Asymptotic stability and sharp decay near homogeneous solutions of a nonlinearly damped Schrödinger equation on the torus

We study the one-dimensional periodic nonlinear Schrödinger equation with a power-type nonlinear damping, both in the focusing and the defocusing regime. For a sufficiently small damping parameter and initial data close in relative $H^1$ norm to a nonzero spatially homogeneous state, we prove global existence and asymptotic convergence, after a constant phase adjustment, to an explicit homogeneous solution. The $L^2$ and $H^1$ norms follow the explicit adiabatic decay law, with the same leading coefficient as the homogeneous solution. We obtain explicit convergence rates, including a $t^{-1\mathrm{/}2}$ bound for the relative spatial perturbation. For every nonconstant initial datum in a small neighborhood of the homogeneous solution, the kinetic energy has a strictly positive leading coefficient and decays one power of time faster than the squared $L^2$ norm.