An algebraic approach to discrete time integrability

Anastasia Doikou, Iain Findlay

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
28 Downloads (Pure)


We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrödinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.
Original languageEnglish
Article number045202
JournalJournal of Physics A: Mathematical and Theoretical
Issue number4
Early online date7 Jan 2021
Publication statusPublished - 29 Jan 2021


  • Classical r-matrix
  • Discrete NLS
  • Discrete time integrability

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modelling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)


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