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## 留言板

Gap solitons and their stabilities in a quasi one-dimensional Bose-Einstein condensate under three-body interaction

## Gap solitons and their stabilities in a quasi one-dimensional Bose-Einstein condensate under three-body interaction

Tang Na, Yang Xue-Ying, Song Lin, Zhang Juan, Li Xiao-Lin, Zhou Zhi-Kun, Shi Yu-Ren
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• #### Abstract

We study the gap solitons and their stability properties in a Bose-Einstein condensation (BEC) under three-body interaction loaded in a Jacobian elliptic sine potential, which can be described by a cubic-quintic Gross-Pitaevskii equation (GPE) in the mean-field approximation. Firstly, the GPE is transformed into a stationary cubic-quintic nonlinear Schrödinger equation (NLSE) by the multi-scale method. A class of analytical solution of the NLSE is presented to describe the gap solitons. It is shown analytically that the amplitude of the gap soliton decreases as the two-body or three-body interaction strength increases. Secondly, many kinds of gap solitons, including the fundamental soliton and the sub-fundamental soliton, are obtained numerically by the Newton-Conjugate-Gradient (NCG) method. There are two families of fundamental solitons: one is the on-site soliton and the other is the off-site soliton. All of them are bifurcated from the Bloch band. Both in-phase and out-phase dipole solitons for off-site solitons do exist in such a nonlinear system. The numerical results also indicate that the amplitude of the gap soliton decreases as the nonlinear interaction strength increases, which accords well with the analytical prediction. Finally, long-time dynamical evolution for the GPE is performed by the time-splitting Fourier spectrum method to investigate the dynamical stability of gap solitons. It is shown that the on-site solitons are always dynamically stable, while the off-site solitons are always unstable. However, both stable and unstable in-phase or out-phase dipole solitons, which are not bifurcated from the Bloch band, indeed exist. For a type of out-phase soliton, there is a critical value $q_c$ when the chemical potential μ is fixed. The solitons are linearly stable as $q>q_c$, while they are linearly unstable for $q<q_c$. Therefore, the modulus q plays an important role in the stability of gap solitons. One can change the dynamical behavior of gap solitons by adjusting the modulus of external potential in experiment. We also find that there exists a kind of gap soliton, in which the soliton is dynamically unstable if only the two-body interaction is considered, but it becomes stable when the three-body interaction is taken into account. This indicates that the three-body interaction has influence on the stability of gap solitons.

