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

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。

Abstract: 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.