In this work we develop the inverse scattering transform (IST) for the defocusing Ablowitz–Ladik (AL) equation with arbitrarily a large nonzero background at space infinity. The IST was developed in previous works under the assumption that the amplitude of the background satisfies a “small norm” condition . On the other hand, Ohta and Yang recently showed that the defocusing AL system, which is modulationally stable for , becomes unstable if , and exhibits discrete rogue wave solutions, some of which are regular for all times. Here, we construct the IST for the defocusing AL with , analyze the spectrum, and characterize the soliton and rational solutions from a spectral point of view.

中文翻译：

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具有任意大非零背景的散焦Ablowitz-Ladik系统的逆散射变换

在这项工作中，我们为空间无穷大的散焦Ablowitz-Ladik（AL）方程开发了逆散射变换（IST），该方程具有任意大的非零背景。IST是在以前的工作中根据背景振幅满足“小范数”条件而开发的。另一方面，Ohta和Yang最近表明，对于调制稳定的散焦AL系统，如果会变得不稳定，并且会显示出离散的无赖波解，其中有些问题在所有时间都是规则的。在这里，我们使用构造用于散焦AL的IST ，分析光谱，并从光谱角度表征孤子和有理解。