DSC  >> Vol. 5 No. 1 (January 2016)

    一类(1 + 2)-维非线性薛定谔方程的Lie-对称分析
    The Lie-Symmetry Analysis of (1 + 2)-Coupled Nonlinear Schrodinger Equations

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作者:  

徐冬冬,朝 鲁:内蒙古大学,内蒙古 呼和浩特

关键词:
非线性薛定谔方程Lie对称优化系统不变解Nonlinear Schrodinger Equation Lie Algebra Optimal System Invariant Solutions

摘要:

本文中,作者用对称方法研究了一类(1 + 2)-维非线性薛定谔方程组。首先,给出了它的无穷维Lie代数及其8-维有限子代数,并计算确定了该有限维8-维子代数的1-维子代数优化系统;其次,用获得的优化系统对原(1 + 2)方程进行了对称约化,化其为一系列低维方程;第三,对已经约化的低维方程再次用对称方法进行约化获得一系列常微分方程;解该常微分方程得到了原(1 + 2)-维薛定谔方程组的精确解。

For a class of (1 + 2)-dimensional nonlinear Schrödinger equations, 8-dimensional subalgebra of the infinite Lie algebra is found and its one optimal system is constructed. By further reduction with its symmetry we obtain the corresponding ordinary differential equations. Solving the ordinary differential equations, one finds some exact invariant solutions of the Schrödinger equations.

文章引用:
徐冬冬, 朝鲁. 一类(1 + 2)-维非线性薛定谔方程的Lie-对称分析[J]. 动力系统与控制, 2016, 5(1): 18-30. http://dx.doi.org/10.12677/DSC.2016.51003

参考文献

[1] Benney, D.J. and Roskes, G.J. (1969) Wave Instabilities. Studies in Applied Mathematics, 48, 377-385.
http://dx.doi.org/10.1002/sapm1969484377
[2] Davey, A. and Stewartson, K. (1974) On Three-Dimensional Packets of Surface Waves. Proceedings of the Royal Society of London. Series A, 388, 101-110.
http://dx.doi.org/10.1098/rspa.1974.0076
[3] Nakamura, A. (1982) Simple Multiple Explode-Decay Mode Solu-tions of a Two-Dimensional Nonlinear Schrödinger Equation. Journal of Mathematical Physics, 23, 417.
http://dx.doi.org/10.1063/1.525361
[4] Nakamura, A. (1982) Explode-Decay Mode Lump Solitons of a Two-Dimensional Nonlinear Schrödinger Equation. Physics Letters A, 88, 55-56.
http://dx.doi.org/10.1016/0375-9601(82)90587-4
[5] Kanth, A.S.V. and Aruna, K. (2009) Two-Dimensional Differential Transform Method for Solving Linear and Non-Linear Schrödinger Equations. Chaos, Solitons & Fractals, 41, 2277-1833.
http://dx.doi.org/10.1016/j.chaos.2008.08.037
[6] Chaolu, T. and Li, A. (2014) Lie Symmetries, One-Dimensional Optimal System and Optimal Reductions of (2+1)- Coupled Nonlinear Schrödinger Equations. Journal of Applied Mathematics and Physics, 2, 667-690.
[7] Ovsyannikov, L.V. (1982) Lectures on the Theory of Group Properties of Differential Equations. Higher Education Press, Beijing.
[8] Olver, P.J. (1998) Applications of Lie Groups to Differential Equations. 2nd Edition, Spronger-Verlag New York Berlin Heidelberg, New York.
[9] Chaolu, T. and Jing, P. (2010) An Algorithm for the Completes 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