arXiv subjects

Friedrich Klaus

Publications and source records attributed to Friedrich Klaus.

8 recordsLinked to original sources

arXiv2026

Global well-posedness of defocusing cubic NLS in $M^{\infty,1}(\mathbb{R})$

We prove global well-posedness of the one-dimensional defocusing cubic nonlinear Schrödinger equation in the modulation space $M^{\infty,1}(\mathbb{R})$. This space imposes no spatial decay and contains $C_b^2(\mathbb{R})$ as well as all absolutely convergent sums of plane waves. The result applies to arbitrary data in this space, including large smooth quasiperiodic profiles and their localized perturbations. The proof constructs a nonnegative density satisfying a local conservation law from forward Weyl ratios, which are defined through half-line square-integrable solutions of the associated spectral problem. A suitable nonlinear combination of localized integrals of this density controls the modulation norm. Finally, choosing the spatial localization scale and NLS scaling in a coordinated way makes the accumulated boundary flux small enough to continue every mild solution globally.

math.AP↗

arXiv2026

Wellposedness for the KdV hierarchy

We prove a version of wellposedness for all equations of the KdV hierarchy in $H^{-1}$. Ingredients are 1) The Miura map which allows to define the Gardner hierarchy through the generating function of the energies so that the $N$th Gardner equation is equivalent to the $N$th KdV equation. 2) A rigorous relation between the generating functions of the energies and the KdV resp. Gardner Hamiltonians. 3) Kato smoothing estimates for weak solutions and approximate flows. Section 2 has been rewritten. Typos corrected-

math.AP↗

arXiv2023

A Strichartz estimate for quasiperiodic functions

In this work we prove a Strichartz estimate for the Schrödinger equation in the quasiperiodic setting. We also show a lower bound on the number of resonant frequency interactions in this situation.

math.AP↗

arXiv2023

A priori estimates for a quadratic dNLS

In this work we consider integrable PDE with higher dimensional Lax pairs. Our main example is a quadratic dNLS equation with a $3 \times 3$ Lax pair. For this equation we show a-priori estimates in Sobolev spaces of negative regularity $H^s(\mathbb{R}), s > -\frac{1}{2}$. We also prove that for general $N \times N$ Lax operators $L$, the transmission coefficient coincides with the $2$-renormalized perturbation determinant.

math.AP↗

arXiv2022

Wellposedness of NLS in Modulation Spaces

We prove new local and global well-posedness results for the cubic one-dimensional nonlinear Schrödinger equation in modulation spaces. Local results are obtained via multilinear interpolation. Global results are proven using conserved quantities based on the complete integrability of the equation, persistence of regularity, and by separating off the time evolution of finitely many Picard iterates.

math.AP↗

arXiv2021

Global wellposedness of NLS in $H^1(\mathbb{R}) + H^s(\mathbb{T})$

We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.

math.AP↗

arXiv2021

A priori estimates for the derivative nonlinear Schrödinger equation

We prove low regularity a priori estimates for the derivative nonlinear Schrödinger equation in Besov spaces with positive regularity index conditional upon small $L^2$-norm. This covers the full subcritical range. We use the power series expansion of the perturbation determinant introduced by Killip--Vişan--Zhang for completely integrable PDE. This makes it possible to derive low regularity conservation laws from the perturbation determinant.

math.AP↗

arXiv2019

Unconditional uniqueness of higher order nonlinear Schrödinger equations

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^σ(\mathbb T), H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T)\}$ and $q\in[1,2]$, $s\geq0$, or $σ\geq0$, or $s_{2}\geq s_{1}\geq0$. Moreover, if $M_{2,q}^{s}(\mathbb R)\hookrightarrow L^{3}(\mathbb R)$, or if $σ\geq\frac16$ or if $s_{1}\geq\frac16$ and $s_{2}>\frac12$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work

math.AP↗