Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients

doi:10.1088/1674-1056/22/1/010303

Abstract In this paper, the extended symmetry transformation of (3+1)-dimensional (3D) generalized nonlinear Schrödinger (NLS) equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS equation.

Keywords: (3+1)-dimensional nonlinear Schrödinger equation extended symmetry exact solution symbolic computation

Received: 30 March 2012
Revised: 25 June 2012
Accepted manuscript online:

PACS:	03.65.Ge	(Solutions of wave equations: bound states)
	11.30.-j	(Symmetry and conservation laws)
	02.70.Wz	(Symbolic computation (computer algebra))

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11041003), the Ningbo Natural Science Foundation, China (Grant No. 2009B21003), and K.C. Wong Magna Fund in Ningbo University, China.

Corresponding Authors: Li Biao
E-mail: biaolee2000@yahoo.com.cn

Cite this article:

Jing Jian-Chun (荆建春), Li Biao (李彪) Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients 2013 Chin. Phys. B 22 010303

[1]	Kivshar Y S and Agrawal G P 2003 Optical Solitons: From Fibers to Photonic Crystals (New York: Academic Press) p. 540
[2]	Hasegawa A 1989 Optical Solitons in Fibers (Berlin: Springer-Verlag)
[3]	Pitaevskii L and Stringari S 2003 Bose-Einstein Condensation (England: Oxford) p. 492
[4]	Hu X H, Zhang X F, Zhao Dun, Luo H G and Liu W M 2009 Phys. Rev. A 79 023619
[5]	Li Z D, Li Q Y, He P B, Liang J Q and Liu W M 2010 Phys. Rev. A 81 015602
[6]	Wang D S, Hu X H, Hu J P and Liu W M 2010 Phys. Rev. A 81 025604
[7]	Yan Z Y and Chao H 2009 Phys. Rev. A 80 063626
[8]	Yan Z Y, Konotop V V and Akhmediev N 2010 Phys. Rev. E 82 036610
[9]	Yan Z Y and Konotop V V 2009 Phys. Rev. E80 036607
[10]	Li B, Zhang X F, Li Y Q, Chen Y and Liu W M 2008 Phys. Rev. A 78 023608
[11]	Liu X B and Li B 2011 Chin. Phys. B 20 114219
[12]	Zhang X F, Zhang P, He W Q and Liu X X 2011 Chin. Phys. B 20 020307
[13]	Li B, Li Y Q and Chen Y 2009 Commun. Theor. Phys. 51 773
[14]	Wang J, Li B and Ye W C 2010 Chin. Phys. B 19 030401
[15]	Wang H and Li B 2011 Chin. Phys. B 20 040203
[16]	Fei J X and Zheng C L 2012 Chin. Phys. B 21 070304
[17]	Zheng C L and Li Y 2012 Chin. Phys. B 21 070305
[18]	Hu X and Li B 2011 Chin. Phys. B 20 050315
[19]	Ablowitz M J and Clarkson P A 1991 Soliton, Nonlinear Evolution Equations and Inverse Scattering (New York: Cambridge University Press)
[20]	Gu C H, Hu H S and Zhou Z X 1999 Darboux Transformation in Soliton Theory and its Geometric Applications (Shanghai: Shanghai Scientific and Technical Press) (in Chinse)
[21]	Hirota R 2004 The Direct Method in Soliton Theory (Cambridge: Cambridge University Press)
[22]	Bluman G W and Kumei S 1989 Symmetries and Differential Equation: Applied Mathematical Sciences 81 (Berlin: Springer)
[23]	Olver P J 1993 Application of Lie Groups to Differential Equation (2nd edn.) (New York: Springer)
[24]	Wazwaz A M 2009 Partial Differential Equations and Solitary Waves Theory (New York: Springer)
[25]	Li Z B and Liu Y P 2002 Comput. Phys. Commun. 148 256
[26]	Fan E 2000 Phys. Lett. A 277 212
[27]	Li B, Chen Y and Zhang H Q 2002 J. Phys. A 35 8253
[28]	Gao Y T and Tian B 2001 Comput. Phys. Commun. 133 158
[29]	Fan E G 2003 J. Phys. A 36 7009
[30]	Li B, Chen Y and Li Y Q 2008 Z. Naturforsch. 63a 763
[31]	Bluman G W and Cole J D 1974 Similarity Methods for Differential Equations (Berlin: Springer)
[32]	Olver P J and Rosenau P 1986 Phys. Lett. A 114 107
[33]	Wang J and Li B 2009 Chin. Phys. B 18 2109
[34]	Yao R X, J X Y and Lou S Y 2009 Chin. Phys. B 18 1821
[35]	Jia M 2007 Chin. Phys. 16 1534
[36]	Ma H C and Lou S Y 2005 Chin. Phys. 14 1495
[37]	Li J H and Lou S Y 2008 Chin. Phys. B 17 0747
[38]	Lian Z J, Cheng L L and Lou S Y 2005 Chin. Phys. 14 1486
[39]	Shi S Y and Fu J L 2011 Chin. Phys. B 20 021101
[40]	Gagnon L and Wtntemitz P 1993 J. Phys. A: Math. Gen. 26 7061
[41]	Mansfield E L, Reid G J and Clarkson P A 1998 Comput. Phys. Commun. 115 460
[42]	Ruan H Y and Chen Y X 2003 J. Phys. Soc. Jpn. 72 1350
[43]	Ruan H Y and Chen Y X 2004 Math. Methods Appl. Sci. 27 1077
[44]	Ruan H Y, Li H J and Lou S Y 2005 J. Phys. A: Math. Gen. 38 3995
[45]	Li Z F and Ruan H Y 2010 Chin. Phys. B 19 040201

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Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients

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