Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2409.04393

Linearized dynamic stability for vortices of Ginzburg-Landau evolutions

José M. Palacios·Fabio Pusateri
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the problem of dynamical stability for the $n$-vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic ``particle plus field'' description of interacting vortices. In this paper we focus on co-rotational perturbations of vortices and establish decay estimates for their linearized evolution in the relativistic case. One of the main ingredients is a construction of the distorted Fourier basis associated to the linearized operator at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag and relies on the spectral analysis of Schrödinger operators with strongly singular potentials. Since one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, additional work is needed for our construction and to control the spectral measure. The decay estimates that we obtain are of both wave and Klein-Gordon type, and are consistent with the general theory for $2$d Schrödinger operators, including those that have an $s$-wave resonance, as in the present case, but faster decaying potentials. Finally, we give a new proof of the absence of unstable spectrum and provide an estimate on the location of embedded eigenvalues by using a suitable Lieb-Thirring inequality due to Ekholm and Frank. In particular, we show that eigenvalues must lie in the interval $(1.332,2)$, where $2$ represents the effective mass of one of the two scalar operators appearing in the linearization.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

José M. Palacios, Fabio Pusateri. 2024-09-06. Linearized dynamic stability for vortices of Ginzburg-Landau evolutions. https://arxiv.org/abs/2409.04393

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Blow-Up Dynamics for the $L^2$ critical case of the $2$D Zakharov-Kuznetsov equation

We study blow-up dynamics for the $L^2$-critical cubic Zakharov--Kuznetsov equation in two dimensions, \[ \partial_tu+\partial_{x_1}(Δu+u^3)=0 \qquad\text{on }\mathbb R^2. \] For a class of localized $H^1$ perturbations of the ground state $Q$, we establish a trichotomy near the soliton manifold: exit from a small $L^2$-tube, global asymptotic stability, or finite-time blow-up. In the stable blow-up regime, the solution concentrates a single bubble and \[ λ(t)\sim \ell_0(T-t)^{1/(3-c)}, \] where $\ell_0>0$ depends on the initial datum and $c\in(1,2)$ is an explicit constant determined by the transverse tail of the first-order approximate profile. Consequently, \[ \|\nabla u(t)\|_{L^2} \sim \frac{\|\nabla Q\|_{L^2}} {\ell_0(T-t)^{1/(3-c)}}. \] After subtraction of the concentrating soliton, the radiation converges strongly in $L^p(\mathbb R^2)$ for every $2\leq p<\infty$ to a common nonzero profile $u^*$, while \[ u^*\notin H^s(\mathbb R^2) \qquad\text{for every }s\geq\frac c2. \] The stable blow-up branch is open in the relative $H^1$ topology of the localized class. Finally, every non-soliton datum in this class with non-positive energy blows up in finite time. Interval-arithmetic computer-assisted proofs certify the numerical inputs to the virial coercivity argument. They also yield a rigorous enclosure of $c$, justifying the polynomial moment of order $21$ imposed on the initial data.

math.AP↗
arXiv2026

Propagation of wave packets close to conical intersections

In this paper, we study the propagation of wave packets close to conical intersections with respect to a system of two Schr{ö}dinger equations presenting a codimension 2 crossing. We focus on the dynamics that occur when the wave packets pass through an area close to the crossing, and our main results provide an explicit formula for the outgoing wave packet in terms of the incoming one, with a complete description of its phase and of the classical trajectories it follows, including a drift.

math.AP↗
arXiv2026

A Volterra Calculus for Lie Groupoids

A pseudodifferential Volterra calculus for inverting parabolic differential equations on Lie groupoids is introduced. This enables the study of fundamental solutions of various cases of heat flows on singular manifolds with corners with non-resonant boundary indicial symbols, such as the $b$-manifolds, as well as other geometric bisection covariant heat flows. We also establish the short time asymptotic expansion for the heat kernel of a positive, elliptic differential operator on a Lie groupoid that acts on suitable Sobolev Hilbert modules and is positive definite with respect to the appropriate $L^2$ inner product.

math.AP↗