Theoretical and Mathematical Physics

, Volume 198, Issue 2, pp 197–214 | Cite as

Strict Versions of Integrable Hierarchies in Pseudodifference Operators and the Related Cauchy Problems

  • G. F. HelminckEmail author
  • V. A. Poberezhny
  • S. V. Polenkova


In the algebra PsΔ of pseudodifference operators, we consider two deformations of the Lie subalgebra spanned by positive powers of an invertible constant first-degree pseudodifference operator Λ0. The first deformation is by the group in PsΔ corresponding to the Lie subalgebra Ps<0 of elements of negative degree, and the second is by the group corresponding to the Lie subalgebra PsΔ≤0 of elements of degree zero or lower. We require that the evolution equations of both deformations be certain compatible Lax equations that are determined by choosing a Lie subalgebra depending on Λ0 that respectively complements the Lie subalgebra PsΔ<0 or PsΔ≤0. This yields two integrable hierarchies associated with Λ0, where the hierarchy of the wider deformation is called the strict version of the first because of the form of the Lax equations. For Λ0 equal to the matrix of the shift operator, the hierarchy corresponding to the simplest deformation is called the discrete KP hierarchy. We show that the two hierarchies have an equivalent zero-curvature form and conclude by discussing the solvability of the related Cauchy problems.


pseudodifference operator Lax equation zero-curvature form Cauchy problem 


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  • G. F. Helminck
    • 1
    Email author
  • V. A. Poberezhny
    • 2
    • 3
  • S. V. Polenkova
    • 4
  1. 1.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Institute for Theoretical and Experimental PhysicsMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.University of TwenteEnschedeThe Netherlands

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