非自治方程到标准NLS方程的转换
Transformation from the nonautonomous to standard NLS equations
摘要：
本文给出了一种求非自治非线性薛定谔方程精确解的系统方法。首先通过Painlevè分析得到了一个可积条件，并证明了该条件与用拉克斯对方法得到的结果是一致的.在此条件下，我们给出了一个一般变换，它可以将标准NLS方程的所有允许的精确解直接转化为非自治NLS方程的相应精确解.该方法是相当强大的，因为标准的NLS方程已经在过去的几十年中得到了很好的研究，其精确解在文献中是巨大的。这一结果为孤子动力学的控制提供了一种有效的途径。最后，以基本亮孤子和基本暗孤子为例说明了它的应用。
英文摘要：
In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrödinger (NLS) equation. An integrable condition is first obtained by the Painlevè analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.
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