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 $t\to \pm \mathrm{\infty }$:$$\underset{t\to \pm \mathrm{\infty }}{\lim }\mathrm{\parallel }{\mathrm{\nabla }}_{t,x}u(t){\mathrm{\parallel }}_{L^2(\mathrm{\mid }x\mathrm{\mid }\ge \mathrm{\mid }t\mathrm{\mid }+R)}=0,$$for some $R\mathrm{\&}gt;0$. 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 \cite{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 \cite{CL24}). We also construct nonlinear solutions with an arbitrary prescribed radiation field.