Achievability of repulsive-case charge-density restrictions under boundary conditions

Determine whether the inequalities 0 ≤ \bar{q}_{+}(n)\,\bar{r}_{+}(n) < 1 and 0 ≤ \bar{q}_{-}(n)\,\bar{r}_{-}(n) < 1 can be enforced globally in space under suitable boundary conditions for the twelve-component semi-discrete nonlinear integrable system with repulsive nonlinearity (σ = −1), possibly analogous to those used for the semi-discrete nonlinear Schrödinger equation with repulsive nonlinearity.

Background

In the repulsive case σ = −1, the physical interpretation of certain local densities as nonpositive charge densities is viable only if the products \bar{q}{+}\bar{r}{+} and \bar{q}{-}\bar{r}{-} remain strictly below one everywhere. The authors note that such global constraints may require special boundary conditions.

The feasibility of achieving these inequalities globally—akin to conditions known for the semi-discrete nonlinear Schrödinger system—remains undetermined and is crucial for a consistent physical picture of charge densities in the repulsive regime.