Discrete Crum’s Theorems and Lattice KdV-Type Equations

Cheng Zhang, Linyu Peng, Da jun Zhang

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)165-182
Number of pages18
JournalTheoretical and Mathematical Physics(Russian Federation)
Issue number2
Publication statusPublished - 2020 Feb 1
Externally publishedYes


  • Darboux transformation
  • discrete Crum’s theorem
  • discrete Schrödinger equation
  • exact discretization
  • lattice KdV equations

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


Dive into the research topics of 'Discrete Crum’s Theorems and Lattice KdV-Type Equations'. Together they form a unique fingerprint.

Cite this