Construction of excited multi-solitons for the 5D energy-critical wave equation

For the 5D energy-critical wave equation, we construct excited $N$-solitons with collinear speeds, i.e. solutions $u$ of the equation such that begin{equation*} lim_{tto+infty}bigg| abla_{t,x}u(t)- abla_{t,x}bigg(sum_{n=1}^{N}Q_{n}(t)bigg)bigg|_{L^{2}}=0, end{equation*} where for $n=1,ldots,N$, $Q_n(t,x)$ is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel-Merle and C^ote-Martel developed for the energy-critical wave and nonlinear Klein-Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.