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

We show that the energy critical Wave Maps equation from ${\mathbb{R}}^{2+1}$ into ${\mathbb{S}}^2$, restricted to the $k=2$ co-rotational setting, admits arbitrarily large numbers of concentrating concentric $n$ bubble profiles. For any $n\in \mathbb{N}$, we construct an $n$-bubble solution concentrating at scales ${\lambda }_1(t)\gg {\lambda }_2(t)\gg \dots \gg {\lambda }_n(t)$, where ${\lambda }_n(t)=t^{-1}\mathrm{\mid }\log t{\mathrm{\mid }}^{\beta }$, and ${\lambda }_j(t)\gtrsim \exp ({\int }_t^{t_0}{\lambda }_{j+1}(s)ds)$, for any $j<n$. Here $\beta >\frac{3}{2}$ is a parameter that can be chosen arbitrarily. This shows that, as far as finite time blow-up case is concerned, the entirety of cases postulated in the soliton resolution theorem indeed occur, provided the concentric collapsing bubbles have alternating signs.