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

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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

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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

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Received: 2017-07-15

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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)

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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.

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Near integrability of kink lattice with higher order interactions

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