Global flows for the cubic nonlinear Schr\"odinger equation without decay assumptions

Abstract

For each positive integer $N$, we construct global spectral group actions for the defocusing and focusing nonlinear Schr\"odinger hierarchies on initial-data classes containing $W^{2N+1,\infty}(\mathbb R)$. For $N\ge2$, the quadratic flow gives global classical cubic NLS solutions. The construction extends the Sato--Segal--Wilson and Kotani frameworks to matrix Dirac systems; a homogeneous tau-function identity supplies the focusing positivity. No decay, smallness, periodicity, arithmetic condition, or prescribed spectral background is required for the initial-data inclusion. As a principal application, we prove global well-posedness on $M_{\infty,1}^{5}(\mathbb R)$ for both signs, with finite-time bounds and locally Lipschitz dependence on initial-data norm balls. The passage to this Banach-space flow uses near-real-axis Weyl asymptotics and continuity of normalized Toeplitz operators on compact initial-data sets to obtain uniform spatial bounds from five bounded derivatives. In the defocusing case, the flow extends to $M_{\infty,1}^{5}(\mathbb R)+H^1(\mathbb R)$. Further consequences include quantitative approximation, preservation of spatial Bohr almost periodicity and its frequency module, conjugacy of spatial translation hulls, and preservation of the full complex Dirac spectrum and spatial transfer growth rates.