Near integrability of kink lattice with higher order interactions

Yun-Guo Jiang; Jia-Zhen Liu; Song He

doi:10.1088/1674-1137/41/11/113107

Chinese Physics C> 2017, Vol. 41> Issue(11) : 113107 DOI: 10.1088/1674-1137/41/11/113107

Near integrability of kink lattice with higher order interactions

Yun-Guo Jiang ^1,2,, ,
Jia-Zhen Liu ³ ,
Song He ^4,5,,

1.
School of Space Science and Physics, Shandong University at Weihai, Weihai 264209, China
2.
Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai 264209, China
3.
Department of Physics, University of Miami, Coral Gables, FL 33126, USA
4.
Max Planck Institute for Gravitational Physics(Albert Einstein Institute) Am Mü
5.
State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China

Abstract
HTML
Reference
Related

PDF

Abstract：
We make use of Manton's analytical method to investigate the force between kinks and anti-kinks at large distances in 1+1 dimensional field theory. The related potential has infinite order corrections of exponential pattern, and the coefficients for each order are determined. These coefficients can also be obtained by solving the equation of the fluctuations around the vacuum. At the lowest order, the kink lattice represents the Toda lattice. With higher order correction terms, the kink lattice can represent one kind of generic Toda lattice. With only two sites, the kink lattice is classically integrable. If the number of sites of the lattice is larger than two, the kink lattice is not integrable but is a near integrable system. We make use of Flaschka's variables to study the Lax pair of the kink lattice. These Flaschka's variables have interesting algebraic relations and non-integrability can be manifested. We also discuss the higher Hamiltonians for the deformed open Toda lattice, which has a similar result to the ordinary deformed Toda.
- integrable system ,
- soliton ,
- Toda lattice

References

[1]	N. S. Manton, Nucl. Phys. B, 150:397 (1979)
[2]	N. S. Manton and P. Sutcliffe, Topological solitons, First Edition (Cambridge, England:Cambridge University Press, 2004), p.114
[3]	T. Vachaspati, Kinks and domain walls:An introduction to classical and quantum solitons, First Edition (Cambridge, England:Cambridge University Press, 2006), p.10
[4]	S. He, Y. Jiang, and J. Liu, arXiv:1605.06867
[5]	M. Moshir, Nucl. Phys. B, 185:318-332 (1981)
[6]	D. K. Campbell, J. F. Schonfld, and C. A. Wingate Physica D, 9:1-32 (1983)
[7]	P. Dorey, K. Mersh, T. Romanczukiewicz, and Y. Shnir, Phys. Rev. Lett., 107:091602 (2011)
[8]	V. A. Gani, A. E. Kudryavtsev, and M. A. Lizunova, Phys. Rev. D, 89(12):125009 (2014)
[9]	V. A. Gani, V. Lensky, and M. A. Lizunova, JHEP, 1508:147 (2015)
[10]	M. Toda, Journal of the Physical Society of Japan, 20(11):2095A (1965)
[11]	H. Flaschka, Phys. Rev. B, 9(4):1924-1925 (1974)
[12]	M. Henon, Phys. Rev. B, 9(4):1921-1923 (1974)
[13]	S. Manakov, Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 67:543-555 (1974); Soviet Journal of Experimental and Theoretical Physics, 40(2):269-274 (1975)
[14]	K. Sawada and T. Kotera, Supplement of the Progress of Theoretical Physics, 59:101-106 (1976)
[15]	L. A. Ferreira and W. J. Zakrzewski, JHEP, 1105:130 (2011)
[16]	L. A. Ferreira, P. Klimas, and W. J. Zakrzewski, JHEP, 1605:065(2016)
[17]	C. Gignoux, B. Silvestre-Brac, Solved problems in Lagrangian and Hamiltonian Mechanics, First Edition (Springer, 2009)
[18]	M. A. Agrotis, P. A. Damianou, and C. Sophocleous, Physica A Statistical Mechanics and its Applications, 365:235-243 (2006)
[19]	O. Babelon, D. Bernard, and M. Talon, Introduction to classical integrable systems, First Edition (Cambridge University Press, 2003)
[20]	E. Gildener and A. Patrascioiu, Phys. Rev. D, 16:423(1977) Erratum:Phys. Rev. D, 16:3616 (1977)
[21]	M. Shifman and A. Yung, Phys. Rev. D, 70:045004 (2004)
[22]	M. Eto, T. Fujimori, S. B. Gudnason, Y. Jiang et al, JHEP, 1112:017 (2011)
[23]	A. Alonso-Izquierdo, M. A. Gonzalez Leon, and J. Mateos Guilarte, Phys. Rev. Lett. 101:131602 (2008)
[24]	D. Harland, J. Math. Phys. 50:122902 (2009)
[25]	M. Arai, F. Blaschke, M. Eto, and N. Sakai, JHEP, 1409:172 (2014)
[26]	D. Tong, JHEP, 0304:031 (2003)

