Construction of multi-bubble blow-up solutions to the $L^2$-critical half-wave equation

This paper concerns the bubbling phenomena for the $L^2$-critical half-wave equation in dimension one. Given arbitrarily finitely many distinct singularities, we construct blow-up solutions concentrating exactly at these singularities. This provides the first examples of multi-bubble solutions for the half-wave equation. In particular, the solutions exhibit the mass quantization property. Our proof strategy draws upon the modulation method in \cite{K-L-R} for the single-bubble case, and explores the localization techniques in \cite{CSZ21,RSZ21} for bubbling solutions to nonlinear Schr\"odinger equations (NLS). However, unlike the single-bubble or NLS cases, different bubbles exhibit the strongest interactions in dimension one. In order to get sharp estimates to control strong interactions, as well as nonlocal effects on localization functions, we utilize the Carlder\'on estimate and the integration representation formula of the half-wave operator, and find that there exists a narrow room between the orders $|t|^{2+}$ and $|t|^{3-}$ for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi-bubble non-local structure.