Soliton Resolution for NLS

Establish the soliton resolution conjecture for the nonlinear Schrödinger equation: prove that asymptotic completeness holds by showing that every global solution decomposes, as time tends to infinity, into a free Schrödinger wave plus a finite sum of solitons (coherent structures), with any remaining component dispersing as radiation.

Background

The review surveys advances on large-data scattering for nonlinear Schrödinger equations, highlighting Tao’s decomposition into free and weakly localized parts and related profile decompositions. Building on these developments, the author notes the emergence of a widely discussed conjecture asserting asymptotic completeness via soliton resolution—namely, that long-time dynamics consist of free radiation plus finitely many solitons.

The text also emphasizes that, due to the existence of coherent structures beyond solitons (e.g., breathers, vortices), soliton resolution is expected to hold only generically, complicating fully general proofs.

References

These works also give rise to a standard conjecture regarding Asymptotic Completeness (AC) for NLS, termed 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, 2024) in Introduction (Nonlinear Dispersive Dynamics)

A much stronger statement about the dynamics of solutions is the Soliton Resolution Conjecture, which predicts that a global, bounded (in an appropriate sense) solution decomposes asymptotically into a sum of energy bubbles with weak interactions (i.e., at different scales) and a radiation term (a solution of the linear Schrödinger equation). This was proved for radial solutions of the focusing, energy-critical NLW in dimension $N \geq 3$ by the concentration-compactness-rigidity argument and the channel of energy by Duyckaerts, Kenig and Merle in three dimension, and in all odd dimensions, Jendrej and Lawrie in all even dimensions, and Duyckaerts, Jia, Kenig and Merle in three dimension for the non-radial case, see also Duyckaerts, Kenig, Martel, and Merle in four dimension, and Collot, Duyckaerts, Kenig and Merle in six dimension. For nonlinear Schr\"odinger equation eq:nls this problem is completely open.

— Construction of two-bubble solutions for the energy-critical NLS in dimension 6  (2608.16186 - Jendrej et al., 17 Aug 2026) in Section 1, subsection “Setting of the problem”