Construction of a multi-soliton blow-up solution to the semilinear wave equation in one space dimension

We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blow-up solution with a characteristic point, we refine the blow-up behavior first derived by Merle and Zaag. We also refine the geometry of the blow-up set near a characteristic point, and show that except may be for one exceptional situation, it is never symmetric with the respect to the characteristic point. Then, we show that all blow-up modalities predicted by those authors do occur. More precisely, given any integer $k\ge 2$ and $\zeta_0 \in \m R$,weconstructablow-upsolutionwithacharacteristicpoint$a$, such that the asymptotic behavior of the solution near $(a,T(a))$ shows a decoupled sum of $k$ solitons with alternate signs, whose centers (in the hyperbolic geometry) have ${\zeta }_0$ as a center of mass, for all times.