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Soliton resolution for the nonlinear Schrödinger equation (NLS)

Establish, in a generic sense, the soliton resolution conjecture for the nonlinear Schrödinger equation, namely that global solutions asymptotically decompose into a finite superposition of solitons plus a dispersive free wave as time tends to infinity.

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Background

Building on decomposition and rigidity methods for NLS, the author notes the widespread belief that long‑time dynamics should resolve into coherent solitons plus radiation. However, the presence of other coherent structures (breathers, vortices, kinks) suggests the need for a generic formulation rather than a universal statement.

Progress in special regimes (e.g., radial, inter‑critical cases) supports aspects of this picture, but a general resolution remains conjectural.

References

These works also give rise to a standard conjecture regarding Asymptotic Completeness (AC) for NLS, termed {\bf Soliton resolution.} As noted by Tao, this conjecture can only be expected to hold in a generic sense, as there exist many coherent states that are not solitons (such as breathers, lumps of various types, vortices, kinks, and their combinations).

— A New Paradigm For Scattering Theory of Linear And Nonlinear Waves: Review And Open Problem (2408.14269 - Soffer, 26 Aug 2024) in Section 1: Introduction (Nonlinear Dispersive Dynamics)