On multi-solitons for the energy-critical wave equation in dimension 5

In this paper, we construct $K$-solitons of the focusing energy-critical nonlinear wave equation in five-dimensional space, i.e. solutions $u$ of the equation such that begin{equation*} | abla_{t,x}u(t)- abla_{t,x}big(sum_{k=1}^{K}W_{k}(t)big)|_{L^{2}}to 0quad mathrm{as} tto infty, end{equation*} where for any $kin {1,dots,K}$, $W_{k}$ is Lorentz transform of the explicit standing soliton $W(x)=(1+|x|^{2}/15)^{-3/2}$, with any speed $boldsymbol{ell}_{k}in mathbb{R}^{5}$ ,$|boldsymbol{ell}_{k}|<1$ ($boldsymbol{ell}_{k} e boldsymbol{ell}_{k}$ for $k e k$) satisfying an explicit smallness condition.