New solutions from nonlocal symmetry of the generalized fifth order KdV equation

doi:10.1088/1674-1056/24/8/080202

中国物理B ›› 2015, Vol. 24 ›› Issue (8): 80202-080202.doi: 10.1088/1674-1056/24/8/080202

New solutions from nonlocal symmetry of the generalized fifth order KdV equation

刘希忠, 俞军, 任博

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

收稿日期:2015-01-22 修回日期:2015-03-06 出版日期:2015-08-05 发布日期:2015-08-05
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).

New solutions from nonlocal symmetry of the generalized fifth order KdV 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:2015-01-22 Revised:2015-03-06 Online:2015-08-05 Published:2015-08-05
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 nonlocal symmetry of the generalized fifth order KdV equation (FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Bäcklund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.

关键词: generalized fifth order KdV equation, localization procedure, nonlocal symmetry, symmetry reduction solution

Abstract: The nonlocal symmetry of the generalized fifth order KdV equation (FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Bäcklund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.

Key words: generalized fifth order KdV equation, localization procedure, nonlocal 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. New solutions from nonlocal symmetry of the generalized fifth order KdV equation[J]. 中国物理B, 2015, 24(8): 80202-080202.

Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博). New solutions from nonlocal symmetry of the generalized fifth order KdV equation[J]. Chin. Phys. B, 2015, 24(8): 80202-080202.

参考文献 30

[1]	Hirota R 1971 Phys. Rev. Lett. 27 1192
[2]	Hu X B and Wang H Y 2006 Inver. Prob. 22 1903
[3]	Gu C H, Hu H and Zhou Z X 2005 Darboux Transformations in Integrable Systems Theory and Their Applications to Geometry, Series: Mathematical Physics Studies, Vol. 26 (Dordrecht: Springer)
[4]	Rogers C and Schief W K 2002 Bäcklund and Darboux transformations geometry and modern applications in soliton theory, Cambridge texts in applied mathematics (Cambridge: Cambridge University Press)
[5]	Gelfand I M and Levitan B M 1955 Am. Math. Soc. Trans. 1 253
[6]	Ablowitz M and Segur H 1981 Solitons and the Inverse Scattering Transform (Philadelphia: SIAM)
[7]	Lie S 1891 Vorlesungen ber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen (Leipzig: Teuber) (reprinted by Chelsea: New York, 1967)
[8]	Olver P J 1993 Application of Lie Groups to Differential Equation, Graduate Texts in Mathematics, 2nd edn. (New York: Springer-Verlag)
[9]	Bluman G W and Kumei S 1989 Symmetries and Differential Equation, Appl. Math. Sci. (Berlin: Springer-Verlag)
[10]	Tang X Y and Lou S Y 2002 Chin. Phys. Lett. 19 1
[11]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[12]	Liu X Z 2010 Chin. Phys. B 19 080202
[13]	Liu X Z 2010 J. Phys. A: Math. Theor. 43 265203
[14]	Liu P, Yang J J and Ren B 2013 Chin. Phys. Lett. 30 100201
[15]	Gao X N, Lou S Y and Tang X Y 2013 JHEP 05 029
[16]	Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607
[17]	Lou S Y, Tang X Y, Liu Q P and Fukuyamae T 2002 Z. Naturforsch. A 57 737
[18]	Kuznetsov E A, Rubenchik A M and Zakharov V E 1986 Phys. Rep. 142 103
[19]	Karpman V I and Vanden-Broeck J M 1995 Phys. Lett. A 200 423
[20]	Hai W and Xiao Y 1995 Phys. Lett. A 208 79
[21]	Pomeau Y, Ramani A and Grammaticos B 1988 Physica D 31 127
[22]	Hunter J K and Vanden-Broeck J M 1983 J. Fluid Mech. 134 205
[23]	Dey B, Khare A and Nagraj Kumar C 1996 Phys. Lett. A 223 449
[24]	Xin X P and Chen Y 2013 Chin. Phys. Lett. 30 100202
[25]	Fuchssteiner B and Fokas A S 1981 Physica D: Nonlinear Phenomena 4 47
[26]	Fuchssteiner B 1979 Application of spectral-gradient methods to nonlinear soliton equations (preprint)
[27]	Lou S Y and Hu X B 1997 J. Phys. A: Math. Gen. 30 95
[28]	Cao C W and Geng X G 1990 J. Phys. A: Math. Gen. 23 4117
[29]	Cheng Y and Li Y S 1991 Phys. Lett. A 157 22
[30]	Zeng Y B, Ma W X and Lin R L 2000 J. Math. Phys. 41 5453

New solutions from nonlocal symmetry of the generalized fifth order KdV equation

New solutions from nonlocal symmetry of the generalized fifth order KdV equation

RichHTML

PDF (PC)

赞

可视化

摘要/Abstract

引用本文

使用本文

参考文献 30

相关文章 9

Metrics

本文评价

推荐阅读 0

[1]	Xi-zhong Liu(刘希忠) and Jun Yu(俞军). A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis[J]. 中国物理B, 2022, 31(5): 50201-050201.
[2]	胡恒春, 刘飞艳. Nonlocal symmetries and similarity reductions for Korteweg-de Vries-negative-order Korteweg-de Vries equation[J]. 中国物理B, 2020, 29(4): 40201-040201.
[3]	黄丽丽, 陈勇. Nonlocal symmetry and exact solutions of the (2+1)-dimensional modified Bogoyavlenskii-Schiff equation[J]. 中国物理B, 2016, 25(6): 60201-060201.
[4]	胡晓瑞, 陈勇. Nonlocal symmetries, consistent Riccati expansion integrability, and their applications of the (2+1)-dimensional Broer-Kaup-Kupershmidt system[J]. 中国物理B, 2015, 24(9): 90203-090203.
[5]	胡晓瑞, 陈勇. Nonlocal symmetries and negative hierarchies related to bilinear Bäcklund transformation[J]. 中国物理B, 2015, 24(3): 30201-030201.
[6]	刘希忠, 俞军, 任博. Explicit solutions from residual symmetry of the Boussinesq equation[J]. , 2015, 24(3): 30202-030202.
[7]	刘希忠, 俞军, 任博, 杨建荣. Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation[J]. , 2015, 24(1): 10203-010203.
[8]	刘希忠, 俞军, 任博, 杨建荣. New interaction solutions of the Kadomtsev-Petviashvili equation[J]. , 2014, 23(10): 100201-100201.
[9]	辛祥鹏, 苗倩, 陈勇. Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation[J]. Chin. Phys. B, 2014, 23(1): 10203-010203.