Soliton resolution conjecture for the Calogero–Moser derivative nonlinear Schrödinger equation

For every solution of the Calogero–Moser derivative nonlinear Schrödinger equation with initial data $u_0\in L^2_+(\mathbb R)$ belonging to the stated natural critical initial-data class, the solution admits a complete asymptotic resolution. If the solution is global and flow–Lax admissible, it decomposes into finitely many explicit modulated solitons plus dispersive radiation. If the solution blows up at a finite time, it resolves into quantized zero-carrier $R$ -bubbles together with a remainder that converges strongly at the blow-up endpoint. The precise definition of the admissible critical-data class and the topology and modulation parameters of the decompositions are those specified in the cited source.

References

Primary source

arXiv

Additional references

Soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation in the scaling-critical space — arXiv — Shou-Fu Tian, Tao Zhang

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed solved

A new preprint claims a broad description of both global and blow-up behavior at critical regularity, but the claim has not been independently checked.

The conjecture asks whether every solution in the relevant critical class resolves into solitons and radiation, including finite-time blow-up. Earlier work treated less general data; the latest claim extends this to a stated class of critical initial data.

Known results

Kim and Kwon, 2024: for blow-up data in $H^{1}$ and global data in $H^{1,1}$ , claimed soliton resolution with an $L^{2}$ remainder, including scattering alternatives and no radial or size assumptions.
Earlier work established global well-posedness under a mass bound, classified traveling solitary waves, and described multisoliton energy cascades.
A 2024 construction produced smooth radial finite-time blow-up solutions with quantized rates.
A 2026 study classified certain single-bubble blow-up regimes under higher regularity, but did not prove full resolution.

September 2026 critical-regularity claim

Shou-Fu Tian and Tao Zhang claim resolution at critical regularity: global solutions are described by modulated solitons plus radiation, while blow-up solutions consist of quantized bubbles and an endpoint remainder. The result is restricted to the stated initial-data class and remains unverified.

Current status (as of October 2026): A critical-regularity resolution is claimed for the stated class, while independent verification of that claim is absent.

Sources

No solutions have been posted yet.