[1]	Yi Ling , Peng Liu , Chao Niu , Jian-Pin Wu . Pseudo-gap phase and duality in a holographic fermionic system with dipole coupling on Q-lattice. Chinese Physics C, 2016, 40(4): 043102. doi: 10.1088/1674-1137/40/4/043102
[2]	ZHANG Hui , SHU Song . Mean-field approximation for the chiral soliton in achiral phase transition. Chinese Physics C, 2015, 39(9): 094104. doi: 10.1088/1674-1137/39/9/094104
[3]	FU Zi-Wen . Lattice calculation of the κ meson. Chinese Physics C, 2012, 36(6): 489-497 . doi: 10.1088/1674-1137/36/6/003
[4]	JIA Duo-Jie , ZHANG Jin-Hu , WANG Qing . A note on the soliton picture in a Skyrme-like model. Chinese Physics C, 2012, 36(9): 797-804. doi: 10.1088/1674-1137/36/9/001
[5]	ZHU Qiao , YANG Zhan-Ying , SHI Kang-Jie , WEN Jun-Qing . Study of Solving a Toda Dynamic System with Loop Algebra. Chinese Physics C, 2006, 30(9): 838-843.
[6]	ZHU Qiao , YANG Zhan-Ying , SHI Kang-Ji . The Study on Generalized Toda Mechanics System with Loop Algebra L(D^r). Chinese Physics C, 2005, 29(1): 19-22.
[7]	LIU Wang-Yun , YANG Zhan-Ying , ZHAO Liu . Generalized Toda Mechanics Associated with Classical Lie Algebras D_r. Chinese Physics C, 2004, 28(4): 359-364.
[8]	YANG ZhanYing , ZHEN Yi . Construction on the Solution of osp(1｜4)Toda Model. Chinese Physics C, 2000, 24(6): 484-489.
[9]	Deng Shenghua , Wang Enke , Li Jiarong . Two-Loop Effective Potential of the Non-topological Soliton Model at Finite Temperature. Chinese Physics C, 1995, 19(S3): 289-304.
[10]	ZHAO Liu . From Integrability to Conformal Symmetry: Bosonic Superconformal Toda Theories. Chinese Physics C, 1993, 17(5): 431-443.
[11]	ZHAO Liu , YANG Wen-Li . Bosonic Superconformal Toda Model and Dressing Transformation. Chinese Physics C, 1993, 17(4): 338-344.
[12]	Zhao Liu , Yang Wenli . Bosonic Superconformal Toda Theory and Dressing Symmetry. Chinese Physics C, 1993, 17(S1): 65-72.
[13]	HOU Bo-Yu , ZHAO Liu . Conformally Reduced WZNW Theory,New Extended Chiral Algebras and Their Associated Toda Type Integrable Systems (Ⅰ) General Framework. Chinese Physics C, 1993, 17(1): 28-37.
[14]	HOU Bo-Yu , ZHAO Liu . Conformally Reduced WZNW Theory,New Extended Chiral Algebras and Their Associated Toda Type Integrable Systems (Ⅱ)An Example. Chinese Physics C, 1993, 17(3): 241-251.
[15]	HOU Bo-Yu , ZHAO Liu , YANG Huan-Xiong . Affine Toda Fields as Restriced WZNW Model. Chinese Physics C, 1991, 15(8): 701-710.
[16]	Zheng Xite , Liu Beishu . Analytical Treatment of SU(2)-Higgs System on Lattice at Finite Temperature. Chinese Physics C, 1990, 14(S4): 367-375.
[17]	Xu Xiaoming , Qiu Xijun . A Chiral Quark-Soliton Model Theory for Nuclear Force. Chinese Physics C, 1990, 14(S4): 381-389.
[18]	Wang Enke , Li Jiarong . Deconfinement Transition of the Nontopological Soliton Model at Finite Temperature. Chinese Physics C, 1989, 13(S4): 399-404.
[19]	Zhang Guangyuan , Cai Weicheng , Li Xinzhou . Fermion Fields in Soliton Background of Six-dimensional Pure Kaiuza-Klein Theories. Chinese Physics C, 1988, 12(S2): 159-164.
[20]	Wang Jie , Tan Zuguo , Zheng Xite . Analytical Study of U(1)-Higgs System On a Lattice. Chinese Physics C, 1988, 12(S3): 237-244.

