Discrete Crum’s Theorems and Lattice KdV-Type Equations
Discrete Crum’s Theorems and Lattice KdV-Type Equations
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DOI：
10.1134/s0040577920020038
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发表时间：
2020-02
影响因子：
1
通讯作者：
Cheng Zhang;Linyu Peng;Da‐jun Zhang
中科院分区：
文献类型：
--
作者：
Cheng Zhang;Linyu Peng;Da‐jun Zhang
我们开发了两个Schrödinger-type差分方程的达布变换（dt）及其相关的Crum公式，这两个差分方程本身就是KdV和修改KdV方程的谱问题的离散化版本。由于dt被视为离散化过程，半离散和完全离散KdV型系统的类别，包括势KdV的晶格版本，势修正KdV和Schwarzian KdV方程，作为微分/差分光谱问题及其dt的一致性条件而出现。基本晶格模型的可积性，如Lax对、多维一致性、τ函数和孤子解，可以很容易地通过直接应用离散Crum公式得到。
We develop Darboux transformations (DTs) and their associated Crum’s formulas for two Schrödinger-type difference equations that are themselves discretized versions of the spectral problems of the KdV and modified KdV equations. With DTs viewed as a discretization process, classes of semidiscrete and fully discrete KdV-type systems, including the lattice versions of the potential KdV, potential modified KdV, and Schwarzian KdV equations, arise as the consistency condition for the differential/difference spectral problems and their DTs. The integrability of the underlying lattice models, such as Lax pairs, multidimensional consistency, τ-functions, and soliton solutions, can be easily obtained by directly applying the discrete Crum’s formulas.