Strict versions of integrable hierarchies in pseudodifference operators and related Cauchy problems

Theoretical and Mathematical Physics. 2019. Vol. 198. No. 2. P. 197–214.

Хельминк Г. Ф., Poberezhny V. A., Поленкова С. В.

In this paper we consider inside the algebra PsΔ of pseudo dierence
operators, two deformations of the Lie subalgebra spanned by the positive powers
of an invertible constant pseudo difference operator 0 of degree one. The rst
deformation is by the group in PsΔ corresponding to the Lie subalgebra PsΔ<0
of elements of negative degree and the second 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 are certain compatible Lax equations that
are determined by choosing a Lie subalgebra, depending of Λ_0, that complements
the Lie subalgebras PsΔ<0 resp. PsΔ≤0. This yields two integrable hierarchies
associated with Λ_0, where the hierarchy of the wider deformation is called the
strict version of the rst because of the form of the Lax equations. For Λ_0 equal
to the matrix of the shift operator the hierarchy corresponding to the simplest
deformation is known as the discrete KP hierarchy. Both hierarchies are shown to
possess an equivalent zero curvature form and we conclude with a discussion of the
solvability of the related Cauchy problems.

Keywords: Zero curvature form, pseudodifference operators, Lax equations, псевдоразностные операторы, уравнения Лакса, условия нулевой кривизны, задачи Коши

Publication based on the results of: Development of combinatorial, homological and geometric methods in the theory of moduli spaces of algebraic curves and their mappings, with applications to problems of mathematical physics (2019)