Global solutions for the Alber equation in $H^1\mathfrak{S}^1(\mathbb{T})$

Abstract

The Alber equation is the mixed-state nonlinear Schr\"odinger equation with singular ($\delta$-interaction) kernel. It is used in the modeling of stochastic ocean waves, where it appears with the focusing sign in the nonlinearity, on $d=1.$ The main result of the paper is global well-posedness for self-adjoint, non-negative data in the Schatten-Sobolev space $H^1\mathfrak{S}^1(\mathbb{T}),$ for both the focusing and defocusing cases. The Schatten class norms achieve control of the position density without derivative loss, and a systematic Fourier-Galerkin argument tailored to the $\delta$ kernel allows us to establish several qualitative properties of the solution, including energy conservation. In the focusing case, Hoffmann-Ostenhof and Gagliardo-Nirenberg estimates yield a global a priori $H^1\mathfrak{S}^1$ bound with no smallness condition. Non-negativity is a structural requirement for the energy argument to work. The propagation of higher Sobolev regularity $H^s\mathfrak{S}^1$ follows. As an application, small perturbations around Penrose-stable backgrounds are shown to grow at most polynomially in $H^1\mathfrak{S}$ over long timescales.