Construction of two-bubble solutions for energy-critical wave equations

We construct pure two-bubbles for some energy-critical wave equations, that is solutions which in one time direction approach a superposition of two stationary states both centered at the origin, but asymptotically decoupled in scale. Our solution exists globally, with one bubble at a fixed scale and the other concentrating in infinite time, with an error tending to $0$ in the energy space. We treat the cases of the power nonlinearity in space dimension $6$, the radial Yang-Mills equation and the equivariant wave map equation with equivariance class $k\geq 3$. The concentration speed of the second bubble is exponential for the first two models and a power function in the last case.