Robert Wegner | Searcharxiv

arXiv2026

Long-wave KdV hierarchy approximation of the NLS hierarchy with nonzero boundary conditions

We study the approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary conditions, in the long-wave regime, by the energies of the KdV hierarchy. We extend this to all $n \in \mathbb{N}$ by proving an approximation result for the transmission coefficient of the Lax operator of the NLS hierarchy, which is a Dirac operator in the nonrelativistic regime, by the transmission coefficient of a Schr\"odinger operator, which is the Lax operator of the KdV hierarchy. This yields a formal approximation result between the hierarchies, which we quantify using energy methods and previously established well-posedness results.

math.AP↗

arXiv2025

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

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).

math.AP↗

arXiv2023

Global-in-time Well-posedness of the One-dimensional Hydrodynamic Gross-Pitaevskii Equations without Vacuum

We establish global-in-time well-posedness of the one-dimensional hydrodynamic Gross-Pitaevskii equations in the absence of vacuum in $(1 + H^s) \times H^{s-1}$ with $s \geq 1$. We achieve this by a reduction via the Madelung transform to the previous global-in-time well-posedness result for the Gross-Pitaevskii equation in arXiv:1801.08386v2 [math.AP] and arXiv:2204.06293v1 [math.AP]. Our core result is a local bilipschitz equivalence between the relevant function spaces.

math.AP↗