Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics
Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics
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量子力学中KdV和非周期单能隙势的有界解
DOI：
10.1007/978-3-319-63594-1_22
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发表时间：
2015
影响因子：
1.2
通讯作者：
V. Zakharov
中科院分区：
文献类型：
--
作者：
D. Zakharov;S. Dyachenko;V. Zakharov
我们描述了一类广泛的新的KdV方程的精确解。一般来说，这些解不会在无穷远处消失，既不是周期的，也不是准周期的。这类包括作为特例的代数几何有限间隙解。相应的薛定谔算符的谱与N-间隙周期势的谱具有相同的结构，不同之处在于无反射性质仅在无穷大带内成立。这些势以一种非唯一的方式由定义在允许带上的2N个实正函数给出。在这封信中，我们将自己限制在负半轴上具有一个允许带的势；然而，我们的结果一般适用。我们用数值计算来支持我们的结果。
We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.