We consider the use of rational basis functions to compute the scattering and inverse scattering transforms associated with the AKNS system. The proposed numerical forward scattering transform computes the solution of the AKNS system that valid on the entire real axis and thereby computes a reflection coefficient at a point by solving a single linear system. The proposed numerical inverse scattering transform makes use of a novel improvement in the rational function approach to the oscillatory Cauchy operator, enabling the efficient solution of certain Riemann--Hilbert problems without contour deformations. The later development enables access to high-precision computations and this is demonstrated on the inverse scattering transform for the one-dimensional Schr\"odinger operator with a $\mathrm{sech}^2$ potential.

翻译：我们考虑使用理性基础函数来计算与AKNS系统相关的散射和反向散射变异。 拟议的数字前向散射变异计算出AKNS系统的解决方案,它在整个实际轴上有效,从而通过解决单一线性系统在一个点计算反射系数。 拟议的数值反向散射变利用了对螺旋管操作员合理功能方法的新的改进,使某些Riemann-Hilbert问题能够在不发生等离子畸形的情况下得到有效解决。 较晚的开发使得能够获取高精度计算,这表现在单维Schr\'odinger操作员的反向散射变换上,它具有$mathrm{sech ⁇ 2$潜能的单维Schr\'oder操作员的反向散射变。