AKNS系统的散射和逆散射：一种有理函数方法
Scattering and inverse scattering for the AKNS system: A rational function approach
摘要：
我们考虑使用有理基函数来计算与AKNS系统相关的散射和逆散射变换。所提出的数值前向散射变换计算在整个实轴上有效的AKNS系统的解，从而通过求解单个线性系来计算在一点处的反射系数。建议的数值逆散射变换利用一个新的改进的有理函数的方法振荡柯西算子，使有效解某些黎曼-希尔伯特问题没有轮廓变形。后者的发展使访问高精度的计算，这是证明了一维薛定谔算子与$\mathrm{sech}^2$潜在的逆散射变换。
英文摘要：
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 is 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 latter development enables access to high-precision computations and this is demonstrated on the inverse scattering transform for the one-dimensional Schrödinger operator with a $\mathrm{sech}^2$ potential.
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