Explicit solutions from residual symmetry of the Boussinesq equation

doi:10.1088/1674-1056/24/3/030202

›› 2015, Vol. 24 ›› Issue (3): 30202-030202.doi: 10.1088/1674-1056/24/3/030202

Explicit solutions from residual symmetry of the Boussinesq equation

刘希忠, 俞军, 任博

Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China

收稿日期:2014-08-23 修回日期:2014-10-12 出版日期:2015-03-05 发布日期:2015-03-05
基金资助:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11347183, 11405110, 11275129, and 11305106) and the Natural Science Foundation of Zhejiang Province of China (Grant Nos. Y7080455 and LQ13A050001).

Explicit solutions from residual symmetry of the Boussinesq equation

Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博)

Institute of Nonlinear Science, Shaoxing University, Shaoxing 312000, China

Received:2014-08-23 Revised:2014-10-12 Online:2015-03-05 Published:2015-03-05
Contact: Liu Xi-Zhong E-mail:liuxizhong123@163.com
Supported by:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11347183, 11405110, 11275129, and 11305106) and the Natural Science Foundation of Zhejiang Province of China (Grant Nos. Y7080455 and LQ13A050001).

摘要/Abstract

摘要： The Bäcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained: a new type of Bäcklund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.

关键词: Boussinesq equation, localization procedure, residual symmetry, symmetry reduction solution

Abstract: The Bäcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained: a new type of Bäcklund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.

Key words: Boussinesq equation, localization procedure, residual symmetry, symmetry reduction solution

中图分类号: (Partial differential equations)

02.30.Jr

02.30.Ik (Integrable systems) 05.45.Yv (Solitons) 47.35.Fg (Solitary waves)

刘希忠, 俞军, 任博. Explicit solutions from residual symmetry of the Boussinesq equation[J]. , 2015, 24(3): 30202-030202.

Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博). Explicit solutions from residual symmetry of the Boussinesq equation[J]. Chin. Phys. B, 2015, 24(3): 30202-030202.

参考文献 24

[1]	Lie S 1891 Vorlesungen ber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen (Leipzig: Teuber) (reprinted by Chelsea: New York; 1967)
[2]	Olver P J 1993 Application of Lie Groups to Differential Equation, Graduate Texts in Mathematics (2nd ed.) (NewYork: Springer-Verlag)
[3]	Bluman G W and Kumei S 1989 Symmetries and Differential Equation, Appl. Math. Sci. (Berlin: Springer-Verlag)
[4]	Tang X Y and Lou S Y 2002 Chin. Phys. Lett. 19 1
[5]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[6]	Liu X Z 2010 Chin. Phys. B 19 080202
[7]	Jing J C and Li B 2013 Chin. Phys. B 22 010303
[8]	Liu X Z 2010 J. Phys. A: Math. Gen. 43 265203
[9]	Lou S Y and Hu X B 1997 J. Phys. A: Math. Gen. 30 95
[10]	Gao X N, Lou S Y and Tang X Y 2013 JHEP 5 029
[11]	Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607
[12]	Xin X P and Chen Y 2013 Chin. Phys. Lett. 30 100202
[13]	Boussinesq J 1871 Comptes Rendus C. R. Acad. Sci Paris 72 755
[14]	Boussinesq J 1872 J. Pure Appl. 7 55
[15]	Whitham G B 1974 Linear and Nonlinear Waves (New York: Wiley)
[16]	Toda M 1975 Phys. Rep. 18 1
[17]	Zabusky N J 1967 A Synergetic Approach to Problems of Nonlinear Dispersive Wave Propagation and Interaction (New York: Academic Press)
[18]	Zakharov V E 1974 Sov. Phys. JETP 38 108
[19]	Infeld E and Rowlands G 1990 Nonlinear Waves, Solitons and Chaos (Cambridge: Cambridge University Press)
[20]	Scott A C 1975 The Application of Bäcklund Transforms to Physical Problems (Berlin: Springer)
[21]	Lou S Y 2013 Residual Symmetries and Bäcklund Transformations arXiv:1308.1140v1
[22]	Cao C W and Geng X G 1990 J. Phys. A: Math. Gen. 23 4117
[23]	Cheng Y and Li Y S 1991 Phys. Lett. A 157 22
[24]	Zeng Y B, Ma W X and Lin R L 2000 J. Math. Phys. 41 5453

Explicit solutions from residual symmetry of the Boussinesq equation

Explicit solutions from residual symmetry of the Boussinesq equation

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