Global well-posedness of the NLS hierarchy with nonzero boundary condition

TL;DR

This paper proves the global well-posedness of the NLS hierarchy with nonzero boundary conditions, using explicit formulas, dispersive estimates, and conserved energies, for high regularity initial data.

Contribution

It introduces a new approach to establish global well-posedness for the NLS hierarchy with nonzero boundary conditions, including explicit formulas and a generalized dispersive theory.

Findings

Global well-posedness for high regularity initial data.

Explicit formulas for all terms in the NLS hierarchy.

Development of a dispersive nonlinear systems theory with local smoothing and maximal function estimates.

Abstract

We consider the NLS hierarchy with the nonzero boundary condition $q(t, x) \rightarrow q_\pm \in \mathbb{S}^1$ as $x \rightarrow \pm \infty$ and prove that it is global well-posedness for initial data of high regularity. Specifically, we prove well-posedness of the problem for the perturbation $p = q - q_\ast$ from a time-independent front $q_\ast$ connecting $q_-$ to $q_+$. The equations in the NLS hierarchy are defined using a recurrence relation derived from the expansion of the logarithmic derivative of the Jost solutions associated to the Lax operator. Using this recurrence relation, we are able to determine explicit formulas for all terms in the NLS hierarchy with at most one factor that is $q_x$, $\bar{q}_x$, or a derivative thereof. We then view the equation for $p$ as part of a large class of dispersive nonlinear systems, for which we develop a local well-posedness theory in weighted Sobolev spaces. This involves certain local smoothing and maximal function estimates, which we establish for a large class of dispersion relations with finitely many critical points. Finally, we globalize the solutions using the conserved energies constructed in [1, 2]. [1] H. Koch and X. Liao. "Conserved energies for the one dimensional Gross-Pitaevskii equation". In: Adv. Math. 377, 107467 (2021). [2] H. Koch and X. Liao. "Conserved energies for the one dimensional Gross-Pitaevskii equation: low regularity case". In: Adv. Math. 420, 108996 (2023).