On the set of non radiative solutions for the energy critical wave equation

Non radiative solutions of the energy critical non linear wave equation are global solutions u that furthermore have vanishing asymptotic energy outside the lightcone at both t → ± ∞:\[ limt → ± ∞ \| ∇t,x u(t) \|L^2(|x| ≥ |t|+R) = 0, \]for some $R \> 0$. They were shown to play an important role in the analysis of long time dynamics of solutions, in particular regarding the soliton resolution: we refer to the seminal works of Duyckaerts, Kenig and Merle, see {DKM:23} and the references therein.We show that the set of non radiative solutions which are small in the energy space is a manifold whose tangent space at $0$ is given by non radiative solutions to the linear equation (described in {CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.