Analytical solutions for ultra-fast precessional switching in inertial magnetization dynamics

The paper derives, under the large-pulse/weak-damping–inertia scalings and a rotation C that aligns the pulse with e3, a two-time-scale reduction of iLLG, proves second-order-in-μ approximations for (θ,φ) in spherical coordinates (Lemma 5.1), and establishes a safe switching window [Tsw−δsw,Tsw+δsw] together with a resonance refinement when Ξ<1 (Theorem 6.2). The candidate solution reproduces the same rotation, scalings, multiple-time-scale construction, and explicit formulas for φ[≤2], θ[≤2] (identical to eq. (38) of the paper), the pole-avoidance threshold μ0 (eq. (39)), the energy decay statement, and the switching window with Ξ and its resonant refinement (eqs. (55)–(59))—hence it follows the paper’s proof strategy closely and reaches the same conclusions. Minor issues in the candidate text include a sign slip in the rotated-rescaled equation (the paper has x''×x + x' = μΛ̂x + μ^2[ x×Êx + x×x'] in eq. (24)) and an over-assertive simplification for m1 when h3=0; however, these do not affect subsequent derivations or the final results, which match the paper’s statements and formulas. Key correspondences: rotation and rescaling (eqs. (11)–(21), (22)–(24) in the paper), spherical reduction (eqs. (29)–(33)), the two-scale expansion (eq. (38)), the error/avoidance threshold (eq. (39)), and the switching theorem with Tsw, δsw, Ξ, and the resonant refinement (eqs. (55)–(59)) all align with the candidate solution. Citations: rotation/scaling and µ, ω̂, Ê definitions per eqs. (22)–(24) ; spherical system and multiple-time-scale setup eqs. (29)–(33) ; explicit φ[≤2], θ[≤2] eq. (38) ; switching theorem and formulas eqs. (55)–(59) ; energy Lyapunov identity (4)–(6) for post-pulse convergence .