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Abstract This paper concerns the bubbling phenomena for the ‐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 Krieger, Lenzmann and Raphaël Arch. Ration. Mech. Anal. 209 (2013), no. 1, 61–129 for the single‐bubble case, and explores the localization techniques in Cao, Su and Zhang Arch. Ration. Mech. Anal. 247 (2023), no. 1, Paper No. 4 and Röckner, Su and Zhang Trans. Amer. Math. Soc., 377 (2024), no. 1, 517–588 for bubbling solutions to non‐linear Schrödinger 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 these interactions, as well as non‐local effects on localization functions, we utilize the Carlderón estimate and the integration representation formula of the half‐wave operator, and find that there exists a narrow room between the orders and for the remainder in the geometrical decomposition. Based on this, a novel bootstrap scheme is introduced to address the multi‐bubble non‐local structure.
Cao et al. (Tue,) studied this question.