Dice Question Streamline Icon: https://streamlinehq.com

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.

Information Square Streamline Icon: https://streamlinehq.com

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, 26 Aug 2024) in Introduction (Nonlinear Dispersive Dynamics)