of the standard nonlinear Schrodinger equation, we obtain exact rogue wave solutions of variable-coefficient Gross-Pitaevskii equation with parabolic potential. In this solution, a Galilean transformation is used such that the center of optical pulse is Xc = v(T-T0) while the Galilean transformation was not used in previous analysis. By the Galilean transformation, the parameter T0 is added into the solution. It is found that the parameter T0 is important to control the excitations of rogue waves. Moreover, the parameters a1 and a2 in solution are complex parameters which can modulate the behaviors of rogue waves. If they are restricted to real numbers, we can obtain some well-known rogue wave solutions. If the parameter a2 =-1/12, we can have a second-order rogue wave solution. If the parameter a2 is a complex number, the solution can describe rogue wave triplets. Here two kinds of rogue wave triplets, namely, rogue wave triplets I and II are presented. For rogue wave triplet I, at first, two first-order rogue waves on each side are excited, and then a first-order rogue wave in the middle is excited with the increase of time. On the contrary, for rogue wave triplet II, a first-order rogue wave in the middle is initially excited, and then two first-order rogue waves on each side are excited with the increase of time.#br#From these solutions, the controls for the excitations of rogue waves, such as the restraint, maintenance and postponement, are investigated in a system with an exponential-profile interaction. In this system, by modulating the relation between the maximum of accumulated time Tmax and the peak time T0 (or TI,TII), we realize the controls of rogue waves. When Tmax > T0 (or TI,TII), rogue wave is excited quickly, and the atom number of condensates increases; when Tmax = T0 (or TI,TII), rogue wave is excited to the maximum amplitude, then maintains this magnitude for a long time, and the atom number of condensates also increases; when Tmax T0 (or TI,TII), the threshold of exciting rogue wave is never reached, thus the complete excitation is restrained, and the atom number of condensates reduces. These results can be used to understand rogue waves better, that is, besides their "appearing from nowhere and disappearing without a trace", rogue waves can be controlled as discussed by a similar way in this paper. These manipulations for rogue waves give edification on theory and practical application.-
Keywords:
- Gross-Pitaevskii equation /
- variable coefficient /
- rogue wave solutions /
- control
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[1] Osborne A R 2009 Nonlinear Ocean Waves (New York: Academic Press)
[2] Kharif C, Pelinovsky E, Slunyaev A 2009 Rogue Waves in the Ocean, Observation, Theories and Modeling(New York: Springer)
[3] Draper L 1965 Marine Observer 35 193
[4] Solli D R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
[5] Dudley J M, Genty G, Eggleton B J 2008 Opt. Express 16 3644
[6] Bludov Yu V, Konotop V V, Akhmedicv N 2009 Opt. Lett. 34 3015
[7] Bludov Yu V, Konotop V V, Akhmedicv N 2009 Phys. Rev. A 80 033610
[8] Yan Z Y 2010 Phys. Lett. A 374 672
[9] Wen L, Li L, Li Z D, Song S W, Zhang X F, Liu W M 2011 Eur. Phys. J. D 64 473
[10] Moslem W M 2011 Phys. Plasmas 18 032301
[11] Stenflo L, Marklund M 2010 J. Plasma Phys. 76 293
[12] Ma Z Y, Ma S H 2012 Chin. Phys. B 21 030507
[13] Tao Y S, He J S, Porsezian K 2013 Chin. Phys. B 22 074210
[14] Wang X, Chen Y 2014 Chin. Phys. B 23 070203
[15] Zhang J F, Jin M Z, He J D, Lou J H, Dai C Q 2013 Chin. Phys. B 22 054208
[16] Hu W C, Zhang J F, Zhao B, Lou J H 2013 Acta Phys. Sin. 62 024216 (in Chinese) [胡文成, 张解放, 赵辟, 楼吉辉 2013 物理学报 62 024216]
[17] Pan N, Huang P, Huang L G, Lei M, Liu W J 2015 Acta Phys. Sin. 64 090504 (in Chinese) [潘楠, 黄平, 黄龙刚, 雷鸣, 刘文军 2015 物理学报 64 090504]
[18] Sun Q H, Pan N, Lei M, Liu W J 2014 Acta Phys. Sin. 63 150506 (in Chinese) [孙庆华, 潘楠, 雷鸣, 刘文军 2014 物理学报 63 150506]
[19] Feshbach H P 1992 Theoretical Nuclear Physics(New York: Wiley)
[20] Li B, Zhang X F, Li Y Q, Chen Y, Liu W M 2008 Phys. Rev. A 78 023608
[21] Zhao L C 2013 Ann. Phys. 329 73
[22] Zhang J F, Yang Q 2005 Chin. Phys. Lett. 22 1855
[23] Pérez García V M, Michinel H, Herrero H 1998 Phys. Rev. A 57 3837
[24] Yang Q, Zhang H J 2008 Chin. J. Phys. 46 457
[25] Ohta Y, Yang J K 2012 Proc. R. Soc. A 468 1716
[26] Akhmediev N, Ankiewicz A 1997 Solitons, Nonlinear Pulses and Beams(London: Chapman and Hall)
[27] Peregrine D H 1983 J. Australian Math. Soc. Ser. B 25 16
[28] Strecker K E, Partridge G B, Truscott A G, Hulet R G 2002 Nature 417 150
[29] Liang Z X, Zhang Z D, Liu W M 2005 Phys. Rev. Lett. 94 050402
[30] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Salomon J 2002 Science 296 1290
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