Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

doi:10.1088/1674-1056/24/1/010203

Abstract In nonlinear physics, it is very difficult to study interactions among different types of nonlinear waves. In this paper, the nonlocal symmetry related to the truncated Painlevé expansion of the (2+1)-dimensional Burgers equation is localized after introducing multiple new variables to extend the original equation into a new system. Then the corresponding group invariant solutions are found, from which interaction solutions among different types of nonlinear waves can be found. Furthermore, the Burgers equation is also studied by using the generalized tanh expansion method and a new Bäcklund transformation (BT) is obtained. From this BT, novel interactive solutions among different nonlinear excitations are found.

Keywords: residual symmetry Bäcklund transformation symmetry reduction solution generalized tanh expansion method

Received: 09 June 2014
Revised: 18 August 2014
Accepted manuscript online:

PACS:	02.30.Jr	(Partial differential equations)
	02.30.Ik	(Integrable systems)
	05.45.Yv	(Solitons)
	47.35.Fg	(Solitary waves)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11347183, 11275129, 11305106, 11365017, and 11405110) and the Natural Science Foundation of Zhejiang Province of China (Grant Nos. Y7080455 and LQ13A050001).

Corresponding Authors: Liu Xi-Zhong
E-mail: liuxizhong123@163.com

Cite this article:

Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博), Yang Jian-Rong (杨建荣) Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation 2015 Chin. Phys. B 24 010203

[1]	Korteweg D J and de Vries F 1895 Philos. Mag. 39 422
[2]	Crighton D G 1995 Acta Appl. Math. 39 39
[3]	Sulem C and Sulem P L 1999 The Nonlinear Schrödinger Equation Self-Focusing and Wave Collapse (New York: Springer-Verlag)
[4]	Kevrekidis P G 2009 Discrete Nonlinear Schödinger Equation-Mathematical Analysis, Numerical Computations and Physical Perspectives (Berlin Heidelberg: Springer-Verlag)
[5]	Rubinstein J 1970 J. Math. Phys. 11 258
[6]	Skyrme T H R 1958 Proc. Roy. Soc. A 247 260
[7]	Skyrme T H R 1961 Proc. Roy. Soc. A 262 237
[8]	Perring J K and Skyrme T H R 1962 Nucl. Phys. 31 550
[9]	Davey A and Stewartson K 1974 Proc. R. Soc. A 338 17
[10]	Clarkson Peter A and Kruskal M D 1989 J. Math. Phys. 30 2201
[11]	Matveev V B and Salle M A 1991 Darboux transformations and Solitons (Berlin Heidelberg: Springer-Verlag)
[12]	Hirota R 1971 Phys. Rev. Lett. 27 1192
[13]	Lou S Y and Ni G J 1989 J. Math. Phys. 30 1614
[14]	Lou S Y 1999 J. Phys. A: Math. Gen. 32 4521
[15]	Lou S Y 1998 Z. Naturforsch 53a 251
[16]	Lie S 1891 Vorlesungen ber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen (Leipzig: Teuber) (reprinted by Chelsea: New York; 1967)
[17]	Wu J L and Lou S Y 2012 Chin. Phys. B 21 120204
[18]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[19]	Liu X Z 2010 Chin. Phys. B 19 080202
[20]	Jing J C and Li B 2013 Chin. Phys. B 22 010303
[21]	Liu P, Yang J J and Ren B 2013 Chin. Phys. Lett. 30 100201
[22]	Xin X P and Chen Y 2013 Chin. Phys. Lett. 30 100202
[23]	Bluman G, Cheviakov A and Anco S 2010 Appl. Math. Sci. vol 168 Symmetry Methods Partial Differential Equations (New York: Springer)
[24]	Ma W X 1992 J. Phys. A: Math. Theor. 25 5329
[25]	Lou S Y 1993 Phys. Lett. B 302 261
[26]	Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607
[27]	Cheng X P, Chen C L and Lou S Y 2012 Interactions among Different Types of Nonlinear Waves Described by the Kadomtsev-Petviashvili Equation [preprint]
[28]	Hong K Z,Wu B and Chen X F 2003 Commun. Theor. Phys. 4 39
[29]	Lou S Y and Lian Z J 2005 Chin. Phys. Lett. 22 1
[30]	Cao C W and Geng X G 1990 J. Phys. A: Math. Gen. 23 4117
[31]	Cheng Y and Li Y S 1991 Phys. Lett. A 157 22
[32]	Zeng Y B, Ma W X and Lin R L 2000 J. Math. Phys. 41 5453

[1]	A nonlocal Boussinesq equation: Multiple-soliton solutions and symmetry analysis Xi-zhong Liu(刘希忠) and Jun Yu(俞军). Chin. Phys. B, 2022, 31(5): 050201.
[2]	Residual symmetries, consistent-Riccati-expansion integrability, and interaction solutions of a new (3+1)-dimensional generalized Kadomtsev—Petviashvili equation Jian-Wen Wu(吴剑文), Yue-Jin Cai(蔡跃进), and Ji Lin(林机). Chin. Phys. B, 2022, 31(3): 030201.
[3]	Painlevé property, local and nonlocal symmetries, and symmetry reductions for a (2+1)-dimensional integrable KdV equation Xiao-Bo Wang(王晓波), Man Jia(贾曼), and Sen-Yue Lou(楼森岳). Chin. Phys. B, 2021, 30(1): 010501.
[4]	Residual symmetry, interaction solutions, and conservation laws of the (2+1)-dimensional dispersive long-wave system Ya-rong Xia(夏亚荣), Xiang-peng Xin(辛祥鹏), Shun-Li Zhang(张顺利). Chin. Phys. B, 2017, 26(3): 030202.
[5]	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). Chin. Phys. B, 2015, 24(8): 080202.
[6]	Explicit solutions from residual symmetry of the Boussinesq equation Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博). Chin. Phys. B, 2015, 24(3): 030202.
[7]	Bäcklund transformations for the Burgers equation via localization of residual symmetries Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博), Yang Jian-Rong (杨建荣). Chin. Phys. B, 2014, 23(11): 110203.
[8]	New interaction solutions of the Kadomtsev-Petviashvili equation Liu Xi-Zhong (刘希忠), Yu Jun (俞军), Ren Bo (任博), Yang Jian-Rong (杨建荣). Chin. Phys. B, 2014, 23(10): 100201.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

Cite this article:

Online attention