We present a method to solve numerically the Cauchy problem for the defocusing nonlinear Schr\"{o}dinger (NLS) equation with a box-type initial condition (IC) having a nontrivial background of amplitude $q_o>0$ as $x\to \pm \infty$ by implementing numerically the associated Inverse Scattering Transform (IST). The Riemann--Hilbert problem associated to the inverse transform is solved numerically by means of appropriate contour deformations in the complex plane following the numerical implementation of the Deift-Zhou nonlinear steepest descent method. In this work, the box parameters are chosen so that there is no discrete spectrum (i.e., no solitons). In particular, the numerical method is demonstrated to be accurate within the two asymptotic regimes corresponding to two different regions of the $(x,t)$-plane depending on whether $|x/(2t)| < q_o$ or $|x/(2t)| > q_o$, as $t \to \infty$.

翻译：本文提出了一种数值求解具有非零背景的散焦非线性薛定谔方程柯西问题的方法，其中初始条件为盒型，且当$x\to \pm \infty$时具有振幅$q_o>0$的非平凡背景。该方法通过数值实现相关的逆散射变换来完成。与逆变换相关的黎曼-希尔伯特问题通过复平面上适当的围道变形进行数值求解，遵循Deift-Zhou非线性最速下降法的数值实现。在本工作中，盒型参数的选择使得不存在离散谱。特别地，数值方法被证明在对应于$(x,t)$平面上两个不同区域的两个渐近区域内是精确的，这两个区域取决于当$t \to \infty$时，$|x/(2t)| < q_o$ 还是 $|x/(2t)| > q_o$。