非线性Schroedinger方程精确解的直接搜索
Direct search for exact solutions to the nonlinear Schroedinger equation
摘要：
首先对非线性Schroedinger方程计算了一个由李点对称性组成的五维对称代数，它与反射不变性一起产生两个五参数解群.然后分析了三种变换形式，并利用它们构造了非线性薛定谔方程的精确解。在仔细分析的基础上，给出了常数、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。非线性薛定谔方程在求解过程中明显地表现出分岔现象。
英文摘要：
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.
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