Access

Get Citation

Yun-Guo Jiang, Jia-Zhen Liu and Song He. Near integrability of kink lattice with higher order interactions[J]. Chinese Physics C, 2017, 41(11): 113107. doi: 10.1088/1674-1137/41/11/113107

Yun-Guo Jiang, Jia-Zhen Liu and Song He. Near integrability of kink lattice with higher order interactions[J]. Chinese Physics C, 2017, 41(11): 113107. doi: 10.1088/1674-1137/41/11/113107 shu

RIS(for EndNote,Reference Manager,ProCite)

BibTex

Txt

Milestone

Received: 2017-07-15

Fund

Supported by Shandong Provincial Natural Science Foundation (ZR2014AQ007), National Natural Science Foundation of China (11403015, U1531105), S. He is supported by Max-Planck fellowship in Germany and National Natural Science Foundation of China (11305235)

Article Metric

Article Views(1634)
PDF Downloads(43)
Cited by(0)

Policy on re-use

To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Near integrability of kink lattice with higher order interactions

Corresponding author: Yun-Guo Jiang,

Corresponding author: Song He,

1. School of Space Science and Physics, Shandong University at Weihai, Weihai 264209, China

2. Shandong Provincial Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute of Space Sciences, Shandong University, Weihai 264209, China

3. Department of Physics, University of Miami, Coral Gables, FL 33126, USA

4. Max Planck Institute for Gravitational Physics(Albert Einstein Institute) Am Mü

5. State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Science, Beijing 100190, China

Received Date: 2017-07-15

Available Online: 2017-11-05

Fund Project: Supported by Shandong Provincial Natural Science Foundation (ZR2014AQ007), National Natural Science Foundation of China (11403015, U1531105), S. He is supported by Max-Planck fellowship in Germany and National Natural Science Foundation of China (11305235)

Abstract: We make use of Manton's analytical method to investigate the force between kinks and anti-kinks at large distances in 1+1 dimensional field theory. The related potential has infinite order corrections of exponential pattern, and the coefficients for each order are determined. These coefficients can also be obtained by solving the equation of the fluctuations around the vacuum. At the lowest order, the kink lattice represents the Toda lattice. With higher order correction terms, the kink lattice can represent one kind of generic Toda lattice. With only two sites, the kink lattice is classically integrable. If the number of sites of the lattice is larger than two, the kink lattice is not integrable but is a near integrable system. We make use of Flaschka's variables to study the Lax pair of the kink lattice. These Flaschka's variables have interesting algebraic relations and non-integrability can be manifested. We also discuss the higher Hamiltonians for the deformed open Toda lattice, which has a similar result to the ordinary deformed Toda.

HTML

Reference (26)

Near integrability of kink lattice with higher order interactions

Abstract：

References

Access

Article Metrics

Metrics

通讯作者: 陈斌, bchen63@163.com

Email This Article