Existence of Ground State and Excited Spinning $Q$-Vortex Solitons on Finite Domains

TL;DR

The paper establishes rigorous existence results for spinning $Q$-vortex solitons in a complex scalar field with a sextic potential on a finite domain by reducing the problem to a radial boundary-value problem and applying variational techniques. It proves a ground state via constrained minimization and an excited saddle state via the Mountain Pass Theorem, along with sharp bounds on the angular frequency $\omega$, the amplitude, and the domain size, plus exponential decay of solutions. A constrained minimization framework further yields a second existence result with a lower bound on the reduced norm $\tilde{Q}_0$ and a quantified frequency range between $\omega_{min}^2$ and $\omega_{max}^2$ for admissible vortices. Numerically, a spectral-Galerkin method confirms amplitude saturation, charts the nonlinear dispersion of $\omega^2$ with $\tilde{Q}_0$, and reveals how the vortex winding number $|N|$ shapes the radial profile and phase structure, aligning with the theoretical predictions and illustrating the topological nature of the solutions.

Abstract

We establish the existence of spinning $Q$-vortex solitons in a complex scalar field theory with a sextic potential on a finite domain. By reducing the governing equation to a nonlinear boundary value problem, we use variational methods to prove the existence of at least two distinct types of solutions: a ground state solution obtained via constrained minimization and an excited state of the saddle-point type obtained via the Mountain Pass Theorem. We derive bounds for the angular frequency $ω$, the wave amplitude, and the domain size $P$, and provide explicit estimates for the exponential decay of the solutions. Furthermore, we implement a spectral-Galerkin formulation to numerically compute the profiles of fundamental $Q$-vortices, illustrating the saturation behavior of the soliton's amplitude and the asymptotic dependence of the frequency on a prescribed reduced norm and vortex winding number, as well as verifying the theoretical results and visualizing the topological phase structure of the solutions.

Figures (5)

• Figure 1: Evolution of the fundamental $Q$-vortex profile ($N=1$) with increasing prescribed norm $\tilde{Q}_0$. The amplitude saturates and flattens as $\tilde{Q}_0$ gets large, strictly respecting the theoretical upper bound $\phi < \sqrt{2a/3}$ (red dashed line).
• Figure 2: Comparison of the spatial structure of fundamental $Q$-vortices ($N=1$) for moderate ($\tilde{Q}_0=100$) and large ($\tilde{Q}_0=500$) prescribed norm. The left panel shows a rounded peak, while the right panel illustrates the "flat-top" saturation predicted by Theorem 1.1, where the amplitude is capped and the soliton widens.
• Figure 3: Asymptotic behavior of the nonlinear frequency shift $\omega^2$ as a function of prescribed norm $\tilde{Q}_0$. The frequency monotonically approaches the critical lower bound $\omega^2_{min}=0.2$ for arbitrarily large prescribed norm.
• Figure 4: Effect of increasing vortex number $N$ on the soliton profile for fixed prescribed norm $\tilde{Q}_0=100$. The centrifugal barrier $N^2/\rho^2$ forces the wavefunction away from the origin, reducing the peak amplitude and increasing the effective radius.
• Figure 5: Phase-colored 3D visualization of the $Q$-vortex for $N=1$ (left) and $N=2$ (right). The height represents the field amplitude $|\Phi|$, while the color represents the phase angle $\arg(\Phi)$. The number of full color cycles around the ring visually confirms the topological winding number $N$. Note the wider core for $N=2$, consistent with the stronger centrifugal barrier.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3