Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation

doi:10.1088/1674-1056/23/1/010203

Abstract The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.

Keywords: nonlocal symmetry optimal system prolonged system explicit solutions

Received: 15 May 2013
Revised: 07 June 2013
Accepted manuscript online:

PACS:	02.30.Jr	(Partial differential equations)
	11.10.Lm	(Nonlinear or nonlocal theories and models)
	02.20.-a	(Group theory)
	04.20.Jb	(Exact solutions)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11075055 and 11275072), the Innovative Research Team Program of the National Natural Science Foundation of China (Grant No. 61021004), the National High Technology Research and Development Program of China (Grant No. 2011AA010101), and the Shanghai Knowledge Service Platform for Trustworthy Internet of Things, China (Grant No. ZF1213).

Corresponding Authors: Chen Yong
E-mail: ychen@sei.ecnu.edu.cn

Cite this article:

Xin Xiang-Peng (辛祥鹏), Miao Qian (苗倩), Chen Yong (陈勇) Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation 2014 Chin. Phys. B 23 010203

[1]	Bluman G W, Cheviakov A F and Anco S C 2010 Applications of Symmetry Methods to Partial Differential Equations (New York: Springer)
[2]	Bluman G W and Reid G I 1988 IMA J. Appl. Math. 40 87
[3]	Akhatov I S, Gazizov R K and Ibragimov N K 1991 J. Sov. Math. 55 1401
[4]	Galas F 1992 J. Phys. A: Math. Gen. 25 L981
[5]	Lou S Y 1997 J. Phys. A: Math. Phys. 30 4803
[6]	Lou S Y and Hu X B 1997 J. Phys. A: Math. Gen. 30 L95
[7]	Lou S Y and Hu X B 1993 Chin. Phys. Lett. 10 577
[8]	Hu X B and Lou S Y 2000 Proceedings of Institute of Mathematics of NAS of Ukraine 30 120
[9]	Guthrie G A 1994 Proc. R. Soc. London, Ser. A 446 107
[10]	Guthrie G A and Hickman M S 1993 J. Math. Phys. 34 193
[11]	Krasilshchik I S and Vinogradov A M 1984 Acta Appl. Math. 2 79
[12]	Bluman G W and Cheviakov A F 2005 J. Math. Phys. 46 123506
[13]	Bluman G W and Kumei S 1989 Symmetries and Differential Equations (New York: Springer-Verlag)
[14]	Lou S Y, Hu X R and Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[15]	Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607
[16]	Lie S 1881 Arch. Math. 6 328
[17]	Ovsiannikov L V 1982 Group Analysis of Differential Equations (New York: Academic)
[18]	Olver P J 1986 Applications of Lie Groups to Differential Equations (Berlin: Springer)
[19]	Ibragimov N H 1985 Transformation Groups Applied to Mathematical Physics (Boston, MA: Reidel)
[20]	Chen Y and Dong Z Z 2009 Nonlinear Anal. 71 e810
[21]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[22]	Dong Z Z, Chen Y and Lang Y H 2010 Chin. Phys. B 19 090205
[23]	Dong Z Z, Chen Y, Kong D X and Wang Z G 2012 Chin. Ann. Math. Ser. B 33 309
[24]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[25]	Zhang H P, Li B and Chen Y 2010 Chin. Phys. B 19 060302
[26]	Nucci M C 1989 J. Phys. A: Math. Gen. 22 2897
[9]	Guthrie G A 1994 Proc. R. Soc. London, Ser. A 446 107
[10]	Guthrie G A and Hickman M S 1993 J. Math. Phys. 34 193
[11]	Krasilshchik I S and Vinogradov A M 1984 Acta Appl. Math. 2 79
[12]	Bluman G W and Cheviakov A F 2005 J. Math. Phys. 46 123506
[13]	Bluman G W and Kumei S 1989 Symmetries and Differential Equations (New York: Springer-Verlag)
[14]	Lou S Y, Hu X R and Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[15]	Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607
[16]	Lie S 1881 Arch. Math. 6 328
[17]	Ovsiannikov L V 1982 Group Analysis of Differential Equations (New York: Academic)
[18]	Olver P J 1986 Applications of Lie Groups to Differential Equations (Berlin: Springer)
[19]	Ibragimov N H 1985 Transformation Groups Applied to Mathematical Physics (Boston, MA: Reidel)
[20]	Chen Y and Dong Z Z 2009 Nonlinear Anal. 71 e810
[21]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[22]	Dong Z Z, Chen Y and Lang Y H 2010 Chin. Phys. B 19 090205
[23]	Dong Z Z, Chen Y, Kong D X and Wang Z G 2012 Chin. Ann. Math. Ser. B 33 309
[24]	Hu X R, Chen Y and Huang F 2010 Chin. Phys. B 19 080203
[25]	Zhang H P, Li B and Chen Y 2010 Chin. Phys. B 19 060302
[26]	Nucci M C 1989 J. Phys. A: Math. Gen. 22 2897

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Nonlocal symmetry, optimal systems, and explicit solutions of the mKdV equation

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