#### References

 [1] Bose S N 1924 Z. Phy. 26 178 [2] Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198 [3] Tsurumi T 2000 Int. J. Mod. Phys. B 14 655 [4] Malomed B A 2006 Soltion Management in Periodic Systems (Vol. 1) (Berlin: Springer) pp 1—6 [5] Wang D S, Hu X H, Hu J, Liu W M 2010 Phys. Rev. A 81 025604 [6] Burger S, Bongs K, Dettmer S, Ertmer W, Sengstock K, Sanpera A, Shlyapnikov G V, Lewenstein M 1999 Phys. Rev. Lett. 83 5198 [7] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr L D, Castin Y, Salomon C 2002 Science 296 1290 [8] Ku M J H, Ji W, Mukherjee B, Guardado S E, Cheuk L W, Yefash T, Zwierlein M W 2014 Phys. Rev. Lett. 113 065301 [9] Yu H Y, Yan J R, Xie Q T 2004 Chin. Phys. Lett. 21 1881 [10] Wang Z X, Zhang X H, Shen K 2008 Chin. Phys. B 17 3270 [11] Li J H, Kuang L M 2003 Commun. Theor. Phys. 39 44 [12] EtienneW B, Alidou M D, Timoléon C K 2008 Phys. Rev. E 77 046216 [13] Zhang W, Wright E M, Pu H, Meystre P 2003 Phys. Rev. A 68 023605 [14] Abdullaev F Kh, Gammal A, Tomio L, Frederico T 2001 Phys. Rev. A 63 043604 [15] Chang N N, Yu Z F, Zhang A X, Xue J K 2017 Chin. Phys. B 26 115202 [16] Etienne W, Porsezian K, Alidou M, Timoléon C K 2013 Phys. Lett. A 377 262 [17] Chin C, Kraemer T, Mark M, Herbig J, Waldburger P, Nagerl H C, Grimm R 2005 Phys. Rev. Lett. 94 123201 [18] Peng P, Li G Q 2009 Chin. Phys. B 18 3221 [19] 阮航宇, 陈一新 2001 物理学报 4 0586 Ruan H Y, Chen Y X 2001 Acta. Phys. Sin. 4 0586 [20] Saito H, Ueda M 2001 Phys. Rev. Lett. 86 1406 [21] Fiho V S, Gammal A, Frederico T, Tomio L 2000 Phys. Rev. A 62 033605 [22] Meng D X, Li K Z 2019 Phys. Rev. Lett. 33 19150123 [23] Wahlquist H D, Estabreok F B 1973 Phys. Rev. Lett. 31 1386 [24] Radhakrishnan R, Sahadcvan R, Lakshmanan M 1995 Chaos, Solitons Fractals 5 2315 [25] Weiss J, Tabor M, Carneval G 1983 J. Math. Phys. 24 6 [26] Wang D S, Liu J 2018 Appl. Math. Lett. 79 211 [27] 文林, 梁毅, 周晶, 余鹏, 夏雷, 牛连斌, 张晓斐 2019 物理学报 68 080301 Wen L, Liang Y, Zhou J, Yu P, Xia L, Niu L B, Zhang X F 2019 Acta Phys. Sin. 68 080301 [28] Dmitry E P, Andrey A S, Yuri S K 2004 Phys. Rev. E 70 036618 [29] Tang N, Yang X Y, Feng W X, Song L, Li X L, Xi Z H, Wang D S, Shi Y R 2019 Physica A 528 1 [30] Akozbek N, John S 1998 Phys. Rev. E 57 2287 [31] Mingaleev S F, Kivshar Yu S 2001 Phys. Rev. Lett. 86 5474 [32] Louis P J Y, Ostrovskaya E A, Kivshar Yu S 2005 Phys. Rev. A 71 023612 [33] Eiermann B, Anker Th, Albiez M, Taglieber M, Treutlein M, Marzlin K P, Oberthaler M K 2004 Phys. Rev. Lett. 92 230401 [34] Zhu J, Bian C L, Wang H C 2019 Chin. Phys. B 28 093701 [35] Li Y, Hai W H 2005 Commun. Theor. Phys. 44 840 [36] Eddy T, Paolo T, Mahir H, Arthur K 1999 Phys. Rep. 315 199 [37] Bao W 2007 Dynamics In Models Of Coarsening, Coagulation, Condensation And Quantization. (Vol. 1) (Singpore: World Scientific) p147—149 [38] Yang J K 2010 Nonlinear Waves in Integrable and Nonintegrable Systems (Vol. 1) (Philadelphia: SIAM) p269−283 [39] Kostov N A, Enolśkii V Z, Gerdjikov V S, Konotop V V and Salerno M 2004 Phys. Rev. E 70 056617 [40] Catarina C, Vitor Nistor, Yu Q 2017 Operator Theory: Advances and Applications (Vol. 1) (Berlin: Springer) p20−24 [41] Yang J K 2009 J. Comput. Phys. 228 7007 [42] 唐恒永 1985 北京工业大学学报 11 69 Tang H Y 1985 J. BJUT. Tech. 11 69 [43] Royer C W, Neill M O’, Wright S J 2018 Math. Program. 1-38 [44] Zhang Q, Li S, Lei Y, Zhang X D 2012 J. Control Theory Appl. 10 223 [45] Allen A J, Zuccher S, Caliari M, Proukakis N, Parker N G, Barenghi C F 2014 Phys. Rev. A 90 013601 [46] Adhikari S F, Malomed B A 2007 Europhys. Lett. 79 50003

#### Cited By

• 图 1  3−5次GPE的单峰带隙孤子($V_0=4$)　(a), (b) $q=0.1$; (c), (d) $q=0.99$. 阴影部分表示外势$V(x)$低处

Figure 1.  Profiles of singl-hump gap solitons of the cubic-quintic GPE ($V_0=4$): (a), (b) $q$ = 0.1; (c), (d) $q$ = 0.99. Shaded regions represent lattice sites, i.e., regions of low potential values $V(x)$.

图 2  单峰带隙孤子的振幅随相互作用强度的变化($V_0=4$)

Figure 2.  Amplitudes of single-hump gap solitons v.s. nonlinear interaction strength.

图 3  on-site孤子三体相互作用能量和两体相互作用能量的比

Figure 3.  The ratio of three-body energy to two-body energy for on-site solitons with different interaction strength.

图 4  3−5次GPE的同相偶极孤子和异相偶极孤子($V_0=4$)　(a), (b) $q=0.1;$ (c), (d) $q=0.99$. 阴影部分表示外势$V(x)$低处

Figure 4.  Profiles of double-hump gap solitons of the cubic-quintic GPE ($V_0=4$) (a), (b) $q$ = 0.1; (c), (d) $q$ = 0.99. Shaded regions represent lattice sites, i.e., regions of low potential values $V(x)$.

