标题： 直接搜索非线性薛定谔方程的精确解 显示英文标题

标题： Direct search for exact solutions to the nonlinear Schroedinger equation

摘要： 首先为非线性薛定谔方程计算了一个由李点对称性组成的五维对称代数，该代数与反射不变性一起生成了两个五参数解群。 其次分析了三种变换假设，并用于构建非线性薛定谔方程的精确解。 经过仔细分析，给出了具有常数、三角函数型、指数函数型和有理函数振幅的各种精确解的例子。 在求解过程中，非线性薛定谔方程中的分岔现象被清晰地展示出来。 显示英文摘要

摘要： A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schroedinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansaetze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schroedinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schroedinger equation is clearly exhibited during the solution process.