Long finite time bubble trees for two co-rotational wave maps

We show that the energy‑critical Wave Maps equation from R²⁺¹ into S², restricted to the k = 2 co‑rotational setting, admits arbitrarily large numbers of concentrating concentric n‑bubble profiles. For any n ∈ ℕ, we construct an n‑bubble solution concentrating at scales λ₁(t) » λ₂(t) » … » λₙ(t), where λₙ(t) ~ t⁻¹ · |log t|ᵝ, and λⱼ(t) ≳ exp( ∫₀ᵗ [1 / λⱼ₊₁(s)] ds ) for any j < n. Here β > 3/2 is a parameter that can be chosen arbitrarily. This shows that, in the finite‑time blow‑up case, all scenarios postulated in the soliton‑resolution theorem do occur, provided the concentric collapsing bubbles have alternating signs.