图 5  不同类型带隙孤子的动力学演化　(a) on-site孤子; (b) 同相偶极孤子; (c), (d) 异相偶极孤子

Figure 5.  Contour plots of $|{\varPsi(x, t)}|$ for perturbed gap solitons: (a) On-site soliton; (b) in-phase dipole soliton; (c), (d) out-phase dipole soliton.

图 6  (a), (c) 第一带隙中的亚基本带隙孤子(红色). 蓝线表示外势; (b), (d) 亚基本带隙孤子的动力学演化

Figure 6.  (a), (c) Profiles of sub-fundamental gap solitons (red lines) lie in the first bandgap. The solid blue lines denote the external potential; (b), (d) contour plots of $|{\varPsi(x, t)}|$ for perturbed gap solitons.

图 7  第一带隙中同相偶极孤子　(a) $\eta=0$和(b) $\eta=-0.2$时的动力学演化

Figure 7.  Contour plots of $|{\varPsi(x, t)}|$ for in-phase dipole solitons lie in the first bandgap with (a) $\eta=0$ and (b) $\eta=-0.2$.

•  [1] Bose S N 1924 Z. Phy. 26 178 [2] Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198 [3] Tsurumi T 2000 Int. J. Mod. Phys. B 14 655 [4] Malomed B A 2006 Soltion Management in Periodic Systems (Vol. 1) (Berlin: Springer) pp 1—6 [5] Wang D S, Hu X H, Hu J, Liu W M 2010 Phys. Rev. A 81 025604 [6] Burger S, Bongs K, Dettmer S, Ertmer W, Sengstock K, Sanpera A, Shlyapnikov G V, Lewenstein M 1999 Phys. Rev. Lett. 83 5198 [7] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr L D, Castin Y, Salomon C 2002 Science 296 1290 [8] Ku M J H, Ji W, Mukherjee B, Guardado S E, Cheuk L W, Yefash T, Zwierlein M W 2014 Phys. Rev. Lett. 113 065301 [9] Yu H Y, Yan J R, Xie Q T 2004 Chin. Phys. Lett. 21 1881 [10] Wang Z X, Zhang X H, Shen K 2008 Chin. Phys. B 17 3270 [11] Li J H, Kuang L M 2003 Commun. Theor. Phys. 39 44 [12] EtienneW B, Alidou M D, Timoléon C K 2008 Phys. Rev. E 77 046216 [13] Zhang W, Wright E M, Pu H, Meystre P 2003 Phys. Rev. A 68 023605 [14] Abdullaev F Kh, Gammal A, Tomio L, Frederico T 2001 Phys. Rev. A 63 043604 [15] Chang N N, Yu Z F, Zhang A X, Xue J K 2017 Chin. Phys. B 26 115202 [16] Etienne W, Porsezian K, Alidou M, Timoléon C K 2013 Phys. Lett. A 377 262 [17] Chin C, Kraemer T, Mark M, Herbig J, Waldburger P, Nagerl H C, Grimm R 2005 Phys. Rev. Lett. 94 123201 [18] Peng P, Li G Q 2009 Chin. Phys. B 18 3221 [19] 阮航宇, 陈一新 2001 物理学报 4 0586 Ruan H Y, Chen Y X 2001 Acta. Phys. Sin. 4 0586 [20] Saito H, Ueda M 2001 Phys. Rev. Lett. 86 1406 [21] Fiho V S, Gammal A, Frederico T, Tomio L 2000 Phys. Rev. A 62 033605 [22] Meng D X, Li K Z 2019 Phys. Rev. Lett. 33 19150123 [23] Wahlquist H D, Estabreok F B 1973 Phys. Rev. Lett. 31 1386 [24] Radhakrishnan R, Sahadcvan R, Lakshmanan M 1995 Chaos, Solitons Fractals 5 2315 [25] Weiss J, Tabor M, Carneval G 1983 J. Math. Phys. 24 6 [26] Wang D S, Liu J 2018 Appl. Math. Lett. 79 211 [27] 文林, 梁毅, 周晶, 余鹏, 夏雷, 牛连斌, 张晓斐 2019 物理学报 68 080301 Wen L, Liang Y, Zhou J, Yu P, Xia L, Niu L B, Zhang X F 2019 Acta Phys. Sin. 68 080301 [28] Dmitry E P, Andrey A S, Yuri S K 2004 Phys. Rev. E 70 036618 [29] Tang N, Yang X Y, Feng W X, Song L, Li X L, Xi Z H, Wang D S, Shi Y R 2019 Physica A 528 1 [30] Akozbek N, John S 1998 Phys. Rev. E 57 2287 [31] Mingaleev S F, Kivshar Yu S 2001 Phys. Rev. Lett. 86 5474 [32] Louis P J Y, Ostrovskaya E A, Kivshar Yu S 2005 Phys. Rev. A 71 023612 [33] Eiermann B, Anker Th, Albiez M, Taglieber M, Treutlein M, Marzlin K P, Oberthaler M K 2004 Phys. Rev. Lett. 92 230401 [34] Zhu J, Bian C L, Wang H C 2019 Chin. Phys. B 28 093701 [35] Li Y, Hai W H 2005 Commun. Theor. Phys. 44 840 [36] Eddy T, Paolo T, Mahir H, Arthur K 1999 Phys. Rep. 315 199 [37] Bao W 2007 Dynamics In Models Of Coarsening, Coagulation, Condensation And Quantization. (Vol. 1) (Singpore: World Scientific) p147—149 [38] Yang J K 2010 Nonlinear Waves in Integrable and Nonintegrable Systems (Vol. 1) (Philadelphia: SIAM) p269−283 [39] Kostov N A, Enolśkii V Z, Gerdjikov V S, Konotop V V and Salerno M 2004 Phys. Rev. E 70 056617 [40] Catarina C, Vitor Nistor, Yu Q 2017 Operator Theory: Advances and Applications (Vol. 1) (Berlin: Springer) p20−24 [41] Yang J K 2009 J. Comput. Phys. 228 7007 [42] 唐恒永 1985 北京工业大学学报 11 69 Tang H Y 1985 J. BJUT. Tech. 11 69 [43] Royer C W, Neill M O’, Wright S J 2018 Math. Program. 1-38 [44] Zhang Q, Li S, Lei Y, Zhang X D 2012 J. Control Theory Appl. 10 223 [45] Allen A J, Zuccher S, Caliari M, Proukakis N, Parker N G, Barenghi C F 2014 Phys. Rev. A 90 013601 [46] Adhikari S F, Malomed B A 2007 Europhys. Lett. 79 50003
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•  Citation:
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• Abstract views:  123
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##### Publishing process
• Received Date:  23 August 2019
• Accepted Date:  12 October 2019
• Available Online:  05 December 2019
• Published Online:  01 January 2020

