Scalar behavior for a complex multi‐soliton arising in blow‐up for a semilinear wave equation

This paper deals with blow‐up for the complex‐valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow‐up point, the solution behaves exactly as in the real‐valued case. Namely, up to a rotation in the complex plane, the solution decomposes into a sum of a finite number of decoupled solitons with alternate signs. The main novelty of our proof is a resolution of a complex‐valued first‐order Toda system governing the evolution of the positions and the phases of the solitons.