arXiv subjects

Jonas Luhrmann

Publications and source records attributed to Jonas Luhrmann.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Spectral Theory and Numerics

This is the first of three papers proving asymptotic stability of the degree-one vortex under equivariant perturbations in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at self-dual coupling. In the orthogonal gauge, the linearized dynamics are governed by a selfadjoint matrix Schr\"odinger operator $\mathbf{M}$. The super-symmetric partner operator is a diagonal matrix whose diagonal entries are strongly singular radial Schr\"odinger operators on $\mathbb{R}^2$. After a conjugation, this reduces the spectral problem to the analysis of two strongly singular scalar half-line operators. Combining analysis with rigorous interval arithmetic, we prove absence of threshold resonances and show that the discrete spectrum of $\mathbf{M}$ consists of exactly one positive gap eigenvalue (internal mode) with a two-dimensional eigenspace. We also certify that the relevant nonlinear Fermi Golden Rule coefficients form a definite quadratic form, yielding effective nonlinear damping of the internal mode. These spectral inputs form the basis of the stability analysis in the subsequent papers.

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arXiv2026

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model: Linear Theory

We study the linearized dynamics near the degree-one vortex of the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, restricted to equivariant perturbations in the orthogonal gauge. The linearized operator is a selfadjoint matrix Schr\"odinger operator $\mathbf{M}$ on radial $L^2_{\mathrm{rad}}(\mathbb{R}^2;\mathbb{R}^4)$ with continuous spectrum $[1,\infty)$ and a two-dimensional internal mode at a unique gap eigenvalue $\lambda^2 \in (0,1)$, as established in Part I of our three-paper series on asymptotic stability of the ground state vortex. In this second part of the series, we prove linear estimates for $\mathbf{M}$ for applications in Part III. Specifically, we prove dispersive and local-energy decay estimates, as well as a transference relation which allows us to implement the space-time resonance method with respect to the flat Klein-Gordon operator in the nonlinear analysis in Part III. The engine for proving linear estimates for $\mathbf{M}$ in our approach is the distorted Fourier transform associated with $\mathbf{M}$. The construction of the distorted Fourier transform together with a detailed analysis of the underlying generalized eigenfunctions occupy the first half of this paper. For this we exploit the super-symmetric factorization of $\mathbf{M}$, and the diagonal structure of the super-symmetric partner operator, to relate the problem to the Weyl-Titchmarsh theory of two strongly singular scalar half-line Schr\"odinger operators.

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arXiv2026

Asymptotic stability of the degree-one vortex in the abelian Yang-Mills-Higgs model

We prove asymptotic stability of the degree-one vortex in the $(1+2)$-dimensional abelian Yang-Mills-Higgs model at the self-dual coupling, for small equivariant perturbations in weighted Sobolev spaces. The abelian Yang-Mills-Higgs model is a classical relativistic field theory on $(1+2)$-dimensional Minkowski space, describing a complex-valued field coupled to an electromagnetic potential and admitting topological solitons known as vortices. This paper is the final and main part of a three-paper series. Under the orthogonal gauge condition used here, perturbations of the vortex are governed by a system of nonlinear Klein-Gordon equations for the dynamical variables, coupled to an elliptic equation for the temporal component of the electromagnetic potential. The linearized operator has continuous spectrum $[1,\infty)$ and a single positive gap eigenvalue (internal mode) of multiplicity two, whose spectral properties, associated distorted Fourier theory, and linear decay estimates are developed in the two companion papers. The main difficulty is the long-time analysis of the coupled radiation--internal-mode dynamics. In two space dimensions the dispersive decay of the Klein-Gordon radiation is relatively weak, while the internal mode decays only on the long time scale dictated by nonlinear radiation damping. At the same time, the Klein-Gordon equations for the radiation contain non-spatially localized variable coefficient quadratic interactions, which cannot be treated perturbatively and require a normal form analysis. We prove decay of the radiation by combining a good-bad decomposition, a flat-sharp decomposition, and a space-time resonance analysis carried out relative to the flat Klein-Gordon flow. The passage between the flat analysis and the Klein-Gordon flow with potential is achieved through ILED and transference estimates derived from the distorted Fourier theory.

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arXiv2025

On the Gross-Pitaevskii evolution linearized around the degree-one vortex

We study the evolution of the Gross-Pitaevskii equation linearized around the Ginzburg-Landau vortex of degree one under equivariant symmetry. Among the main results of this work, we determine the spectrum of the linearized operator, uncover a remarkable $L^2$-norm growth phenomenon related to a zero-energy resonance, and provide a complete construction of the distorted Fourier transform at small energies. The latter hinges upon a meticulous analysis of the behavior of the resolvent in the upper and lower half-planes in a small disk around zero-energy.

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arXiv2025

Stability of the Catenoid for the Hyperbolic Vanishing Mean Curvature Equation Outside Symmetry

We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the $n$-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for $n \geq 5$. The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.

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arXiv2024

Asymptotic stability of the sine-Gordon kink

We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects combined with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the moving kink. Our analysis crucially relies on two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term and of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving solitons in relativistic scalar field theories on the line.

