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Discrete Crum’s Theorems and Lattice KdV-Type Equations

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Shanghai University, Shanghai, China

    Cheng Zhang & Da-jun Zhang

  2. Waseda Institute for Advanced Study, Waseda University, Tokyo, Japan

    Linyu Peng

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  1. Cheng Zhang
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  2. Linyu Peng
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  3. Da-jun Zhang
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Corresponding authors

Correspondence to Cheng Zhang, Linyu Peng or Da-jun Zhang.

Additional information

This research is supported by the National Natural Science Foundation of China (Grant Nos. 11601312, 11631007, and 11875040), the Shanghai Young Eastern Scholar program (2016–2019), the JSPS (Grant-in-Aid for Scientific Research No. 16KT0024), Waseda University (Special Research Project Nos. 2017K-170, 2019C-179, 2019E-036, and 2019R-081), and the MEXT Top Global University Project.

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 202, No. 2, pp. 187–206, February, 2020.

Conflicts of interest

The authors declare no conflicts of interest.

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Zhang, C., Peng, L. & Zhang, Dj. Discrete Crum’s Theorems and Lattice KdV-Type Equations. Theor Math Phys 202, 165–182 (2020). https://doi.org/10.1134/S0040577920020038

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  • DOI: https://doi.org/10.1134/S0040577920020038

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