Soliton resolution for Calogero--Moser derivative nonlinear Schr\"odinger equation

TL;DR

This paper proves the soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation, including blow-up and global solutions, without symmetry or size restrictions, advancing understanding of non-integrable Schrödinger equations.

Contribution

It provides the first non-integrable proof of full soliton resolution for Schrödinger-type equations, applicable to both finite-time blow-up and global solutions without symmetry constraints.

Findings

Established soliton resolution for blow-up solutions

Proved continuous-in-time soliton resolution

Developed energy bubbling estimate for outer radiation

Abstract

We consider soliton resolution for the Calogero--Moser derivative nonlinear Schr\"odinger equation (CM-DNLS). A rigorous PDE analysis of (CM-DNLS) was recently initiated by G\'erard and Lenzmann, who demonstrated its Lax pair structure. Additionally, (CM-DNLS) exhibits several symmetries, such as mass-criticality with pseudo-conformal symmetry and a self-dual Hamiltonian. Despite its integrability, finite-time blow-up solutions have been constructed. The purpose of this paper is to establish soliton resolution for both finite-time blow-up solutions and global solutions in a fully general setting, \emph{without imposing radial symmetry or size constraints}. To our knowledge, this is the first non-integrable proof of full soliton resolution for Schr\"odinger-type equations. A key aspect of our proof is the control of the energy of the outer radiation after extracting a soliton, referred to as the \emph{energy bubbling} estimate. This benefits from two levels of convervation laws, mass and energy, and self-duality. This approach allows us to directly prove continuous-in-time soliton resolution, bypassing time-sequential soliton resolution. Importantly, our proof does not rely on the integrability of the equation, potentially offering insights applicable to other non-integrable models.