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Rogue waves for a \((2+1)\)-dimensional Gross-Pitaevskii equation with time-varying trapping potential in the Bose-Einstein condensate. (English) Zbl 1443.81040
Summary: The Bose-Einstein condensates (BECs) are seen during the studies in atomic optics, cavity opto-mechanics, cavity quantum electrodynamics, black-hole astrophysics, laser optics and atomtronics. Under investigation in this paper is a \((2+1)\)-dimensional Gross-Pitaevskii equation with time-varying trapping potential which describes the dynamics of a \((2+1)\)-dimensional BEC. Based on the Kadomtsev-Petviashvili hierarchy reduction, we construct the bilinear forms and the \(N\) th order rogue-wave solutions in terms of the Gramian. With the help of the analytic and graphic analysis, we exhibit the first- and second-order rogue waves under the influence of the strength of the interatomic interaction, \(\alpha(t)\), and of \(\Omega(t)=\omega_{\widetilde{R}}/\omega_Z\), where \(t\) is the scaled time, \(\omega_{\widetilde{R}}\) and \(\omega_Z\) are the confinement frequencies in the radial and axial directions: When \(\Omega(t)=0\), the first-order rogue wave exhibits as an eye-shaped distribution; When \(\Omega(t)\) is a periodic function, the rogue wave periodically raises; When \(\Omega(t)\) is an exponential function, the rogue wave appears on the exponentially increasing background; When \(\alpha(t)\) decreases, background and amplitude of the rogue wave both increase. The second-order rogue waves reach the maxima only once or three times when \(\Omega(t)\) is a constant. With \(\Omega(t)\) being the exponential function, the backgrounds of the second-order rogue waves exponentially increase with \(t\) increasing.
MSC:
81Q80 Special quantum systems, such as solvable systems
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q82 PDEs in connection with statistical mechanics
78A60 Lasers, masers, optical bistability, nonlinear optics
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References:
[1] Anderson, M. H.; Ensher, J. R.; Matthews, M. R.; Wieman, C. E.; Cornell, E. A., Science, 269, 198 (1995)
[2] Abdullaev, F. K.; Kamchatnov, A. M.; Konotop, V. V.; Brazhnyi, V. A., Phys. Rev. Lett., 90, Article 230402 pp. (2003)
[3] Brennecke, F.; Ritter, S.; Donner, T.; Esslinger, T., Science, 322, 235 (2008)
[4] Brennecke, F.; Donner, T.; Ritter, S.; Bourdel, T.; Köhl, M.; Esslinger, T., Nature, 450, 268 (2007)
[5] Garay, L. J.; Anglin, J. R.; Cirac, J. I.; Zoller, P., Phys. Rev. Lett., 85, 4643 (2000)
[6] Sakhel, R. R.; Sakhel, A. R.; Ghassib, H. B.; Balaz, A., Eur. Phys. J. D, 70, 66 (2016)
[7] Weiss, P., Sci. News Online, 157, 104 (2000)
[8] Javanainen, J.; Yoo, S. M., Phys. Rev. Lett., 76, 161 (1996)
[9] Pitaevskii, L. P.; Stringari, S., Bose-Einstein Condensation (2003), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 1110.82002
[10] Busch, T.; Anglin, J. R., Phys. Rev. Lett., 87, Article 010401 pp. (2001)
[11] Trombettoni, A.; Smerzi, A., Phys. Rev. Lett., 86, 2353 (2001)
[12] Salasnich, L.; Parola, A.; Reatto, L., Phys. Rev. Lett., 91, Article 080405 pp. (2003)
[13] Strecker, K. E.; Partridge, G. B.; Truscott, A. G.; Hulet, R. G., Nature, 417, 150 (2002)
[14] Akhmediev, N.; Ankiewicz, A.; Soto-Crespo, J. M., Phys. Rev. E, 80, Article 026601 pp. (2009)
[15] Baronio, F.; Degasperis, A.; Conforti, M.; Wabnitz, S., Phys. Rev. Lett., 109, Article 044102 pp. (2012)
[16] Manikandan, K.; Muruganandam, P.; Senthilvelan, M.; Lakshmanan, M., Phys. Rev. E, 90, Article 062905 pp. (2014)
[17] Dai, C. Q.; Zheng, C. L.; Zhu, H. P., Eur. Phys. J. D, 66, 112 (2012)
[18] Khawaja, U. A.; Taki, M., Phys. Lett. A, 377, 2944 (2013)
[19] Zhao, L. C., Ann. Phys., 329, 73 (2013)