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arXiv2024

Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation under even perturbations

We establish the full asymptotic stability of solitary waves for the focusing cubic Schrödinger equation on the line under small even perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the symmetries of the problem, namely the invariance under scaling and phase shifts. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix Schrödinger operator around the solitary waves. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

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arXiv2024

On codimension one stability of the soliton for the 1D focusing cubic Klein-Gordon equation

We consider the codimension one asymptotic stability problem for the soliton of the focusing cubic Klein-Gordon equation on the line under even perturbations. The main obstruction to full asymptotic stability on the center-stable manifold is a small divisor in a quadratic source term of the perturbation equation. This singularity is due to the threshold resonance of the linearized operator and the absence of null structure in the nonlinearity. The threshold resonance of the linearized operator produces a one-dimensional space of slowly decaying Klein-Gordon waves, relative to local norms. In contrast, the closely related perturbation equation for the sine-Gordon kink does exhibit null structure, which makes the corresponding quadratic source term amenable to normal forms [76]. The main result of this work establishes decay estimates up to exponential time scales for small "codimension one type" perturbations of the soliton of the focusing cubic Klein-Gordon equation. The proof is based upon a super-symmetric approach to the study of modified scattering for 1D nonlinear Klein-Gordon equations with Pöschl-Teller potentials from [76], and an implementation of a version of an adapted functional framework introduced in [39].

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arXiv2023

The wave maps equation and Brownian paths

We discuss the $(1+1)$-dimensional wave maps equation with values in a compact Riemannian manifold $\mathcal{M}$. Motivated by the Gibbs measure problem, we consider Brownian paths on the manifold $\mathcal{M}$ as initial data. Our main theorem is the probabilistic local well-posedness of the associated initial value problem. The analysis in this setting combines analytic, geometric, and probabilistic methods.

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arXiv2023

Asymptotic stability of the sine-Gordon kink under odd perturbations

We establish the asymptotic stability of the sine-Gordon kink under odd perturbations that are sufficiently small in a weighted Sobolev norm. Our approach is perturbative and does not rely on the complete integrability of the sine-Gordon model. Key elements of our proof are a specific factorization property of the linearized operator around the sine-Gordon kink, a remarkable non-resonance property exhibited by the quadratic nonlinearity in the Klein-Gordon equation for the perturbation, and a variable coefficient quadratic normal form introduced in [53]. We emphasize that the restriction to odd perturbations does not bypass the effects of the odd threshold resonance of the linearized operator. Our techniques have applications to soliton stability questions for several well-known non-integrable models, for instance, to the asymptotic stability problem for the kink of the $ϕ^4$ model as well as to the conditional asymptotic stability problem for the solitons of the focusing quadratic and cubic Klein-Gordon equations in one space dimension.

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arXiv2022

Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry

We establish the conditional asymptotic stability in a local energy norm of the unstable soliton for the one-dimensional quadratic Klein-Gordon equation under even perturbations. A key feature of the problem is the positive gap eigenvalue exhibited by the linearized operator around the soliton. Our proof is based on several virial-type estimates, combining techniques from the series of works [23-26, 28], and an explicitly verified Fermi Golden Rule. The approach hinges on the fact that even perturbations are orthogonal to the odd threshold resonance of the linearized operator.

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arXiv2022

On modified scattering for 1D quadratic Klein-Gordon equations with non-generic potentials

We consider the asymptotic behavior of small global-in-time solutions to a 1D Klein-Gordon equation with a spatially localized, variable coefficient quadratic nonlinearity and a non-generic linear potential. The purpose of this work is to continue the investigation of the occurrence of a novel modified scattering behavior of the solutions that involves a logarithmic slow-down of the decay rate along certain rays. This phenomenon is ultimately caused by the threshold resonance of the linear Klein-Gordon operator. It was previously uncovered for the special case of the zero potential in [51]. The Klein-Gordon model considered in this paper is motivated by the asymptotic stability problem for kink solutions arising in classical scalar field theories on the real line.

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arXiv2021

Asymptotics for 1D Klein-Gordon equations with variable coefficient quadratic nonlinearities

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.

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arXiv2020

Probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation

We establish probabilistic small data global well-posedness of the energy-critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge for scaling super-critical random initial data. The proof relies on an induction on frequency procedure and a modified linear-nonlinear decomposition furnished by a delicate "probabilistic" parametrix construction. This is the first global existence result for a geometric wave equation for random initial data at scaling super-critical regularity.

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arXiv2020

Decay and asymptotics for the 1D Klein-Gordon equation with variable coefficient cubic nonlinearities

We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein-Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic nonlinearities. Vector-field techniques to deal with the long-range nature of the cubic nonlinearity become problematic in the presence of variable coefficients. We introduce a novel approach based on pointwise-in-time local decay estimates for the Klein-Gordon propagator to overcome this impasse.

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arXiv2019

Local smoothing estimates for Schrödinger equations on hyperbolic space

We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space $\mathbb{H}^d$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a means to localize in frequency, we develop a general Littlewood-Paley machinery on $\mathbb{H}^d$ based on the heat flow. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on $\mathbb{H}^{d}$. Specifically, some of the estimates established in this paper play a crucial role in the authors' proof of the nonlinear asymptotic stability of harmonic maps under the Schrödinger maps evolution on the hyperbolic plane; see [29]. As a testament of the robustness of approach, which is based on the positive commutator method and a heat flow based Littlewood-Paley theory, we also show that the main results are stable under small time-dependent perturbations, including polynomially decaying second order ones, and small lower order nonsymmetric perturbations.

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arXiv2019

Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{3} < s < 1$. The main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear Schrödinger equation, which is inspired by certain function spaces used in the study of the Schrödinger maps problem, and is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing, and maximal function spaces. Additionally, we prove an almost sure scattering result for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{2} < s < 1$.

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arXiv2018

Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

We consider the energy-critical defocusing nonlinear wave equation on $\mathbb{R}^4$ and establish almost sure global existence and scattering for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4) \times H^{s-1}_x(\mathbb{R}^4)$ for $\frac{1}{2} < s < 1$. This is the first almost sure scattering result for an energy-critical dispersive or hyperbolic equation with scaling super-critical initial data. The proof is based on the introduction of an approximate Morawetz estimate to the random data setting and new large deviation estimates for the free wave evolution of randomized radially symmetric data.

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