Soliton resolution conjecture for 1D NLS

Establish the soliton resolution conjecture for the one-dimensional nonlinear Schrödinger equation i ∂_t v − ∂_x^2 v − F'(|v|^2) v = 0 on the real line: prove that for all (or almost all, in an appropriate sense) initial data, the solution decomposes as t → ∞ into a finite sum of solitary waves plus a decaying dispersive term, up to the symmetries of the equation.

Background

Within the review’s framework, full asymptotic stability addresses small perturbations of a single solitary wave and proves a decomposition into a solitary wave plus radiation. The soliton resolution conjecture is a stronger, global-in-data statement seeking a decomposition into a finite sum of solitary waves plus radiation for all (or almost all) initial data.

The authors emphasize that, for the nonlinear Schrödinger equation considered here, such a global decomposition is currently out of reach except in completely integrable cases, highlighting its status as a central unresolved problem beyond the scope of current techniques. Full asymptotic stability is presented as a first step toward this conjecture.