Asymptotic stability conjecture for the discrete NLS soliton manifold

Consider the discrete nonlinear Schrödinger evolution and its soliton family $e^{-i(-\mu t+\nu)}\alpha_\mu$, parametrized by $(\mu,\nu)$. The associated soliton manifold is the set of these states as $(\mu,\nu)$ vary. Asymptotic stability conjecture for the discrete NLS soliton manifold. The soliton manifold specified by the coordinates $(\mu,\nu)$ with respect to $u(t)=e^{-i(-\mu t+\nu)}\alpha_\mu$ is asymptotically stable under perturbed evolution via the discrete NLS. This is presented as the goal toward which the paper's spectral and dispersive estimates point; the source gives no resolution of the full stability assertion.