A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation
A continuous analog of the binary Darboux transformation for the Korteweg–de Vries equation
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DOI：
10.1111/sapm.12578
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发表时间：
2022-08
影响因子：
2.7
通讯作者：
A. Rybkin
中科院分区：
文献类型：
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作者：
A. Rybkin
在 Korteweg–de Vries 方程 (KdV) 背景下，我们提出了二元达布变换的连续版本（又名双交换法）。我们的方法基于黎曼-希尔伯特问题，并产生一个新的显式公式，用于扰动各种阶跃型势的负谱，而不改变其余的散射数据。这将先前已知的用于插入/删除有限多个束缚态的公式扩展到任意性质的负谱的任意集合。在 KdV 背景下，我们的方法提供了与经典二元达布变换相同的优点。
In the Korteweg–de Vries equation (KdV) context, we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann–Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step‐type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context, our method offers same benefits as the classical binary Darboux transformation does.