Description
We investigate the long-time behavior of a resonance-based low-regularity integrator for the cubic nonlinear Schrödinger equation (NLS). Specifically, we analyze the cubic NLS with a weak nonlinearity characterized by a dimensionless parameter ε ∈ (0, 1]. Through rescaling, this equation is equivalent to the NLS with small initial data. We provide rigorous error estimates for rough initial data ϕ ∈ H¹, valid up to times of order O(ε^−α), where α can be chosen up to 4 in one dimension and arbitrarily large in two dimensions.
Notably, in dimension three—and also in dimension two for initial data in the weighted space Σ—we establish uniform-in-time estimates with the help of scattering theory. These results highlight the capability of low-regularity integrators to accurately capture the long-time dynamics of weakly nonlinear dispersive equations with low regularity initial data.
