Discrete analog of Darboux’s procedure for explicit solutions of reduced systems

Develop a discrete analog of Darboux’s procedure that yields explicit general solutions for the symmetry-reduced systems (the discrete counterparts of the B- and C-type reductions of the open Toda lattice) in closed form.

Background

Darboux’s classical procedure provides explicit determinant formulae for general solutions of reduced continuous systems associated with B- and C-type Cartan matrices when the Laplace series is finite. This paper proves discrete and semi-discrete Darboux formulae for general hyperbolic operators with finite Laplace series, but does not treat the special reduced systems.

The authors explicitly defer constructing a discrete analog of Darboux’s reduction procedure that would produce explicit general solutions for these reduced (B- and C-series) systems, identifying it as a direction for future research.