Construction of multi-soliton solutions for the energy critical wave equation in dimension 3

We study the energy-critical wave equation in three dimensions, focusing on its ground state soliton, denoted by $W$. Using the Poincar\'e symmetry inherent in the equation, boosting $W$ along any timelike geodesic yields another solution. The slow decay behavior of $W$, $W\sim r^{-1}$, indicates a strong interaction among potential multi-soliton solutions. In this paper, for arbitrary $N\geq0$, we provide an algorithmic procedure to construct approximate solutions to the energy critical wave equation that: (1) converge to a superposition of solitons, (2) have no outgoing radiation, (3) their error to solve the equation decays like $(t-r)^{-N}$. Then, we show that this approximate solution can be corrected to a real solution.