Darboux transformations for linear operators on two-dimensional regular lattices

Darboux Transformations for Linear Operators on Two Dimensional Regular Lattices

Darboux transformations for linear operators on regular two dimensional lattices are reviewed. The six point scheme is considered as the master linear problem, whose various specifications, reductions, and their sublattice combinations lead to other linear operators together with the corresponding Darboux transformations. The second part of the review deals with multidimensional aspects of (bas...

1 Darboux Transformations for Nonlocal Two – Dimensional Toda Lattice

The technique of Darboux transformation is applied to nonlocal partner of two–dimensional periodic A n−1 Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with quantized spectral parameter. The generalization of the Darboux transformation technique on linear equations of such a kind is given. The solutio...

Discrete spectral symmetries of low-dimensional differential operators and difference operators on regular lattices and two-dimensional manifolds

Discrete Bispectral Darboux Transformations from Jacobi Operators

We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n)T where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We conjecture that along with operators previously constructed by Grünbaum, Haine, Horozov, and Iliev they exhaust all bispectral regular (i.e. a(n) 6= 0, c(n) 6= 0,∀n ∈ Z) ...

Darboux Transformations, Infinitesimal Symmetries and Conservation Laws for Nonlocal Two–Dimensional Toda Lattice

The technique of Darboux transformation is applied to nonlocal partner of two– dimensional periodic An−1 Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with quantized spectral parameter. The generalization of the Darboux transformation technique on linear equations of such a kind is given. The connect...