## Gap solitons and their stabilities in a quasi one-dimensional Bose-Einstein condensate under three-body interaction

###### Corresponding author: Shi Yu-Ren, shiyr@nwnu.edu.cn
• 1. College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, China
• 2. Laboratory of Atomic Molecular Physics and Functional Material, Lanzhou 730070, China
• 3. Department of Basic Course, Lanzhou Institute of Technology, Lanzhou 730050, China

Abstract: We study the gap solitons and their stability properties in a Bose-Einstein condensation (BEC) under three-body interaction loaded in a Jacobian elliptic sine potential, which can be described by a cubic-quintic Gross-Pitaevskii equation (GPE) in the mean-field approximation. Firstly, the GPE is transformed into a stationary cubic-quintic nonlinear Schrödinger equation (NLSE) by the multi-scale method. A class of analytical solution of the NLSE is presented to describe the gap solitons. It is shown analytically that the amplitude of the gap soliton decreases as the two-body or three-body interaction strength increases. Secondly, many kinds of gap solitons, including the fundamental soliton and the sub-fundamental soliton, are obtained numerically by the Newton-Conjugate-Gradient (NCG) method. There are two families of fundamental solitons: one is the on-site soliton and the other is the off-site soliton. All of them are bifurcated from the Bloch band. Both in-phase and out-phase dipole solitons for off-site solitons do exist in such a nonlinear system. The numerical results also indicate that the amplitude of the gap soliton decreases as the nonlinear interaction strength increases, which accords well with the analytical prediction. Finally, long-time dynamical evolution for the GPE is performed by the time-splitting Fourier spectrum method to investigate the dynamical stability of gap solitons. It is shown that the on-site solitons are always dynamically stable, while the off-site solitons are always unstable. However, both stable and unstable in-phase or out-phase dipole solitons, which are not bifurcated from the Bloch band, indeed exist. For a type of out-phase soliton, there is a critical value $q_c$ when the chemical potential μ is fixed. The solitons are linearly stable as $q>q_c$, while they are linearly unstable for $q<q_c$. Therefore, the modulus q plays an important role in the stability of gap solitons. One can change the dynamical behavior of gap solitons by adjusting the modulus of external potential in experiment. We also find that there exists a kind of gap soliton, in which the soliton is dynamically unstable if only the two-body interaction is considered, but it becomes stable when the three-body interaction is taken into account. This indicates that the three-body interaction has influence on the stability of gap solitons.

Reference (46)

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