Existence and Stability of Hopf Solitons in the Faddeev-Niemi Model

Title: Existence and Stability of Hopf Solitons in the Faddeev-Niemi Model

Author: Alexander Novickis (alex.novickis@gmail.com)

We prove that the Faddeev-Niemi energy functional $E[\mathbf{n}] = \kappa_2 \int |\nabla \mathbf{n}|^2 + \kappa_4 \int |F[\mathbf{n}]|^2$ on maps $\mathbf{n}: \mathbb{R}^3 \to S^2$ with finite energy admits a minimiser in each non-trivial Hopf sector $H \in \mathbb{Z} \setminus \{0\}$. The proof uses the direct method of the calculus of variations with concentration-compactness (Lions) to prevent loss of topological charge at infinity, the Vakulenko-Kapitanski bound for coercivity, and elliptic regularity for smoothness. The minimiser is smooth, exponentially localised, and the second variation is non-negative with kernel containing at least the 6 zero modes from translations and rotations. This establishes the existence of the Hopf soliton as a mathematical theorem, not merely a numerical observation.

Keywords: math, variational calculus, topology, solitons, PDE, Hopf

Series: Paper CIV in the Hopf Soliton Programme