Global perturbation of isolated equivariant chiral skyrmions from the Bogomol'nyi case

Abstract

Isolated skyrmion solutions to the two-dimensional Landau-Lifshitz equation with Dzyaloshinskii-Moriya interaction, Zeeman term, and easy-plane anisotropy of various strengths are studied. In the full range of parameter values for which the energy is a positive variation of the Bogomol'nyi case, we construct solutions to the corresponding Euler-Lagrange equation and analyze their qualitative properties, including monotonicity, exponential decay, and stability. Our analysis is global and non-perturbative. Moreover, we derive precise estimates qualifying the difference between these solutions and those in the Bogomol'nyi regime. A key ingredient of our approach is a novel resolvent estimate for the linearized operator, which remains uniform with respect to additional implicit potentials arising in the problem.