General A_N open Toda chain yields DS I solutions

Determine whether, for every integer N ≥ 1, the general solution of the open two-dimensional Toda lattice associated with the simple Lie algebra A_N generates an explicit solution of the Davey–Stewartson I equation iu_t + Δu + u v = 0 and v_{ξη} + (1/2) Δ |u|^2 = 0, where Δ = ∂^2/∂ξ^2 + ∂^2/∂η^2.

Background

The paper develops an algorithm for constructing explicit solutions to the Davey–Stewartson type equations using a two-dimensional dressing chain. A key step is employing finite-field reductions of the two-dimensional Toda lattice that are compatible with higher symmetries and Lax pairs. The authors work out the case corresponding to the simple Lie algebra A2, deriving explicit formulas for solutions of the coupled system related to DS I and then obtaining a class of explicit solutions to the Davey–Stewartson I equation depending on two arbitrary functions.

Motivated by the successful construction in the A2 case, the authors propose extending this methodology to higher-rank open Toda chains associated with the simple Lie algebras A_N. The conjectural extension asks whether the general solution of the open Toda chain for arbitrary N can similarly generate explicit solutions of the Davey–Stewartson I equation, thereby broadening the class of multidimensional integrable NLS-type solutions accessible via dressing chains.