The full Kostant-Toda hierarchy on the positive flag variety

We study some geometric and combinatorial aspects of the solution to the full Kostant-Toda (f-KT) hierarchy, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety of SL_n(R). The f-KT flows on the tnn flag variety are complete, and their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by Rietsch. We define the f-KT flow on the weight space via the moment map, and show that the closure of each f-KT flow forms an interesting convex polytope generalizing the permutohedron which we call a Bruhat interval polytope. We also prove analogous results for the full symmetric Toda hierarchy, by mapping our f-KT solutions to those of the full symmetric Toda hierarchy. In the Appendix we show that Bruhat interval polytopes are generalized permutohedra, in the sense of Postnikov, and that their edges correspond to cover relations in the Bruhat order.