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 JAMP  Vol.2 No.7 , June 2014
Lie Symmetries, One-Dimensional Optimal System and Optimal Reduction of (2 + 1)-Coupled nonlinear Schrödinger Equations
A. Li*, Chaolu Temuer
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Abstract: For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.
Keywords: Nonlinear Schrödinger Equations, Lie Aymmetry Group, Lie algebra, Optimal System
Cite this paper: Li, A. and Temuer, C. (2014) Lie Symmetries, One-Dimensional Optimal System and Optimal Reduction of (2 + 1)-Coupled nonlinear Schrödinger Equations. Journal of Applied Mathematics and Physics, 2, 677-690. doi: 10.4236/jamp.2014.27075.
References

[1]   Benney, D.J. and Roskes, G.J. (1969) Wave Instabilities. Studies in Applied Mathematics, 48, 377.

[2]   Davey, A. and Stewartson, K. (1974) On Three-Dimensional Packets of Surface Waves. Proceedings of the Royal Society of London. Series A, 338, 101-110. http://dx.doi.org/10.1098/rspa.1974.0076

[3]   Djordjevic, V.D. and Redekopp, L.G. (1977) On Two-Dimensional Packets of Capillary-Gravity Waves. Journal of Fluid Mechanics, 79, 703-714.

[4]   Freeman, C.N. and Davey, A. (1975) On the Soliton Solutions of the Davey-Stewartson Equation for Long Waves. Proceedings of the Royal Society of London. Series A, 344, 427-433.
http://dx.doi.org/10.1098/rspa.1975.0110

[5]   Ablowitz, M.J. and Haberman, R. (1975) Nonlinear Evolution Equations?? Two and Three Dimensions. Physical Review Letters, 35, 1185. http://dx.doi.org/10.1103/PhysRevLett.35.1185

[6]   Satsuma, J. and Ablowitz, M.J. (1979) Two-Dimensional Lumps in Nonlinear Dispersive Systems. Journal of Mathematical Physics, 20, 1496. http://dx.doi.org/10.1063/1.524208

[7]   Ablowitz, M J. and Segur, H. (1979) On the Evolution of Packets of Water Waves. Journal of Fluid Mechanics, 92, 691-715.

[8]   Anker, D. and Freeman, N.C. (1978) On the Soliton Solutions of the Davey-Stewartson Equation for Long Waves. Proceedings of the Royal Society of London. Series A, 360, 529-540.
http://dx.doi.org/10.1098/rspa.1978.0083

[9]   Nakamura, A. (1982) Explode-Decay Mode Lump Solitons of a Two-Dimensional Nonlinear Schrodinger Equation. Physics Letters A, 88, 55-56. http://dx.doi.org/10.1016/0375-9601(82)90587-4

[10]   Nakamura, A. (1982) Simple Multiple Explode-Decay Mode Solutions of a Two-Dimensional Nonlinear Schrodinger Equation. Journal of Mathematical Physics, 23, 417. http://dx.doi.org/10.1063/1.525361

[11]   Tajiri, M.J. (1983) Similarity Reductions of the One and Two Dimensional Nonlinear Schrodinger Equations. Journal of the Physical Society of Japan, 52, 1908-1917. http://dx.doi.org/10.1143/JPSJ.52.1908

[12]   Tajiri, M. and Hagiwar, M. (1983) Similarity Solutions of the Two-Dimensional Coupled Nonlinear Schrodinger Equation. Journal of the Physical Society of Japan, 52, 3727-3734.
http://dx.doi.org/10.1143/JPSJ.52.3727

[13]   Lbragimov, N.H. and Kovalev, V.F. (2009) Approximate and Renormgroup Symmetries. Higher Education Press, Beijing, 1-72.

[14]   Bluman, G.W. and Kumei, S. (1991) Symmetries and Differential Equations. In: Applied Mathematical Sciences, Vol. 81, Springer-Verlag/World Publishing Corp, New York, 173-186.

[15]   Ovsiannikov, L.V. (1982) Group Analysis of Differential Equations (Translation Edited by Ames, W.F.). Academic Press, New York, 183-209.

[16]   Temuer, C.L. and Pang, J. (2010) An Algorithm for the Complete Symmetry Classification of Differential Equations Based on Wu’s Method. Journal of Engineering Mathematics, 66, 181-199.
http://dx.doi.org/10.1007/s10665-009-9344-5

[17]   Temuer, C. and Bai, Y.S. (2010) A New Algorithmic Theory for Determining and Classifying Classical and NonClassical Symmetries of Partial Differential Equations (in Chinese). Scientia Sinica Mathematica, 40, 331-348.

[18]   Bluman, G.W. and Cole, J. D. (1974) Similarity Method for Differential Equations. Springer, Berlin.
http://dx.doi.org/10.1007/978-1-4612-6394-4

[19]   Olver, P.J. (1993) Applications of Lie Groups to Differential Equations. Spring-Verlag. New York/Berlin/Hong Kong.

[20]   Bluman, G.W. and Anco, S.C. (2002) Symmetry and Integration Methods for Differential Equation. 2nd Edition, Springer-Verlag, New York.

 
 
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