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

TL;DR

This paper proves the existence of spinning $Q$-vortex solitons in a finite domain using variational methods, derives bounds for key parameters, and numerically computes their profiles to confirm theoretical predictions.

Contribution

It introduces a rigorous mathematical framework for existence and characterization of $Q$-vortex solitons, including bounds and numerical validation, in a finite domain setting.

Findings

Existence of at least two types of $Q$-vortex solutions: ground and excited states.

Explicit bounds for frequency, amplitude, and domain size.

Numerical profiles confirming theoretical bounds and illustrating topological structures.

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 $\omega$, 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.