[20] Yang, Z. J.; Zhang, S. M.; Li, X. L.; Pang, Z. G., Appl. Math. Lett., 82, 64 (2018); Gao, X. Y., Appl. Math. Lett., 3, 143 (2017); Gao, X. Y., Appl. Math. Lett., 91, 165 (2019); Guo, R.; Hao, H. Q.; Zhang, L. L., Nonlinear Dynam., 74, 701 (2013)
[21] Xie, X. Y.; Meng, G. Q., Nonlinear Dynam., 93, 779 (2018); Su, J. J.; Gao, Y. T.; Ding, C. C., Appl. Math. Lett., 88, 201 (2019); Su, J. J.; Gao, Y. T., Wave. Random Complex, 28, 708 (2018); Xie, X. Y.; Meng, G. Q., Eur. Phys. J. Plus, 134, 359 (2019); Xie, X. Y.; Meng, G. Q., Chaos Solitons Fractals, 107, 143 (2018)
[22] Xu, T.; Lan, S.; Li, M.; Li, L. L.; Zhang, G. W., Physica D, 390, 47 (2019); Li, M.; Shui, J. J.; Xu, T., Appl. Math. Lett., 83, 110 (2018)
[23] Li, M.; Xu, T.; Meng, D. X., J. Phys. Soc. Japan, 85, Article 124001 pp. (2016); Jia, T. T.; Gao, Y. T.; Feng, Y. J.; Hu, L.; Su, J. J.; Li, L. Q.; Ding, C. C., Nonlinear Dyn., 96, 229 (2019); Jia, T. T.; Chai, Y. Z.; Hao, H. Q., Superlattice. Microstruct., 105, 172 (2017); Li, M.; Xu, T., Phys. Rev. E, 91, 033202 (2015)
[24] Chen, F. P.; Chen, W. Q.; Wang, L.; Ye, Z. J., Appl. Math. Lett., 96, 33 (2019); Ding, C. C.; Gao, Y. T.; Hu, L.; Jia, T. T., Eur. Phys. J. Plus, 133, 406 (2018); Ding, C. C.; Gao, Y. T.; Li, L. Q., Chaos, Solitons Fract., 120, 259 (2019); Wang, L.; Liu, C.; Wu, X.; Wang, X.; Sun, W. R., Nonlinear Dynam., 94, 977 (2018)
[25] Cai, L. Y.; Wang, X.; Wang, L.; Li, M.; Liu, Y.; Shi, Y. Y., Nonlinear Dynam., 90, 2221 (2017); Deng, G. F.; Gao, Y. T.; Gao, X. Y., Wave. Random Complex, 28, 468 (2018); Deng, G. F.; Gao, Y. T.; Su, J. J.; Ding, C. C., Appl. Math. Lett., 98, 177 (2019); Feng, Y. J.; Gao, Y. T.; Yu, X., Nonlinear Dyn., 91, 29 (2018); Wang, X.; Wang, L., Comput. Math. Appl., 75, 4201 (2018)
[26] Onorato, M.; Osborne, A. R.; Serio, M.; Bertone, S., Phys. Rev. Lett., 86, 5831 (2001)
[27] Szameit, A.; Dreisow, F.; Heinrich, M.; Nolte, S.; Sukhorukov, A. A., Phys. Rev. Lett., 106, Article 193903 pp. (2011)
[28] Erkintalo, M.; Genty, G.; Dudley, J. M., Opt. Lett., 34, 2468 (2009)
[29] Solli, D. R.; Ropers, C.; Koonath, P.; Jalali, B., Nature, 450, 1054 (2007)
[30] Bludov, Y. V.; Konotop, V. V.; Akhmediev, N., Phys. Rev. A, 80, Article 033610 pp. (2009)
[31] Chabchoub, A.; Hoffmann, N.; Onorato, M.; Akhmediev, N., Phys. Rev. X, 2, Article 011015 pp. (2012)
[32] Mančić, A.; Maluckov, A.; Hadžievski, L., Phys. Rev. E, 95, Article 032212 pp. (2017)
[33] Ohta, Y.; Yang, J., Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468, 1716 (2012)
[34] Ohta, Y.; Yang, J., Phys. Rev. E, 86, Article 036604 pp. (2012)
[35] Mu, G.; Qin, Z. Y., Nonlinear Anal., 13, 1130 (2014)
[36] Ohta, Y.; Yang, J., J. Phys. A, 47, Article 255201 pp. (2014)
[37] Hu, X. H.; Zhang, X. F.; Zhao, D.; Luo, H. G.; Liu, W. M., Phys. Rev. A, 79, Article 023619 pp. (2009)
[38] Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S., Rev. Modern Phys., 71, 463 (1999)
[39] Saito, H.; Ueda, M., Phys. Rev. Lett., 90, Article 040403 pp. (2003)
[40] Liu, L.; Tian, B.; Wu, X. Y.; Sun, Y., Phys. A, 492, 524 (2018)
[41] He, X. G.; Zhao, D.; Li, L.; Luo, H. G., Phys. Rev. E, 79, Article 056610 pp. (2009)
[42] Hirota, R., The Direct Method in Soliton Theory (2004), Cambridge Univ. Press: Cambridge Univ. Press Cambridge
[43] Ohta, Y.; Wang, D. S.; Yang, J., Stud. Appl. Math., 127, 345 (2011)
[44] Chao, L. M.; Pan, G.; Zhang, D.; Yan, G. X., Chaos Solitons Fractals, 129, 260 (2019); Chao, L. M.; Pan, G.; Zhang, D.; Yan, G. X., J. Fluid. Struct., 85, 27 (2019); Chao, L. M.; Pan, G.; Cao, Y. H.; Zhang, D.; Yan, G. X., J. Fluid. Struct., 82, 610 (2018); Chao, L. M.; Zhang, D.; Pan, G., Fluid Dyn. Res., 49, 044501 (2017)
[45] Hayata, K.; Koshiba, M., Phys. Rev. E, 48, 2313 (1993)
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