Global uniform smoothness conjecture for soliton dressing and undressing maps

Let $\mathbf V_U^0$ and $\mathbf V_U^N$ be the phase spaces with zero and $N$ solitons, respectively, and let $\mathbf M_U^N$ be the pure $N$-soliton manifold, identified with the soliton-parameter space $\mathbf S_U^N$. Let $\mathbf B_+^N$ and $\mathbf B_-^N$ denote the soliton addition and removal maps.

Global uniform smoothness conjecture. Identifying $\mathbf S_U^N$ and $\mathbf M_U^N$, the soliton addition and removal maps are uniformly smooth globally,

$$\mathbf B_+^N: \mathbf V_U^0 \times \mathbf M_U^N \to \mathbf V_U^N, \qquad \mathbf B_-^N: \mathbf V_U^N \to \mathbf V_U^0 \times \mathbf M_U^N$$

for $u$ restricted to a bounded set in $H^s$ and the spectral parameters ${\mathbf z}$ restricted to a compact subset of $U$.

The local smoothness and inverse-map results are proved in the paper. This conjecture asks for global uniform smoothness with the phase space equipped with the Riemannian metric induced from the pure $N$-soliton manifold; the global statement remains open.