Kink dynamics for the Yang-Mills field in an extremal Reissner-Nordström black hole

TL;DR

This work analyzes the Yang–Mills field in the exterior of an extremal Reissner–Nordström black hole, focusing on a 1D kink in a slowly decaying inhomogeneous medium. The authors develop a coupled pair of virial estimates, including a transformed problem, to overcome a threshold resonance and a degenerate energy, ultimately constructing a finite-codimensional stable manifold around the distorted kink and proving local asymptotic stability in the energy space. The approach combines careful spectral analysis of the linearized operators and weighted coercivity with nonlinear virial controls to manage quartic nonlinearities without smallness in the $L^{\infty}$-norm. The results illuminate the long-time dynamics of topological solitons in curved backgrounds and provide a framework for analyzing soliton stability in gauge theories with variable coefficients and polynomial tails.

Abstract

Considered in this work is the Yang-Mills field in an extremal Reissner-Nordström black hole, a physically motivated mathematical model introduced by Bizoń and Kahl. The kink is a fundamental, strongly unstable stationary solution in this non-perturbative, variable coefficients model, with a polynomial tail and no explicit form. In this paper, we introduce and extend several virial techniques, adapt them to the inhomogeneous medium setting, and construct a finite codimensional manifold of the energy space where the kink is asymptotically stable. In particular, we handle, using virial techniques, the emergence of a weak threshold resonance in the description of the stable manifold.

Figures (5)

• Figure 1: Left: Graph of $\phi_0$ (not rescaled to have unit norm), with associated eigenvalue $\sim -0.658$ and $\mu_0\sim 0.811$ (see Lemma \ref{['valor mu0']}). Right: Graph of $\phi_1$ solution to ${\widetilde{L}}\phi_1=\phi_0$, $\phi_1$ even, obtained with $\phi_1(0)=-0.907$.
• Figure 2: Left: Comparison between the potentials $2{\widetilde{Q}}^2(x)(1 - {\widetilde{Q}}(x))$ (blue line) and $- 0.845Q_{9/2}^{7/2}(x)$ (yellow line) in the region $[0,1.1]$. Right: Plot of the difference $2{\widetilde{Q}}^2(x)(1 - {\widetilde{Q}}(x))+ 0.845Q_{9/2}^{7/2}(x)$ in the considered region.
• Figure 3: Left: Numerical computation of $V(\alpha(x)), V'(\alpha(x)), V"(\alpha(x))$ where their roots are explicitly plotted in dashed vertical lines. In particular we observe that $0 < x_{2,1} < x_0 < x_1 < x_{2,2}$. Right: Numerical computation of auxiliary functions $G(s)$ and $V(\alpha(s)) + Q^2(s) - R^2(\alpha(s)$. In particular we observe that $G \leq V + Q^2 + R$ for $x\in(0, x_{2,1})$.
• Figure 4: Left: Numerical computation of $j_1(s)$, lower bound for $h_0"$ for $s$ in $(x_{2,1}, x_0)$, and $j_2(s)$, lower bound for $s$ in $(0, x_{2,1}$). Right: Numerical computation of $k_1(s)$, lower bound for $K(\alpha^{-1}(s))$ with $s$ in $(x_{2,1}, x_0)$, and $k_2(s)$, lower bound for $K(\alpha^{-1}(s))$ with $s$ in $(0, x_{2,1}$).
• Figure 5: Left: Numerical computation of the bounds for $I(\alpha(x))$ in the intervals $(\alpha^{-1}(x_0), \alpha^{-1}(x_1))$, $(\alpha^{-1}(x_1), \alpha^{-1}(x_{2,2}))$, and $(\alpha^{-1}(x_{2,2}), \infty)$. Right: Numerical computation of the bounds for $I(\alpha(x))$ in the intervals $(0, \alpha^{-1}(x_{2,1}))$ and $(\alpha^{-1}(x_{2,1}), \alpha^{-1}(x_0))$.

Theorems & Definitions (93)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Claim 3.2
• proof : Proof of Claim