Soliton resolution along a sequence of times with dispersive error for type II singular solutions to focusing energy critical wave equation

In this paper, we study the soliton resolution conjecture for Type II singular solutions $\overrightarrow{u}(t)$ to the focusing energy critical wave equation in $R^d\times [0,T_{+})$, with $3\le d\le 5$. Suppose that $u$ has a singularity at $(x,t)=(0,T_{+})$, we show that along a sequence of times $t_n\uparrow T_{+}$ and in a neighborhood of $(0,T_{+})$, $\overrightarrow{u}(t_n)$ can be decomposed as the sum of a regular part in the energy space, a finite combination of modulated solitons (translation, scaling and Lorentz transform of steady states) and a residue term which goes to zero in the Strichartz norm. In addition, the residue term is asymptotically radial and can concentrate energy only in a thin annulus near $\mathrm{\mid }x\mathrm{\mid }=T_{+}-t_n$. Our main tools include a Morawetz estimate, similar to the one used for wave maps, and the use of a virial identity to eliminate dispersive energy in $\mathrm{\mid }x\mathrm{\mid }<\lambda (T_{+}-t_n)$ for any $\lambda \in (0,1)$.