On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions $d\ge 2$

TL;DR

This paper proves that in certain dimensions, the only global, non-scattering solutions with ground state mass for the focusing mass-critical NLS are solitary waves, confirming a long-standing conjecture under symmetry conditions.

Contribution

It establishes the conjecture for $H^1_x$ initial data in dimensions 2 and 3 with spherical symmetry, and for higher dimensions with specific symmetry assumptions.

Findings

Confirmed the conjecture for $d=2,3$ with spherical symmetry.

Extended results to $d\ge 4$ with splitting-spherically symmetric data.

Showed that ground state mass solutions are solitary waves under these conditions.

Abstract

For the focusing mass-critical NLS $iu_t+\Delta u=-|u|^{\frac 4d}u$, it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for $H^1_x$ initial data in dimensions $d=2,3$ with spherical symmetry and $d\ge 4$ with certain splitting-spherically symmetric initial data.