Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system

doi:10.1088/1674-1056/24/1/010202

中国物理B ›› 2015, Vol. 24 ›› Issue (1): 10202-010202.doi: 10.1088/1674-1056/24/1/010202

Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system

刘萍^{a b}, 曾葆青^{a b}, 杨建荣^c, 任博^d

a College of Electron and Information Engineering, University of Electronic Science and Technologyof China Zhongshan Institute, Zhongshan 528402, China;
b School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China;
c Department of Physics and Electronics, Shangrao Normal University, Shangrao 334001, China;
d Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China

收稿日期:2014-04-13 修回日期:2014-08-06 出版日期:2015-01-05 发布日期:2015-01-05
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11305031, 11365017, and 11305106), the Natural Science Foundation of Guangdong Province, China (Grant No. S2013010011546), the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ13A050001), the Science and Technology Project Foundation of Zhongshan, China (Grant Nos. 2013A3FC0264 and 2013A3FC0334), and the Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province, China (Grant No. Yq2013205).

Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system

Liu Ping (刘萍)^{a b}, Zeng Bao-Qing (曾葆青)^{a b}, Yang Jian-Rong (杨建荣)^c, Ren Bo (任博)^d

a College of Electron and Information Engineering, University of Electronic Science and Technologyof China Zhongshan Institute, Zhongshan 528402, China;
b School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China;
c Department of Physics and Electronics, Shangrao Normal University, Shangrao 334001, China;
d Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China

Received:2014-04-13 Revised:2014-08-06 Online:2015-01-05 Published:2015-01-05
Contact: Liu Ping, Zeng Bao-Qing E-mail:liuping49@uestc.edu.cn;bqzeng@uestc.edu.cn
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11305031, 11365017, and 11305106), the Natural Science Foundation of Guangdong Province, China (Grant No. S2013010011546), the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ13A050001), the Science and Technology Project Foundation of Zhongshan, China (Grant Nos. 2013A3FC0264 and 2013A3FC0334), and the Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province, China (Grant No. Yq2013205).

摘要/Abstract

摘要：

The residual symmetries of the Ablowitz-Kaup-Newell-Segur (AKNS) equations are obtained by the truncated Painlevé analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symmetries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calculation on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Bäcklund transformations between the AKNS equations and the Schwarzian AKNS equation.

关键词: residual symmetries, Ablowitz-Kaup-Newell-Segur equation, exact solution, Bä, cklund transformation

Abstract:

Key words: residual symmetries, Ablowitz-Kaup-Newell-Segur equation, exact solution, Bä, cklund transformation

中图分类号: (Partial differential equations)

02.30.Jr

02.20.Sv (Lie algebras of Lie groups) 47.35.Fg (Solitary waves) 02.30.Ik (Integrable systems)

刘萍, 曾葆青, 杨建荣, 任博. Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system[J]. 中国物理B, 2015, 24(1): 10202-010202.

Liu Ping (刘萍), Zeng Bao-Qing (曾葆青), Yang Jian-Rong (杨建荣), Ren Bo (任博). Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system[J]. Chin. Phys. B, 2015, 24(1): 10202-010202.

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Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system

Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system

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