Norming Constants of Embedded Bound States and Bounded Positon Solutions of the Korteweg-de Vries Equation
Norming Constants of Embedded Bound States and Bounded Positon Solutions of the Korteweg-de Vries Equation
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DOI：
10.1007/s00220-023-04691-z
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发表时间：
2021-12
影响因子：
2.4
通讯作者：
A. Rybkin
中科院分区：
文献类型：
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作者：
A. Rybkin
在全线薛定谔方程的背景下，我们回顾了二进制达布变换(双对易方法)，它在不改变其余散射数据的情况下插入或移除嵌入到绝对连续谱中的任意数量的正本征值。然后，我们证明了嵌入的特征值在KdV解中产生一个额外的显式项。这一项看起来类似于多孤子解，描述了波在与孤子相反的方向上传播。它也类似于已知的(奇异)多正子解的公式，但仍然是有界的，这肯定地回答了Matveev关于有界正子存在的问题。
In the context of the full line Schrodinger equation, we revisit the binary Darboux transformation (double commutation method) which inserts or removes any number of positive eigenvalues embedded into the absolutely continuous spectrum without altering the rest of scattering data. We then show that embedded eigenvalues produce an additional explicit term in the KdV solution. This term looks similar to multi-soliton solution and describes waves traveling in the direction opposite to solitons. It also resembles the known formula for (singular) multi-positon solutions but remains bounded, which answers in the affirmative Matveev’s question about existence of bounded positons.