Global solutions to nonconservative NLS with non-decaying initial data

We investigate the long-time behavior of solutions to nonconservative nonlinear Schrödinger equations (NLS) of the form $$i\partial_t u = -\partial^2_xu-u^p,\quad p\in\mathbb{N}, p\ge2.$$
We focus on initial data that are neither localized nor periodic. Our approach is based on the work [13] by Jaquette, Lessard and Takayasu, where they used a perturbative analysis around the explicit spatially homogeneous solution of the associated ODE. We establish global existence results and asymptotic decay of solutions in a general Banach algebra setting. Applications include small data global well-posedness results for almost periodic initial data, which seem to be the first in a nonconservative NLS framework. Applications to (almost) periodic initial data with localized perturbations are also presented.