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

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.

💡 Research Summary

The paper investigates rotating (spinning) Q‑vortex solitons in a (2+1)‑dimensional complex scalar field theory with a sextic self‑interaction potential. Starting from the Lagrangian L = ∂_μΦ ∂^μΦ* − U(|Φ|) with U(ϕ) = λ(ϕ⁶ − a ϕ⁴ + b ϕ²) (λ, a, b > 0, b > a²/4), the authors impose cylindrical symmetry and the ansatz Φ(t,ρ,θ) = ϕ(ρ) e^{i(ωt+Nθ)} where ω is the temporal frequency and N ∈ ℤ is the winding (vortex) number. This reduces the field equation to a radial nonlinear boundary‑value problem on a finite disc D_P = { |x| < P } with Dirichlet condition ϕ(P)=0. The finite domain serves both as a physical confinement and as a computational proxy for the infinite‑space problem.

Three main theorems are proved. Theorem 1.1 provides necessary frequency bounds: 2λ(b − a²/3) + N²/P² < ω², and, if ω² < 2λb + N²/P², a uniform amplitude bound ϕ²(ρ) < 2a³ for all ρ∈