In this article, the Nonlinear Schrödinger (NLS) equation, which is very important in physics, have been discussed. The analytical solution of this equation with the $(1/G^{\prime })$-expansion method and its numerical solutions with the Finite Difference Method have been presented. $(1/G^{\prime })$-expansion method has been successfully applied to NLS equation and different type complex hyperbolic travelling wave solutions have been produced. An initial condition for the NLS equation has been created using this solution. This solution has been used to create the initial condition required for FDM. Truncation Error, Stability analysis and $L_2$ and $L_{\mathrm{\infty }}$ norm errors of the numerical results obtained have been examined. Besides, FDM has been successfully applied to NLS equation and numerical solutions have been produced. Besides, both analytical and numerical solutions are supported with graphics and tables. The results obtained by this numerical method have been compared with the exact solution.

本文讨论了在物理学中非常重要的非线性薛定ding（NLS）方程。该方程的解析解与$（\mathrm{1个}/G^{\prime }）$提出了扩展方法及其有限差分法的数值解。 $（\mathrm{1个}/G^{\prime }）$展开法已成功地应用于NLS方程，并产生了不同类型的复双曲行波解。使用此解决方案已为NLS方程式创建了初始条件。此解决方案已用于创建FDM所需的初始条件。截断误差，稳定性分析和${\mathrm{大号}}_{\mathrm{2个}}$ 和 ${\mathrm{大号}}_{\mathrm{\infty }}$检验了所得数值结果的范数误差。此外，FDM已成功应用于NLS方程并产生了数值解。此外，图形和表格都支持解析和数值解决方案。通过此数值方法获得的结果已与精确解进行了比较。