Intermediate focusing-regime dichotomy

Determine whether uniform polynomial-in-time approximation bounds hold for the focusing magnetic nonlinear Schrödinger lattice and its continuum limit in the intermediate regime between the small-mass focusing extension and the large-data modulational-instability construction.

Background

The paper proves uniform polynomial-in-time approximation for focusing data with sufficiently small mass and separately constructs large bounded focusing data for which every uniform polynomial error bound fails. These results leave unresolved the behavior of the focusing continuum limit in the parameter regime between those two cases, so a complete focusing–defocusing dichotomy has not been established.